Regularity of Solutions and Parameter Estimation for Spde’S with Space-Time White Noise

REGULARITY OF SOLUTIONS AND PARAMETER ESTIMATION FOR SPDE’S WITH SPACE-TIME WHITE NOISE

Igor Cialenco

A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulﬁllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS)

May 2007

To my wife Angela, and my parents.

ii Acknowledgements

I would like to acknowledge my academic adviser Prof. Sergey V. Lototsky who introduced me into the Theory of Stochastic Partial Differential Equations, suggested the interesting topics of research and guided me through it. I also wish to thank the members of my committee - Prof. Remigijus Mikulevicius and Prof. Aris Protopapadakis, for their help and support.

Last but certainly not least, I want to thank my wife Angela, and my family for their support both during the thesis and before it.

iii Table of Contents

Dedication ii

Acknowledgements iii

List of Tables v

List of Figures vi

Abstract vii

Chapter 1: Introduction 1 1.1 Sobolev spaces ...... 1 1.2 Diffusion processes and absolute continuity of their measures . . . . . 4 1.3 Stochastic partial differential equations and their applications . . . . 7 1.4 Ito’sˆ formula in Hilbert space ...... 14 1.5 Existence and uniqueness of solution ...... 18

Chapter 2: Regularity of solution 23 2.1 Introduction ...... 23 2.2 Equations with additive noise ...... 29 2.2.1 Existence and uniqueness ...... 29 2.2.2 Regularity in space ...... 33 2.2.3 Regularity in time ...... 38 2.3 Equations with multiplicative noise ...... 41 2.3.1 Existence and uniqueness ...... 41 2.3.2 Regularity in space and time ...... 48 2.4 Covariance functional and relation with some known results . . . . . 57 2.5 Examples ...... 60

Chapter 3: Parameter estimation problems for some classes of SPDE’s 65 3.1 Introduction ...... 65 3.2 Special case: space-time white noise ...... 69 3.3 General case. Approximation of the solution ...... 75 3.4 The estimate and its properties ...... 79 3.5 Applications to stochastic parabolic differential equations ...... 88 3.6 Examples and some numerical results ...... 91

References 100

iv List of Tables

2.1 Existence and Regularity of Solution. Summary ...... 64

3.1 Errors of estimated parameter ...... 94

v List of Figures

2.1 Realization of approximated solution of stochastic heat equation with additive noise ...... 32

2.2 Realization of approximated solution of stochastic heat equation with multiplicative noise ...... 48

3.1 Estimated Parameter. Example 3.6.1 ...... 93

3.2 Estimated Parameter. Example 3.6.2...... 96 ˆ 3.3 Errors |θ − θ0| vs number of Fourier coefﬁcients. Example 3.6.2 . . . 97 3.4 Estimators vs number of Fourier coefﬁcients. Example 3.6.3.a . . . . 98

3.5 Estimators vs number of Fourier coefﬁcients. Example 3.6.3.b . . . . 98

3.6 Estimators vs number of Fourier coefﬁcients. Example 3.6.3.c . . . . 99

vi Abstract

In this work we discuss two problems related to stochastic partial differential equations (SPDEs): analytical properties of solutions and parameter estimation for SPDE’s. We address the problem of existence, uniqueness and regularity of solutions of some parabolic SPDE’s driven by space-time white noise, either additive or multiplicative. The novelty in our study is the special form of the noise term which depends on a real parameter. We establish existence and uniqueness of weak solution in the scale of Sobolev spaces. Regularity properties of the solution are stated in terms of the real parameter involved in the noise term and spectral properties of the elliptic operator which generates the scale of Sobolev spaces.

We study the parameter estimation problem for some parabolic SPDEs with multiplicative stochastic part. Maximum Likelihood Estimators of the parameter, based on ﬁnite-dimensional approximation of the solution are found. Consistency, both in time and space variables, and asymptotic normality of these estimators are established. All theoretical results are followed by numerical simulations.

vii Chapter 1

Introduction

In this section we discuss some general and known results which we will need for further presentation. An overview of Sobolev spaces, Kolmogorov’s criterion, absolute continuity of measures generated by the solutions of diffusion processes, Ito’sˆ formula in Hilbert space, existence and uniqueness of solution of parabolic SPDEs and some applications of SPDEs are presented. All results are presented without proofs, except Ito’sˆ formula in Hilbert space for which we present a simpler proof than that from the know literature.

1.1 Sobolev spaces

Let M be a d-dimensional compact, orientable, C∞ manifold with a smooth positive measure dx. Let us consider an elliptic positive deﬁnite selfadjoint differential operator

1 L of order 2m on M. Then the operator Λ = (L) 2m is well-deﬁnite, elliptic of order

s 1 and generates the scale {H (M)}s∈R of Sobolev spaces on M (see for instance [1], [76]). In particular, for positive integer n the space Hn(M) consists of all generalized functions on M whose derivative up to and including the nth order belong to L2(M) and the norm is deﬁned to be

  1 X Z 2 α 2 kukHn(M) =  |D u| dx , |α|≤nM

1 α where D denotes the partial derivative of order α = (α1, . . . , αd) (see [1],[9] for

s s more details). We will write {H }s∈R instead of {H (M)}s∈R if there is no ambiguity in context. In what follows we will usually use an alternative characterizations of Sobolev spaces. It is well-known (see for instance [76], Theorem 8.4) that the operator L has a complete orthonormal system (CONS) of eigenvectors {hk}k∈N in the space

L2(M, dx). Then for every f ∈ L2(M, dx) we have the representation

X∞ X∞ f(x) = hf, hki0hk(x) = fkhk(x) , (x ∈ M) (1.1) k=1 k=1 R where h·, ·i0 denotes the inner product in L2(M, dx), and fk = f(x)hk(x)dx, k ∈ N M are Fourier coefﬁcients of f with respect to {hk}k∈N.

If µk > 0 is the eigenvalue of the operator L corresponding to the eigenvec- 1 2m s tor hk and λk := µk , then for all s ≥ 0 we have H = {f ∈ L2(M, dx): ∞ P 2s 2 s λk |fk| < ∞} and for s < 0, H is the closure of L2(M, dx) in the norm k=1 ¡ ∞ ¢ P 2s 2 1/2 s ||f||s = λk |fk| . Hence, every element of the space H , s ∈ R, can be k=1 ∞ P 2s 2 identiﬁed with a sequence {fk}k∈N such that λk |fk| < ∞. Note that the space k=1 Hs, equipped with the inner product

X∞ 2s s hf, gis = λk fkgk (f, g ∈ H ) k=1 is a Hilbert space. Moreover, for s > 0, H−s is the dual space of Hs relative to inner

s −s product in L2(M, dx). It is equivalent to say that H = Λ (L2(M, dx)), s ∈ R. Note that

s s s s Λ hk = λkhk and ||hk||s = ||Λ hk|| = λk (k ∈ N, s ∈ R). (1.2)

2 −s s Moreover, the functions hk,s := λk hk, k ∈ N, form a CONS in H for s ∈ R. Similarly, the Sobolev Spaces Hs(Rd)(s ∈ R) are deﬁned. Namely,    Z  Hs(Rd) = f ∈ L (Rd): (1 + |y|2)s|fˆ(y)|2dy < ∞  2  R

R 2 2 s ˆ 2 ˆ with norm kfks = R(1 + |y| ) |f(y)| dy, where f denotes the Fourier Transform in R d ˆ 1 R , i.e. f(y) = (2π)d/2 Rd exp(−ixy)f(x)dx. Also, we can characterize the spaces s d Pd s d H (R ) using the Laplace operator ∆(u) = k=1 uxkxk . Namely, H (R ) = (I −

− s d ∆) 2 L2(R ). For more details on structure and some properties of operator (I − ∆)s, s ∈ R, see for instance [1], [43].

γ d For p ≥ 1 and γ ∈ R, we deﬁne the space Hp (R ) as collection of all gene-

−γ/2 d ralized functions f such that (I − ∆) f ∈ Lp(R ), and we put kfkγ,p := k(I −

−γ/2 γ γ d ∆) fkLp(R ). In our notations H = H2 . The triple (V,H,V 0) of Hilbert spaces is called normal triple if: (a) V,→ H,→ V 0 and both inclusions are dense and continuous; (b) The space V 0 is dual to V relative to inner product in H; (c) There exists a constant C > 0 such that

|(h, v)H | ≤ C · ||v||V · ||u||V 0 (v ∈ V, u ∈ H). For example the Sobolev spaces (Hl+γ(Rd), Hl(Rd), Hl−γ(Rd)), γ > 0, l ∈ R, form a normal triple.

γ1 γ2 Note that Hp ⊂ Hp and k · kγ2,p ≤ k · kγ1,p if γ2 < γ1. Also, let us recall that by

γ d γ−d/p d Sobolev embedding theorem Hp (R ) ⊂ C (R ) if pγ > d (see [9], Chapter 5). Also, we recall the interpolation theorem for spaces Hs [40], [41]. For every u ∈

γ+2 θ 1−θ s−γ H and s ∈ [γ, γ + 2], we have kuks ≤ Ckukγ+2kukγ , where θ = 2 and C depends only on M, γ, s.

|a|p |b|q We will refer to inequality ab ≤ ε p + qεp−1 , as ε-inequality, where a, b ∈ R, 1/p+ 1/q = 1, ε > 0. It follows directly from Holder¨ inequality.

3 Another central result in our investigation is Kolmogorov’s criterion. To some extends, Kolmogorov’s criterion is a version of Sobolev embedding theorem. For sake of completeness of our presentation, we will state this result here too. For the proof, see for instance [79], Corollary 1.2, or [8], Theorem 3.3.

1.1.1 Theorem. [Kolmogorov0s criterion] Let (Ω, F, P) be a probability space, G ⊂

Rd a bounded domain, and X = X(x), x ∈ G, a measurable map from Ω × G to R so that X is continuous in x and ω. Assume that there exist positive real numbers K, p and q so that p > 1 and

¯ ¯ ¯ ¯p E¯X(x) − X(y)¯ ≤ K|x − y|d+q .

Then there exist a positive real number A and a random variable Y so that, with probability one,

¯ ¯ µ ¶2/p ¯ ¯ A ¯X(x) − X(y)¯ ≤ Y |x − y|q/p ln , (1.3) |x − y| for all x, y ∈ G, q ≤ p.

1.2 Diffusion processes and absolute continuity of their

measures

For a given potability space (Ω, F, P), we consider a family of random random processes ξt(ω) on this space with t ∈ [0,T ], where t usually is understood as time variable, and T is a ﬁxed parameter, called also terminal time. For a ﬁxed ω ∈ Ω, the time function ξt(ω), t ∈ [0,T ] is called a trajectory or realization corresponding to an

ξ elementary event ω. The σ-algebra Ft := {ξs : s ≤ t}, being the smallest σ-algebra

4 with respect to which the random variables ξs, s ≤ t are measurable, are naturally

ξ associated with random process ξt. Ft is called the ﬁltration generated by the random process ξt.

Let {Ft}t∈[0,T ] be a nondecreasing family of σ-algebras, Fs ⊆ Ft ⊆ F, s ≤ t.

We say that a random process ξt, t ∈ [0,T ], is adapted to a family of σ-algebras

Ft, t ∈ [0,T ], if for any t ∈ [0,T ] the random variables ξt are Ft-measurable. Also, we will say that ξt is Ft-adapted or nonanticipative.

1.2.1 Deﬁnition. Let (Ω, F, P) be a probability space. The random process W = (W (t)), 0 ≤ t ≤ T , is called a standard Brownian Motion on the probability space

(Ω, F, P) if: (a) W (0) = 0; (b) for every partition 0 = t0 < ··· < ti < ti+1 < . . . tn =

T , the random variables W (ti+1) − W (ti) and W (tj+1) − W (tj) are independent for every i 6= j (independent increments); (c) W (t) − W (s) has a Gaussian standard normal distribution with E(W (t)−W (s)) = 0, Var(W (t)−W (s)) = 1; (d) for almost all ω ∈ Ω the functions W (t) = W (t, ω) are continuous on the interval 0 ≤ t ≤ T .

For more details about existence of Brownian motion see for instance [13], [47]. The Brownian motion is also called Winer Process.

Let (Ω, F, {Ft}t∈[0,T ], P) be a stochastic basis, W (t) be a Wiener Process and ξ(t) a random process, both adapted to the ﬁltration Ft. Denote by Pξ the measure on

C(0,T ) corresponding to the process ξt,

Pξ(B) = P{ω : ξ(ω) ∈ B} .

In this section we will discuss the problem of the absolute continuity of measures

P and P , where ξ and η are some diffusion processes. As usually, by Pξ we denote ξ η Pη the Radon-Nykodim derivative of the measure Pξ w.r.t. the measure Pη.

5 In what follows all stochastic integrals are understood in sense of Ito.ˆ Let ξ =

(ξ1(t), . . . , ξn(t)) and η = (η1(t), . . . , ηn(t)), t ∈ [0,T ], be vector-processes, with Itoˆ differentials

dξt = At(ξ)dt + bt(ξ)dWt ,

dηt = at(η)dt + bt(η)dWt , (1.4)

ξ0 = η0 ,

where Wt = (W1(t),...,Wk(t)) is a k-dimensional Winer process with respect to Ft,

At(x) = (A1(t, x),...,An(t, x)), at(x) = (a1(t, x), . . . , an(t, x)), bt(x) = [bij(t, x)] is a matrix of order n × k, and η0 = (η1(0), . . . , ηn(0)) is a vector of initial values such ¡ Pn ¢ that P |ηj(0)| < ∞ = 1. We suppose that equations (1.4) have a unique strong j=1 solution (for more details about conditions on the coefﬁcients see [62], Theorem 5.5, or [47], Theorem 4.6).

We denote by A∗ the transpose matrix of a given matrix A. By b+ we denote the pseudoinverse matrix with respect to the matrix b. Recall that the matrix b+ is called pseudoinverse matrix with respect to the matrix b, if b+bb+ = b+ and bb+b = b. Note that if b is a square invertible matrix, then b+ = b−1. For more details about pseudoinverse matrices see for instance [4].

The following result about absolute continuity of measures Pξ and Pη holds true:

1.2.2 Theorem. Assume that the system of algebraic equations

bt(x)yt(x) = At(x) − at(x) (1.5)

6 has a solution with respect to yt(x) for every t ∈ [0,T ], x ∈ R, and

ZT ³ ´ ∗ ∗ + ∗ ∗ + At (x)(bt(x)bt (x)) At(x) + at (x)(bt(x)b (x)) at(x) dt < ∞ Pξ − a.s. 0

Then Pξ ∼ Pη and

n ZT dPξ ¡ ¢∗¡ ∗ ¢+ (t, η) = exp As(η) − as(η) bt(η)b (η) dη(s) Pη 0 (1.6) ZT 1 ¡ ¢ ¢¡ ¢ ¡ ¢ o − A (η) − a (η) ∗ b (η)b∗(η) + A (η) + a (η) ds . 2 s s t s s 0

For the proof see for instance Chapter 7 in [47]. More general results are discussed in [26], [27], [28].

1.3 Stochastic partial differential equations and their

applications

The theory of Stochastic Differential Equations (SDEs) is dealing with finite- dimensional noise, namely the perturbation consists of finite number of Wiener Processes. In the case of Stochastic Partial Differential Equation (SPDEs) besides of considering functions of multivariables, usually the perturbation part represents an infinite-dimensional noise, i.e. infinitely many Wiener Processes. A general theory of infinite-dimensional noise is based on the Hilbert space-valued Wiener process (see for instance [8], [70]). For sake of completeness we present here briefly the general idea and construction of infinite-dimensional noise.

7 Let H be a separable Hilbert space, and Q a selfadjoint none-negative operator on H. Suppose that Q is a trace (nuclear) operator. Then there exists a complete orthonormal system (CONS) hk, k ≥ 1, in H, and a sequence of non-negative real P numbers {qk}k≥1 such that Qhk = qkhk, k ≥ 1 and Tr(Q) = qk < ∞. In other words, Q is a selfadjoint, trace operator on H, with eigenvalues qk and eigenvectors hk, k ≥ 1.

1.3.1 Deﬁnition. A H-valued stochastic process W(t), t ≥ 0 is called a Q-Wiener Process if (i) W(0) = 0; (ii) W has continuous trajectories; (iii) W has independent increments; (iv) W(t) − W(s) ∼ N(0, (t − s)Q), 0 ≤ s ≤ t, where N denotes a Gaussian process in Hilbert space H (for more details see [8], Chapter 1-4).

One can show that E(W(t)) = 0, Cov(W(t)) = tQ, t ≥ 0, and for arbitrary t the following expansion holds

X∞ W(t) = λkhkWk(t) , (1.7) k=1

√ where λk = qk, k ∈ N, Wk(t) are real valued standard Wiener processes, mutually independent on some probability space (Ω, F, P), and the last series converges in

L2(Ω, F, P). For complete proof see for instance [8], Proposition 4.1. It turns out that the representation (1.7) is very convenient to use in the theory of SPDEs. Moreover, we can assume that Q is a bounded, selfadjoint, non-negative operator on H, with pure discrete spectrum (non necessary trace operator). In this case W(t) will take value in a larger Hilbert space, having the same type of repre- P sentation (1.7). If λk = ∞, then inﬁnite-dimensional Wiener process W(t) is k≥1 called cylindrical Wiener process or cylindrical Brownian motion. In special case, if

d P we take H = L2(R ) and λk = 1, k ≥ 1, the formal sum dW(t) = hk(x)dWk(t) k≥1

8 is called space-time white noise. In this work, we will mainly focuss on the case

α λk = k , α ∈ R.

0 Let (V,H,V ) be a triple of Hilbert spaces and (Ω, F, {Ft}, P) a stochastic basis.

0 Suppose that A(t): V → V , Mk(t): V → H are linear bounded operators for all k ∈ N, t ∈ [0,T ], and for some ﬁnite terminal time T . A general Linear Stochastic Partial Differential Equation is written as

du(t) = (Au(t) + f(t))dt + (Mku(t) + gk(t))dWk(t) , u(0) = u0, (1.8)

where t ∈ [0,T ], u0 ∈ L2(Ω; H) and u0 is F0-measurable. We assume that Wk (k ∈

N) are independent standard Wiener processes, f, gk (k ∈ N) are Ft-adapted random

0 processes such that f ∈ L2(Ω × (0,T ); V ) and gk ∈ L2(Ω × (0,T ); H) , k ∈ N. Here and in what follows the summation over repeated indices is assumed. Depending on the noise term the equation (1.8) is classiﬁed as follows:

- Equations with additive noise, if Mk = 0;

- Equations with multiplicative noise, if Mk 6= 0.

Similar to classical deterministic PDEs, the solution of equation (1.8) can be spec- iﬁed in different ways. Since the solution is a stochastic process, besides of PDE part, where we have classical solution, strong/weak generalized solution and mild solution, also we can specify strong and weak probabilistic solution. In this work we consider only solution which are strong in probabilistic sense and weak/strong/mild in PDE sense.

9 We say that Ft-adapted function u ∈ L2(Ω × (0,T ); V ) is a weak solution of equation (1.8), if for every ϕ ∈ V and all t ∈ [0,T ], the equality

Zt Zt

(u(t), ϕ)H = (u0, ϕ)H + [Au(s) + f(s), ϕ]ds + (Mku(s) + gk(s), ϕ)H dWk(s) 0 0 (1.9) holds with probability one.

We say that Ft-adapted function u ∈ L2(Ω × (0,T ); V ) is a mild solution of equation (1.8) if

Zt Zt u(t) = S(t)u0+ S(t−s)f(s)ds+ S(t−s)(Mku(s)+gk(s))dWk(s), t ∈ [0,T ], 0 0 holds true with probability one, where S is a strongly continuous semigroup with infinitesimal generator A (for definition of semigroup and infinitesimal operator see for instance [20]).

As we observed, stochastic evolution equations in infinite dimensions are natural generalizations of classical PDEs and Systems of Stochastic Ordinary Differential Equations. The theory related to all these equations has motivations coming from physics, chemistry, biology, medicine, finance etc. Although, the theory of SPDEs is already established and widely developed field in mathematics, and problems arising in this theory represent an interest for mathematics itself, we will mention here several classical problems related to some particular SPDEs which we are going to study latter on in Chapter 2 and 3.

10 Interesting and important examples represent Zakai and Kushner equations (see [8], [70], [80]) which come from diffusion ﬁltering problem. Let X(t) be the unob- served d-dimensional process, and Y (t) be the r-dimensional observed process deﬁned by the following SODEs

dX(t) = b(X(t))dt + σ(X(t))dW (t) + ρ(X(t))dV (t) , (1.10)

dY (t) = h(X(t))dt + dV (t) , 0 < t ≤ T ; X(0) = X0,Y (0) = 0,

d d×r r where b(x) ∈ R , σ(x) ∈ R , h(x) ∈ R , and X0 has a density u0. Let f = f(x)

d 2 be a scalar measurable function on R such that sup0≤t≤T E|f(X(t))| < ∞. The ﬁltering problem is to ﬁnd the best square mean estimate f˜ of f(X(t)), t ∈ [0,T ],

˜ Y given the observations Y (s), s ∈ [0, t]. One can show that f = E(f(X(t))|Ft ) and R f(x)u(t, x)ds d f˜(t) = R R , u(t, x)dx Rd where u satisﬁes the following equation

Xr d du(t, x) = Au(t, x)dt+ Bku(t, x)dYk(t) , 0 < t ≤ T, x ∈ R , u(0, x) = u0(x), k=1 (1.11) where Ak,Bk are some partial differential operators determined by the functions b, σ, ρ, h. Equation (1.11) is called Zakai filtering equation, which is a particular case R of equation (1.8). If we put p(t, x) = u(t, x)/ Rd u(t, x)dx then p satisfies Kushner filtering equation which is a non-linear SPDE.

Helmholtz equation

X∞ du(t, x) = (i∆u(t, x) + au(t, x))dt + iu(t, x) qkhk(x)dWk(t) , k=1

11 2 2 where x ∈ R , t > 0, a > 0, qk > 0, k ≥ 1, hk CONS in L2(R ), is used to describe and study random media (see [70]).

The equation of the form du(t, x) = (F (t, x, u(t, x))ux(t, x))xdt + G(t, x, u(t, x))dW (t, x) appears as a limit of a branching diffusion model, and has applications in biology [10], chemistry and theory of super-processes [58], [59], [63].

The simple SPDE vt = vxx − v + dW with proper initial values and boundary conditions was proposed to model electrical potential at point (x, t) generated by neurons (see for instance [79]). We conclude this section with some application of SPDEs to finance. The great success of stochastic calculus in description of stock markets and valuation of options on stocks strongly influenced the research related to fixed income market, in particular modelling the term structure of interest rates. Before going into details, let us recall some notions from bond market (for more details see for instance [5], [6], [24]).

A zero coupon bond with maturity date T , is a contract which guarantees the holder 1 dollar to be paid on the date T . We denote by by p(t, T ) the price of a bond at time t. We assume that P (t, T ) exists for every T > t, P (t, T ) > 0,P (t, t) = 1, and there exists ∂P (t, T )/∂T .

Instantaneous forward rates at time t for all time to maturity x > 0, f(t, x), are deﬁned by ∂ log P (t, t + x) f(t, x) = − , (1.12) ∂x which is the rate that can be contracted at time t for instantaneous borrowing or lending at time t + x. The spot interest rate at time t is deﬁned as

r(t) = f(t, 0) . (1.13)

12 If we want to make a model for bond market, it sufﬁces to specify the dynamics for one of these variables: Bond Price P (t, T ), forward rate f(t, x), spot rate r(t). Knowing the dynamics of one of them, by the Ito’sˆ formula (see for instance [62]) we can get the dynamics of rest of them (see [5], Chapter 15, page 230). There are many models for spot rate, among which we want to mention Vasicekˇ model dr = √ (b−ar)dt+σdW, a > 0, Cox-Ingersoll-Ross (CIR) model dr = a(b−r)dt+σ rdW , Ho-Lee model dr = θ(t)dt+σdW . A popular model for forward rate is Heath-Jarrow- Morton (HJM) model by which the forward rate follows the dynamics

µ ¶ ∂f(t, x) df(t, x) = + a(t, x) dt + σ(t, x)dW (t) , (1.14) ∂x

R x where a(t, x) = σ(t, x)( 0 σ(t, y)dy + ϕ(t)), and ϕ is a known function (actually ϕ is the market price of risk). We note that in all these models the shock is a one dimensional Brownian motion, which means that the same set of shocks affect all forward rates, which constrains the correlations between bond prices. A natural mathematical generalization of the HJM model for forward rates, since we have a function of two variables, is to consider the noise term to be an inﬁnite-dimensional Brownian motion W(t, x). In other words, this means that every instantaneous forward rate is driven by its own noise, and the equation becomes

µ ¶ ∂f(t, x) df(t, x) = + a(t, x) dt + σ(t, x)dW(t, x) , (1.15) ∂x

This idea was proposed by Kennedy [34], [35], where the forward rate was modelling as a Gaussian ﬁeld, and Goldstein [14], where the noise term is a speciﬁc space-time white nose. P. Santa-Clara and D. Sornette [71] generalize this approach, presenting various types of noise term, as well as, application to pricing bond derivatives.

13 Motivated by statistical properties of interest rates, R. Cont [7] propose a model in which the forward rate curve is decomposed in factors as follows

f(t, x) = f(t, xmin) + s(t)(Y (x) + u(t, x)) (1.16)

where xmin, xmax are shortest and longest maturity available on the market, s(t) = f(t, xmin)−f(t, xmax),Y is a deterministic shape function defining the average profile of the term structure, and u(t, x) an adapted process describing the random deviations of the term structure from its long term average shape. Empirical studies identify the level of interest rates, the steeptness (slope) of term structure and its curvature as three significant parameters in the geometry of the yield (see for instance [49]). Then, under some technical assumptions on the market properties, R. Cont [7] deduces that u satisfies the following parabolic SPDE µ ¶ ∂u(t, x) k ∂2u(t, x) du(t, x) = + b(t, x, u(t, x)) + 2 dt ∂x 2 ∂x (1.17) + σ (t, x, u(t, x)) dW (t, x) , where b and σ are some functions, and k a real parameter. In [7], author discussed the case b ≡ 0, σ ≡ const.

1.4 Ito’sˆ formula in Hilbert space

In this section we will present a version of the Ito’sˆ formula in Hilbert spaces. Although this result is known and can be found, for example, in [70], Chapter 2, Theorem 4.2, we are going to present here a simpler proof, using the one-dimensional Ito’sˆ formula.

14 Let F = (Ω, F, {Ft}t∈[0,T ], P) be a stochastic basis. A random variable τ taking values in R+ is called a stopping time with respect to the filtration Ft, t ∈ R+, if the random event {ω : τ(ω) ≤ t} ∈ Ft, for every t ∈ R+. The real-valued stochastic process M(t) is called martingale relative to the filtration Ft, if it Ft-adaptive, P- integrable, and E[M(t)|Fs] = M(s) for every s, t ∈ R+, s ≤ t. The real-valued process M(t) is called local martingale relative to the filtration Ft, if there exists a sequence of stoping times {τn}n∈N, τn ↑ ∞, such that for every n, the process

c M(t ∧ τn) is a martingale relative to Ft. We denote by Mloc(R+, R) the set of all real-

c valued continuous local martingales. One can show that for every M ∈ Mloc(R+, R), there exists a unique (up to a version) continuous increasing stochastic process hMit

2 c such that |M(t)| − hMit ∈ Mloc(R+, R). The process hMit is called the quadratic variation process of the local martingale M(t). For a predictable process X(t) one Rt may define the integral X(t)dM(t) in sense of Ito.ˆ For more on these definitions, 0 properties and proofs see for instance [48], [75]. All these definitions can be extended naturally to the stochastic processes with values in some Hilbert space (for more details see Chapter 2, Section 1.4 in [70]). For sake of completeness we briefly recall some of them. Let V be a separable Hilbert space and we denote by V 0 its conjugate space. An V -process M(t) is called martingale

(local martingale) relative to the ﬁltration Ft if (a) E||M(t)||V < ∞, for every t ∈ R+

(E||M(t ∧ τn)||, for every t ∈ R+ and some sequence of stoping times {τn}n∈N such

∗ 0 ∗ that τn ↑ ∞); (b) for every h ∈ V , the process h M(t) is a martingale (local martingale) relative to the ﬁltration Ft. The set of all continuous local martingales taking

c values in V will be denoted by Mloc(R+,V ). Similarly to real-valued local martin-

c gales, given M(t) ∈ Mloc(R+,V ), we denote by hMit the increasing process such

15 2 c that kM(t)k − hMit ∈ Mloc(R+,V ). We say that hMit is the quadratic variation of the M(t). Let us consider a triplet of Hilbert spaces (V,H,V 0) and suppose that M(t) is a continuous local martingale relative to ﬁltration {Ft}, which takes values in the

Hilbert space V . Since V is a separable Hilbert space, there exists a CONS {hk}k∈N in this space. We denote by hMiit the quadratic variation of the local continuous martingale Mk(t) := (M(t), hk). To prove the main result of this section we will need the following result.

1.4.1 Lemma. For all ω ∈ Ω1 ⊂ Ω, where P(Ω1) = 1,

dimXV hMit = hMiit . k=1

For the proof see Lemma 8, Section 2.1.8 in [70].

For a predictable V -process f we put

t t Z XN Z (f(s), dM(s)) := lim (f(s), hk)dMk(s) . N→∞ 0 k=1 0

The last integral is well-deﬁned [70]. Suppose that x(ω, t) and y(ω, t) are given functions on Ω×[0,T ] and taking values in V and V 0 respectively. We assume that these functions are predictable and satisfy the inequality

ZT ¡ 2 2 ¢ kx(ω, t)kV + ky(ω, t)kV 0 dt < ∞ (P − a.s.). (1.18) 0

Let x(·, 0) be an F0-measurable function on Ω, and let τ be a stopping time.

16 1.4.2 Theorem. If for every v ∈ V the following equality

(x(t), v) = (x(0), v) + [v, y(s)]V,V 0 ds + (M(t), v) (1.19) 0 holds P × B-a.s. on the set {(ω, t): t < τ(ω)}, then

Zt Zt 2 2 kx(t)kV = kx(0)kV + 2 [x(s), y(s)]V,V 0 ds + 2 (x(s), dM(s)) + hMit (1.20) 0 0

Proof. Since V is a separable Hilbert space, there exists a CONS {hk}k∈N in this space.

Then, equation (1.19) for v = hk implies

xk(t) = xk(0) + [hk, y(s)]ds + Mk(t) , (1.21) 0

where xk(t) = (x, hk)V ,Mk(t) = (M, hk)V , and k ∈ N.

Note that, since x(ω, t):Ω × [0,T ] → V is an adaptive process, and xk(ω, t) =

(x(ω, t), hk)V , using the continuity of inner product in V , we ﬁnd that xk is Ft-adapted for every k ∈ N. By the same arguments, we conclude that Mk(t) = (M(t), hk)V is a local martingale. Also, since [·, ·] is a bounded bilinear form on V × V 0 we get that

[v, y(t)]V,V 0 is Ft-adaptive, and by (1.18) we deduce that   Zt ¯ ¯ ¯ ¯2 P  ¯[v, y(s)]¯ ds < ∞ for all t > 0 = 1 .

17 Thus, we can apply one dimensional Ito’sˆ formula to the process xk given by (1.21), by which we get

Zt Zt 2 2 xk(t) = xk(0) + 2 xk(s)[hk, y(s)]ds + 2 xk(s)dMk(s) + hMk(t)i , (k ∈ N). 0 0 (1.22)

Summing up both parts of last equalities with respect to k = 1,...,N, we ﬁnd

t XN XN Z XN 2 2 xk(t) = xk(0) + 2 xk(s)[hk, y(s)]ds k=1 k=1 0 k=1 t Z XN XN + 2 xk(s)dMk(s) + hMk(t)i . (1.23) 0 k=1 k=1

In the last equality for every (ω, t) ∈ Ω × [0,T ] such that t ≤ τ(ω) we pass to the limit ∞ P 2 2 2 with N → ∞. By Parseval’s equality xk(t) = kx(t)k . Similarly we get kx(0)k . k=1 Rt PN Taking into account the continuity of inner product, lim xk(s)[hk, y(s)]ds = N→∞ 0 k=1 Rt Rt PN Rt [x(s), y(s)]V,V 0 ds. By the deﬁnition lim xk(s)dMk(s) = (x(s), dM(s)). 0 N→∞ 0 k=1 0 Finally, by Lemma 1.4.1 the last term from LHS of (1.23) converges to hMit. Hence we resume (1.19). Theorem is proved.

1.5 Existence and uniqueness of solution

For every differential equation, deterministic or stochastic, ordinary or with partial derivatives, the ﬁrst and fundamental question is existence and uniqueness of solution. A general theory of this problem for SPDEs can be found in [8], [43], [70], [79]. In this section we present some known results about existence and uniqueness of solution of parabolic SPDEs, which will need for future studying.

18 0 Let (V,H,V ) be a triple of Hilbert spaces, (Ω, F, {Ft}, P) a stochastic basis,

0 A(t): V → V , Mk(t): V → H, k ≥ 1, t ∈ [0,T ] some linear bounded operators. We consider the following SPDE

du(t) = (Au(t) + f(t))dt + (Mku(t) + gk(t))dWk(t) , (1.24)

where t ∈ [0,T ], u(0) = u0 ∈ L2(Ω; H) and u0 is F0-measurable. We assume that

Wk (k ∈ N) are independent standard Wiener processes, f, gk (k ∈ N) are Ft-adapted

0 random processes such that f ∈ L2(Ω × (0,T ); V ) and gk ∈ L2(Ω × (0,T ); H)(k ∈ N). Let V be a dense subset of the Hilbert space V relative to the topology generated by the norm k · kV . The following theorem holds true. For complete proof see for instance [70], Chapter 3.

1.5.1 Theorem. In addition to all assumptions mentioned above, suppose that

∞ RT P 2 (i) E||gk(t)||H dt < ∞; k=1 0

(ii) For every ϕ ∈ V, the processes Aϕ(t) and Mkϕ(t) are Ft-adapted;

∞ RT P 2 (iii) E||Mkϕ(t)||H dt < ∞ for all ϕ ∈ V; k=1 0

(iv) There exist a positive constant δ and a real number C0 such that, for all t ∈ [0,T ], all ω ∈ Ω and all v ∈ V, we have

X∞ 2 2 2 2[Aϕ(t), ϕ] + ||Mkϕ(t)||H ≤ −δ||ϕ||V + C0||ϕ||H ; (1.25) k=1

(v) There exist a positive number CA so that, for all t ∈ [0,T ], ω ∈ Ω, ϕ ∈ V,

||Aϕ||V 0 ≤ CA||ϕ||V .

19 Then there exists a unique weak solution u of equation (1.24), the solution belongs to

L2(Ω; C((0,T ); H)) ∩ L2(Ω × [0,T ]; V ) and

³ ´ ³ ZT ´ 2 δ 2 E sup ||u(t)||H + E ||u(t)||V ≤ (1.26) 0

We want to mention that conditions (i)-(iii), (v) mostly are related to the deﬁnition of solution and come naturally from the structure of equation (1.24). The condition (iv) represents the (strong or super-) parabolicity or coercivity assumption and the equation in this case is called parabolic SPDE. This is due to the fact that in the case when operator A is an elliptic differential operator and Mk a differential operator subordi- nated, in some sense, to A, the condition (iv) is fulﬁlled. For example classical heat equation is the simplest deterministic parabolic equation and a stochastic counterpart can be taken as du(t, x) = auxxdt + σuxdW (t), which is a particular case of equation

∂2 ∂ (1.24) with A = a ∂x2 , f(t) = g(t) = 0, M = σ ∂x . This equation is solvable in the triple (H1(R), H0(R), H−1(R)) if δ = 2a − σ2 > 0 or in equivalent form

Z Z σ2 2[Aϕ(t), ϕ] + ||Mϕ(t)||2 = a ϕ ϕdx + ϕ2dx H xx 2 x R Z R 2 2 2 2 = −(2a − σ ) ϕxdx = −δkϕk1 + c0kϕk0 , R which is exactly condition (iv) from Theorem 1.5.1. Note that δ = 0 corresponds to the degenerate case 2a = σ2, and although the corresponding SPDE is a parabolic type equation, special studies have to be done in this case.

20 Similarly to deterministic equations the estimate (1.26) of the solution is called energy estimate which implies uniqueness of the solution. For all equations considered in our research, ﬁrst we will establish the existence and uniqueness of the solution by applying Theorem 1.5.1. Here we will show how this theorem can be applied to stochastic heat equation. Besides the fact that this equation is one of the simplest SPDE, it will also play the role of benchmark in our investigations.

1.5.2 Example. Stochastic heat equation with additive space-time white noise.

1 Let (Ω, F, {Ft}, P) be a stochastic basis and assume that γ < − 2 . In the triple of Hilbert spaces (Hγ+1(0, π), Hγ(0, π), Hγ+1(0, π)) we consider the following SPDE

  P  du(t, x) = uxxdt + hk(x)dWk(t), k≥1 (1.27)   u(0, x) = u(t, 0) = u(t, π) = 0, x ∈ [0, π], t ∈ [0,T ] ,

q 2 where hk(x) = π sin(kπx) and Wk are independent standard Brownian motions.

Note that hk forms a CONS in L2(0, π) and also are all eigenvalues of the Laplace operator ∆ = ∂xx with zero boundary conditions on (0, π). Using the notations from

γ+1 γ 0 γ+1 Theorem 1.5.1 we have V = H ,H = H ,V = H , A = ∆, f = 0, Mk =

0, gk = hk. Now let us check the conditions (i)-(v) from Theorem 1.5.1. Since

1 P 2 P 2γ γ < − 2 we have khkkγ = k < ∞, and hence (i) is satisﬁed. Opera- k≥1 k≥1 tors A and Mk are deterministic and hence (ii) holds true. Mk = 0 and (iii) obviously is satisﬁed. Coercivity condition (iv): 2[Aϕ, ϕ] = 2[ϕxx, ϕ] = −2[ϕx, ϕx] =

2 −2kϕxkγ ≤ −δkϕkγ+1, with δ = 2, so (iv) is veriﬁed. Since A is a differential operator of second order, from general theory of Sobolev spaces, we have that the operator

A : Hs → Hs−2 is a linear bounded operator for every s ∈ R. Thus assumption

21 1 (v) is veriﬁed, and we conclude that for every γ ≤ − 2 there exists a unique solution γ γ+1 u ∈ L2(Ω; C((0,T ); H )) ∩ L2(Ω × [0,T ]; H ) of equation (1.27), and it satisﬁes the following estimate (energy type estimate)

  ZT  2 2  E sup ku(t)kγ + ku(t)kγ+1dt ≤ C. t∈(0

We conclude this chapter with a version of Burkholder-Davis-Gundy (BDG) inequality [8]. Let W (t) be a standard Brownian Motion, and Ft the ﬁltration generated by this Brownian Motion.

1.5.3 Theorem. Let ξ be a square integrable Ft adapted process. For every p > 0 there exists constants cp and Cp, such that

  ¯ ¯ p  ¯ ¯p ¯ZT ¯ 2 ¯Zt ¯ ¯ ¯ ¯ ¯ ¯ 2 ¯   ¯ ¯  cpE ¯ |ξ(s)| ds¯  ≤ E sup ¯ ξ(s)dW (s)¯ ¯ ¯ 0

22 Chapter 2

Regularity of solution

2.1 Introduction

For a given evolution equation, besides existence and uniqueness of solution, it is important to know regularity properties of solution. By regularity or smoothness we mean continuity, Holder¨ continuity, differentiability, etc. of the solution with respect to both time and space variables. This problem have been widely studied by many authors, and various classes of SPDEs have been considered. Here we recall some results related to parabolic evolution equations. As an application to neurophysiology, J.B.Walsh [79] considered the stochastic heat equation on a ﬁnite interval, namely the equation of the form    ∂v = ∂2v − v + f(v, t)W˙ t > 0, 0 < x < L;  ∂t ∂x2  ∂v (0, t) = ∂v (L, t) = 0 t > 0; (2.1)  ∂x ∂x   v(x, 0) = v0(x), x ∈ [0,L],

2 where v0 is F0-measurable, E(v0(x)) is bounded and f is uniformly Lipschitz continuous. In [79] was proven that the mild solution of equation (2.1) (t, x) → v(t, x) is

1 1 Holder¨ continuous in t with exponent 4 −ε, and in x with exponent 2 −ε, for all ε > 0. A general linear parabolic evolution equation of the form

du = (aijuxixj + biuxi + cu + f)dt + (σikuxi + νku + gk)dWk(t), t > 0, (2.2)

23 with proper conditions on the coefﬁcients, boundary values and initial conditions, has been considered in [37]-[45], [50],[51]. In these papers the authors studied the Cauchy

d d problem for the whole space (x ∈ R ) and half space (x ∈ R , x1 > 0), the Dirichlet problem for a bounded domain, developing Lp theory for these equations. By Lp theory we mean that suitable Sobolev spaces and sufﬁcient conditions on the coefﬁcients are found such that equation (2.2) has a unique solution in these spaces. From this theory, in particular, one can conclude about regularity properties of the solution. For example, N.V.Krylov [42] studied the Cauchy problem for equation (2.2) is case of

d whole space R , and proved, as an application of Lp theory, that for d = 1, the solution is Holder¨ continuous in x of order 1/2 − ε and in t of order 1/4 − ε. The Dirich- let problem on bounded domain for the same equation was discussed by N.V.Krylov, S.Lototsky in [44],[45], where the weighted Sobolev spaces were constructed. In particular the equation (2.1) is considered, up to the boundary conditions (Dirichlet in

[45] and Neumann in [79]), and the result for d = 1 agrees with that obtained in [79]. In R.Mikulevicius [54] and R.Mikulevicius, H.Pragarauskas [55] the second-order linear parabolic SPDE with deterministic leading coefﬁcients driven by a cylindrical Brownian motion in some Hilbert space were considered. Following the ideas given in B.Rozovskii [69], R.Mikulevicius [54] studied the Cauchy problem on the whole space (x ∈ Rd) in the scale of Holder¨ spaces, while in [55] the Cauchy-Dirichlet problem on the half-space in suitable weighted Holder¨ spaces is investigated. Since solution belongs to Holder¨ space the conclusion about the smoothness of the solution follows.

Finally we want to mention the results presented in DaPrato [8], a general linear evolution equations is investigated. More precisely, the coefﬁcients of equation are some linear operators which generates a strongly continuous semigroup, and the noise, either additive (Chapter 5) or multiplicative (Chapter 6), is driven by a cylindrical

24 Brownian Motion in some Hilbert space. It is shown that the weak (mild) solution has a continuous path and some particular examples are considered too. In this chapter we will study the regularity properties of equation (1.24) driven by space-time white noise, either additive or multiplicative. Similar to Chapter 1 we consider M being a d-dimensional compact orientable C∞ manifold with a smooth positive measure dx. Let A be a differential, elliptic operator of order 2m on M, formally selfadjoint and semi-lower bounded. Without loss of generality, using the notations from Section 1.1, we consider L = −A and apply all results mentioned

1 therein. The operator Λ = (−A) 2m is well-deﬁnite, elliptic of order 1 and generates

s the scale {H (M)}s∈R of Sobolev spaces. We denote by hk and λk, k ∈ N, the eigenfunctions and corresponding eigenvalues of the operator Λ, and by lk the eigenvalues of the operator A. Recall that {hk}k∈N forms a CONS in L2(M, dx) (for more details, see Section 1.1).

In the triple (Hγ+m, Hγ, Hγ−m), for some γ ∈ R, we consider the following equation   P∞ du(t) = Au(t)dt + α B (u)h (x)dW (t);  k k k k  k=1 (2.3) u(t, ·)|M = 0, 0 < t < T ;    u(0, x) = u0(x), x ∈ M,

∞ γ where t ∈ [0,T ], x ∈ M, u0 ∈ C (0, π), Mk are operators acting in H and

α αk = k for some real parameter α. Using results from Chapter 1, we will ﬁnd sufﬁcient conditions on parameters α and γ, and operators A, Bk, which guarantee existence and uniqueness of solution in the scale of Sobolev spaces Hγ(M).

A major novelty is the factor αk in the noise term; in all previous works α = 0. The speciﬁc form of this factor is motivated by the asymptotic behavior of the eigenvalues of the operator A. It turns out that the parameter α controls the smoothness

25 of the solution u of equation (2.3). As one may expect, as α decreases the solution becomes smoother both in time and space. We state the regularity properties (number of derivatives and Holder¨ continuity) of the solution in both space and time in terms of the parameter α, the L∞ norm and some regularity properties of the eigenfunctions of the operator A.

In connection with this we make the following reasonable assumptions

ρ (A1) khkk∞ ≤ c |lk| , for some ρ ∈ R+ and every k ∈ N;

δ(β) β (A2) |hk(x) − hk(y)| ≤ |lk| |x − y| , where 0 < β ≤ 1, x, y ∈ M, k ∈ N, and δ is an increasing function, with positive values.

We rely on some results from spectral theory of selfadjoint differential operators, which is a separate topic in mathematics, with many fundamental results. Even the particular question of pointwise bounds of eigenfunctions have been studied widely. There are many papers written on this subject, where various operators are considered, and this question represents a well explored ﬁeld. We refer the reader to classical work [61], and as well as to some recent papers [15], [19], [17], [18], [29], [30], [31], [32], [72], [73].

Of course the estimates vary from case to case and depend on the operator A, as well as on the geometry of the domain M and boundary conditions. One general result holds true.

2.1.1 Remark. Without making any geometric assumptions, if A is the Laplacian on

d−1 compact manifolds with boundary, (A1) holds true with ρ = 4 (see the proof in d−1 d [15]). In particular, khkk∞ < c |lk| 4 holds true for a general bounded domain in R . This result is sharp for the sphere M = Sd.

26 We recall that regardless of the domain M, the eigenvalues µk of the operator A with zero boundary conditions have the following asymptotic

2m/d lk ∼ k , (2.4)

1/d and consequently λk ∼ k (see for instance [76]). We want to mention that generally speaking the conditions (A1) and (A2) are hard to check, and represent difﬁcult mathematical problems, however in many applications the eigenvalues and eigenfunctions are computable explicitly and these conditions are veriﬁed by simple arithmetic evaluations. Let us consider the one dimensional Laplace operator with zero boundary conditions.

2.1.2 Example. One dimensional Laplacian.

Assume that d = 1, Au = uxx, m = 1,M = [0, π] and zero boundary conditions.

2 Then hk(x) = π sin(kx), and hence khkk∞ ≤ const. Thus (A1) is satisﬁed with ρ = 0. 2 Also note, that lk = −k , and thus (2.4) also holds true. Using well-known inequality

β β | sin(x)| ≤ |x|, and a trivial trigonometric identity, we have |hk(x)−hk(y)| ≤ k |x−

β y| for all x, y ∈ [0, π] and every β > 0. Obviously we have that |hk(x) − hk(y)| ≤ 2, x, y ∈ [0, π]. From here we conclude

β 1−β |hk(x) − hk(y)| = |hk(x) − hk(y)| |hk(x) − hk(y)| (2.5)

β β β (1−β) 2 β ≤ k |x − y| 2 ≤ clk |x − y| ,

β where c is a constant independent of β. Hence (A2) is veriﬁed with δ(β) = 2 .

More concrete examples will be considered at the end of this chapter.

27 2.1.3 Remark. Conditions (A2) in fact represents the Holder¨ continuity of the eigen-

∞ functions hk. Since hk ∈ C , the constant ζ from inequality |hk(x) − hk(y)| ≤

β β ζ |x−y| corresponds to the Lipschitz constant of the function hk, which can be associated with the ”ﬁrst derivative”. On the other hand, ”the ﬁrst derivative” essentially is

1 2m Λ which is a differential operator of order one. Since Λ(hk) = λkhk and λk = lk , we β get δ(β) = 2m . These completely heuristic arguments turn out to hold true in many concrete example. However, we do not know if this is true in general.

The results obtained here agree with those known and mentioned above. We would like to note that equation (2.3) is a particular case of equations studied in [50], however, it turns out that to check all abstract conditions on coefﬁcients and spaces of solution is not easier than establishing these results directly. Also, we want to note that equation (2.3) does not belong to the class of equations considered in [54] (whole space), [55] (half plane). Not only because we consider the bounded domain, which to some extend is just a technical problem as long as there are theories for the whole space and half plane, but also the noise term is different.

A similar approach to control the smoothness of the solution through some parameters of the noise term are considered in [60] where the nonlinear SPDE du =

1 √ 2 ∆udt + udW (t, x) is studied. Mytnik, Perkins and Sturm in a recent paper [60] study regularity properties of

1 √ the solution of the equation du = 2 ∆udt + udW (t, x), where x ∈ R, t ∈ [0,T ] and W (t, x) is a space-time white noise on R+ × R. The method of duality is used to establish the existence and uniqueness of the solution. The solution itself represents the density function for one-dimensional super-Brownian motion. The regularity properties of the solution are described in terms of asymptotic behavior of the kernel of the covariance functional on the main diagonal y = x. It turns out that there is

28 a connection between our results and results stated in [60]. We will discuss these in more details in Section 2.2. In terms of regularity properties, the results obtained here are the same for both additive and multiplicative noise, however, the spaces in which equation (2.3) has solution differ. Also, the technical evaluations in multiplicative case are more involved. Thus, we consider separately the equations with additive noise, Section 2.2, and equations with multiplicative noise in Section 2.3. The main results of this chapter are contained in Theorem 2.2.5, 2.2.11 and 2.3.8. Every theorem will be followed by a simple example, usually stochastic heat equation. In the last section of this chapter, as applications of general results, we consider various concrete examples.

2.2 Equations with additive noise

2.2.1 Existence and uniqueness

In this section we consider equation (2.3) with additive noise, which means that we take Bku ≡ 1. Thus equation (2.3) becomes

X∞ du(t) = Au(t)dt + αkhk(x)dWk(t) , (2.6) k=1 where t ∈ [0,T ], x ∈ M, the same initial values and boundary conditions as in

α equation (2.3), and αk = k for some real parameter α. Initially we will establish some results about existence and uniqueness of the solution. For this, we make the following assumption:

1 (A3) Assume that α, γ ∈ R such that γ < −d(α + 2 ).

29 2.2.1 Theorem. If the assumption (A3) is satisﬁed, then equation (2.6) has a unique weak solution

γ+m γ u ∈ L2((0,T ) × Ω; H ) ∩ L2(Ω; C((0,T ); H )).

Proof. We apply Theorem 1.5.1 with f = 0, Mk = 0, and gk = αkhk, k ∈ N. Conditions (ii), (iii), (v) of this theorem are obviously satisﬁed. In order to check condition (i) we write

T X∞ Z X∞ X∞ 2 2 2 2α 2γ E||gk(t)||γdt = c αkkhkkγ = c k λk . k=1 0 k=1 k=1

1 Since λk ∼ k d , from the last equality we deduce

∞ ZT ∞ X X 2γ 2 2α /d E||gk(t)||γdt = c k k . k=1 0 k=1

The last series converges in virtue of Assumption (A3).

Finally, we verify condition (iv). It sufﬁces to check it for basis functions hk, which leads to

2 −2m 2 2 2[Ahk, hk] = 2lk[hk, hk] = 2lkkhkkγ = 2lkλk khkkγ+m = −2khkkγ+m .

Thus (iv) from Theorem 1.5.1 is satisﬁed with δ = 2 > 0. Theorem is proved.

2.2.2 Remark. Note that for α = 0, which corresponds to standard space-times white noise, the solution exists if γ < −d/2. In particular, for d = 1, we have that the solution exists if γ < −1/2 (this covers well know result, see for instance [79]). Also, we want to mention that the equations driven by additive space-time white noise have

30 solution in some Hγ regardless of space dimension d. Namely, for every d ∈ N and

γ every α ∈ R there exists γ ∈ R such that u ∈ L2((0,T ) × Ω; H ). As will see latter on, this is not the case for the equations with multiplicative space-time white noise.

2.2.3 Example. Assume that d = 1,M = [0, π] and let A be the one dimensional Laplace operator with zero boundary conditions. In the triple of Hilbert spaces

(Hγ+1, Hγ, Hγ+1) let us consider the following SPDE

X − 2 du(t, x) = uxx(t, x)dt + k 3 sin(kx)dWk(t) , (2.7) k≥1

where x ∈ [0, π], t ∈ [0, 1] and Wk are independent standard Brownian motions. As we already mentioned (see Example 1.5.2), the functions hk are the eigenfunctions of

2 the operator A and forms a CONS in L2. Note that α = − 3 . By Theorem 2.2.1, for 1 γ+1 every γ < 6 , there exists a unique strong solution of equation (2.7) in H .

Looking forward for regularity properties of the solution u, we present here the plot of one realizations of the ﬁeld u given by (2.6) for different values of the parameter α. In the Figure 2.1, Panel (a), the solution corresponds to the equation discussed in Remark 2.2.2, i.e. α = 0, d = 1. The solution of equation (2.7) from Example 2.2.3

2 (α = − 3 ) is shown in Panel (b). We used Crack-Nicolson ﬁnite difference scheme (see for instance [38], [57]) to simulate the ﬁeld u. Note that the solution u looks smoother,

2 Panel (b), for α = − 3 than that for the parameter α = 0, Panel (a). How smother it is, we will answer in the next subsections.

31 Panel (a). α=0 Panel (b). α= −2/3

1 0.4

0.5 0.2

0 0 u(t,x) u(t,x) −0.5 −0.2

−1 −0.4 1 1 4 3 0.5 0.5 2 2 1 t 0 0 x t 0 0 x

Figure 2.1: Realization of approximated solution of stochastic heat equation with additive noise

In what follows we will use a convenient representation for the solution u of equation (2.6). Denote by Gt(x, y) the fundamental solution of the corresponding deterministic parabolic equation ut = Au(t) with the same boundary values and initial conditions as (2.3). Then (see for instance [9], Chapter 2,7)

X lkt Gt(x, y) = hk(x)hk(y)e . (2.8) k≥1

Similarly to deterministic case, it can be shown that the weak solution of equation (2.6) satisﬁes the equality

π t Z X Z Z u(t, x) = Gt(x, y)u0(y)dy + αkGt−s(x, y)hk(y)dWk(s) . (2.9) 0 k≥1 0 M

The proof can be found in [8], Section 5.2, page 121. Note that by (2.9), u actually is a mild solution of equation (2.6), and by speciﬁc structure of this equation it turns out that the mild solution coincides with weak solution, existence of which is provided by Theorem 2.2.1.

32 From (2.9) we see that without loss of generality we can suppose that u0 = 0 and study the regularity of the second term in (2.9). Thus using (2.8), from (2.9) we get

t X Z Z X lj (t−s) u(t, x) = αk hj(x)hj(y)e hk(y)dydWk(s) , k≥1 0 M j≥1

and taking into account that {hk}k≥1 forms an orthonormal basis in L2(M), we have the representation

t Z X lk(t−s) u(t, x) = αkhk(x)e dWk(s) . (2.10) 0 k≥1

2.2.2 Regularity in space

To establish the regularity properties of solution u in space variable x, we will prove an auxiliary result.

2.2.4 Lemma. Under assumption (A2), for any β ∈ (0, 1] and α ∈ R such that

m(1 − 2δ(β)) 1 α < − , (2.11) d 2 the solution u of equation (2.6) satisﬁes the following inequality

¯ ¯ ¯ ¯p E¯u(t, x) − u(t, y)¯ ≤ C|x − y|βp , (2.12) where x, y ∈ M, t ∈ [0,T ], p > 2.

Proof. From (2.10) one deduces ¯ ¯ ¯Zt ¯p ¯ ¯p ¯ X ³ ´ ¯ ¯ ¯ ¯ lk(t−s) ¯ E¯u(t, x) − u(t, y)¯ = E ¯ hk(x) − hk(y) αke dWk(s)¯ . ¯ ¯ 0 k≥1

33 From here by the BDG inequality (1.28), we continue

¯ ¯ ¯ Zt ¯p/2 ¯ ¯p ¯X ¯ ¯2 ¯ ¯ ¯ ¯ ¯ ¯ 2 2lk(t−s) ¯ E¯u(t, x) − u(t, y)¯ ≤ ¯ ¯hk(x) − hk(y)¯ αk e ds¯ ¯ ¯ k≥1 0 ¯ ¯ ¯ ¯ ¯p/2 X ¯ 2 α2 ³ ´ ¯ ¯ k 2lkt ¯ = cp ¯ ¯hk(x) − hk(y)¯ 1 − e ¯ . ¯ −2lk ¯ k≥1

2lkt Note that lk < 0, k ∈ N, and thus 1 − e is uniformly bounded in t and k, hence

¯ ¯p/2 ¯ ¯p ¯X 2 ¯ ¯ ¯ ¯ αk 2¯ E¯u(t, x) − u(t, y)¯ ≤ cp ¯ |hk(x) − hk(y)| ¯ . (2.13) ¯ |lk| ¯ k≥1

From here, by assumption (A2), we get

¯ ¯ ¯ ¯ ¯ ¯p/2 ¯ ¯p ¯X ¯ E¯u(t, x) − u(t, y)¯ ≤ c |x − y|βp ¯ k2α|l |−1|l |2δ(β)¯ . p ¯ k k ¯ k≥1

2m/d Finally, since lk ∼ k , we ﬁnd that the last series converges if

2m 2α + (2δ(β) − 1) < −1 , d which essentially is assumption (2.11) on parameters β and α. Lemma is proved.

Now we are ready to prove the theorem about regularity of the solution in space variable x.

34 1 2.2.5 Theorem. Suppose that Assumptions (A2) and (A3) are satisﬁed, and α < − 2 + m d . Let u be the solution of equation (2.6) and r the biggest integer such that there exists β0 ∈ (0, 1] which satisﬁes the following inequality

2dα + 2r + 4mδ(β0) − 2m < −d .

Then Λr(u) is Holder¨ continuous in x of order

n d dα 1 r o ζ := sup β | δ(β) < − − + − , β ∈ (0, 1] − ε , 4m 2m 2 2m for every ε > 0, as long as ζ > 0;

Proof. By the initial assumption, the solution u of the equation (2.6) exists and satisﬁes Lemma 2.2.4 for some β > 0. Let r be the integer as stated in the theorem. Note that

1 m α < − 2 + d implies that r ≥ 0. By (2.10) we have

t h Z X i r r lk(t−s) Λ (u(t, x)) = Λ αkhk(x)e dWk(s) (2.14) 0 k≥1 t X ³ ´ Z r lk(t−s) = αkΛ hk(x) e dWk(s) k≥1 0 t X Z r lk(t−s) = αkλkhk(x) e dWk(s) k≥1 0

1 as long as the last series converges absolutely. Since λk ∼ k d , we apply Lemma 2.2.4

r r to Λ u with α˜ = α + d and conclude that

¯ ¯p ¯ r r ¯ βp ¯Λ (u(t, x)) − Λ (u(t, y))¯ ≤ cp|x − y| , (2.15)

35 1 r dα d for every β > 0 such that δ(β) < 2 − 2m − 2m − 4m . Taking into account the definition of ζ we have that the inequality (2.15) is satisfied for every β ≤ ζ. By Kolmogorov’s criterion (Theorem 1.1.1) with d := d, p := p, q := βp − d, we get that the process Λru(t, x) satisfies the following inequality

¯ ¯ ³ ´2/p ¯ r r ¯ β− d A ¯Λ u(t, x) − Λ u(t, y)¯ ≤ Y |x − y| p ln (x, y ∈ M, p > 2) |x − y| where Y is a positive random variable, and A is a positive real number. For every ε > 0 ³ ´2/p ε A r the function |x| ln |x| is continuous at x = 0. Hence, the process Λ u(t, x) is 1 Holder¨ continuous in x of order β − p − ε, for every p > 2 and ε > 0. Consequently, Λru(t, x) is Holder¨ continuous in x of order less than β ≤ ζ. Theorem is proved.

2.2.6 Remark. As we already mentioned, the operator Λr, r ∈ N, plays the role of the differential operator of order r. Generally speaking, Theorem 2.2.5 tells how many derivatives has the solution u and what is the order of continuity of the r-th derivative. Moreover, the theorem remains true if we take any other operator Ψ instead of Λr, as

r long as one can show that Ψhk = cλkhk (see equality (2.14)).

β One special case that occurs in many application is δ(β) = 2m (for more motivations about this see also Remark 2.1.3). For this particular situation Theorem 2.2.5 implies:

2.2.7 Theorem. Suppose that Assumptions (A2) and (A3) are satisﬁed, and assume

1 m β that α < − 2 + d and δ(β) = 2m , β ∈ (0, 1]. Let u be the solution of equation (2.6), d and denote by η := m − dα − 2 and r := bηc (bηc denotes the biggest integer less than or equal to η).

(i) if η 6= N then Λr(u) is Holder¨ continuous of order ζ = η − r − ε, for every ε > 0, as long as ζ > 0;

36 (ii) if η ∈ N, then Λr−1(u) is Holder¨ continuous of order ζ, for every ζ ∈ (0, 1).

2.2.8 Example. As an application of Theorem 2.2.7 we will consider the following

P − 2 SPDE: du(t, x) = uxx(t, x)dt + k 3 sin(kx)dWk(t) under the same setup as in k≥1 Example 2.2.3, i.e. the stochastic heat equation with Dirichlet boundary conditions,

2 zero initial values, α = − 3 , d = 1, x ∈ M = [0, π], t ∈ [0, 1]. By Example 2.2.3 γ+1 1 we have that the solution u ∈ H for every γ < 6 . As we concluded in Example β 2.1.2, Assumption (A2) is veriﬁed and δ(β) = 2 . Hence we can apply Theorem 2.2.7. 2 ∂r r ˜ ˜ Recall that hk(x) = π sin(kx) and λk = k. Since ∂xr hk = cλkhk, where hk is either sin(kx) or cos(kx), which essentially obey the same regularity properties, by Remark

∂ 7 2.2.6, we can take Λ = ∂x . Hence, by Theorem 2.2.7 with η = 6 we conclude: the solution u has one derivative in x, and the ﬁrst derivative ux is Holder¨ continuous of

1 order less than 6 . By the same arguments, we have that the solution u(t, x) of equation du(t, x) = P uxx(t, x)dt+ sin(kx)dWk(t) (see Example 1.5.2) is Holder¨ continuous in x of order k≥1 1 1 1 less than η = m − dα − 2 = 1 − 0 · 1 − 2 = 2 . This resume the classical result by Walsh [79]. Compare the graphs of these two stochastic ﬁelds shown on Figure 2.1.

2.2.9 Remark. We want to mention that the Theorem 2.2.5 about regularity of the solution can not be obtained by directly the Sobolev embedding theorems. For example, by Theorem 2.2.7 we have that the solution exists, is unique and u ∈ Hγ+m, for

d d m− 2 −dα+ε every γ < −dα− 2 , i.e. u ∈ H for every ε > 0. By Sobolev embedding the- s s− d m−dα−d−ε orem (see for instance see [9], Chapter 5) H ⊂ C 2 , and hence u ∈ C . We

m−dα− d −ε proved that u ∈ C 2 , which obviously is a stronger result than that obtained by Sobolev embedding theorems. However, if one can develop the Lp theory for this

γ+m equation, namely to show that the solution exists in Hp for every p > 2, then m− d −dα+ε m− d −dα+ε− d by Sobolev embedding theorem u ∈ H p ⊂ C p p . Taking p large

37 m−dα− d −ε enough we obtain our result, namely u ∈ C 2 for every ε > 0, as long as

d m − dα − 2 − ε > 0. This is not of big surprise, since the main tool in our proof has been the Kolmogorov’s criterion 1.1.1, which to some extends is a version of Sobolev embedding theorem. In other words, one may conclude that we actually developed implicitly the Lp theory.

2.2.3 Regularity in time

Regularity in time is obtained in the same way as regularity in space. Initially we will prove an auxiliary result similar to Lemma 2.2.4.

2.2.10 Lemma. Under Assumption (A1), for every α ∈ R and β ∈ (0, 1] such that

d(1+2α) β < 1 − 2ρ − 2m , the solution u of equation (2.6) satisﬁes the following inequality

¯ ¯ ¯ ¯p βp E¯u(t1, x) − u(t2, x)¯ ≤ C|t1 − t2| 2 , (2.16)

where t1, t2 ∈ [0,T ], x ∈ M, p > 2.

Proof. Let 0 < t1 < t2 < T . By (2.10)

¯ ¯ ¯ ¯p E¯u(t1, x) − u(t2, x)¯ ¯ ¯p ¯Zt1 Zt2 ¯ ¯ X X ¯ ¯ lk(t1−s) lk(t2−s) ¯ = E ¯ hk(x)αke dWk(s) − hk(x)αke dWk(s)¯ ¯ ¯ 0 k≥1 0 k≥1 ¯ ¯p ¯Zt2 ¯ ¯ X ³ ´ ¯ ¯ lk(t2−s) lk(t1−s) ¯ = E ¯ hk(x)αk e − e I(s≤t1) dWk(s)¯ . ¯ ¯ 0 k≥1

38 By the BDG inequality, Theorem 1.5.3, we continue

¯ ¯ ¯ ¯p E¯u(t1, x) − u(t2, x)¯

¯ ¯p/2 ¯Zt2 ¯ ¯ X ¯ ¯2 ¯ ¯ 2 2¯ lk(t2−s) lk(t1−s) ¯ ¯ ≤ E ¯ hk(x)αk¯e − e I(s≤t1)¯ ds¯ ¯ ¯ 0 k≥1 ¯ ¯p/2 ¯ Zt2 ¯ ¯X ¯ ¯2 ¯ ¯ 2 2 ¯ lk(t2−s) lk(t1−s) ¯ ¯ ≤ E ¯ hk(x)αk ¯e − e I(s≤t1)¯ ds¯ . ¯ ¯ k≥1 0

After direct evaluations one shows

Zt2 ¯ ¯2 ¯ lk(t2−s) lk(t1−s) ¯ ¯e − e I(s≤t1)¯ ds 0 µ ¶ 1 ³ ´ ³ ´2 = 2 1 − elk(t2−t1) − elkt2 − elkt1 −2lk 1 − elk(t2−t1) ≤ . −lk

Thus, ¯ ¯p/2 ¯ ¯p ¯X lk(t2−t1) ¯ ¯ ¯ ¯ 2 2 1 − e ¯ E¯u(t1, x) − u(t2, x)¯ ≤ c ¯ hk(x)αk ¯ . ¯ |lk| ¯ k≥1

−µ|t2−t1| β β Since |1−e | ≤ µ |t2 −t1| for every 0 < β ≤ 1 and µ > 0, using Assumption (A1) the last inequality can be continued

¯ ¯ ¯ ¯ ¯X ¯p/2 ¯X ¯p/2 ¯ 2 2 β β −1¯ βp ¯ 2ρ 2 β−1¯ ≤ ¯ h (x)α l |t − t | l ¯ ≤ |t − t | 2 ¯ l α l ¯ . ¯ k k k 2 1 k ¯ 2 1 ¯ k k k ¯ k≥1 k≥1

39 α By asymptotic property (2.4), and the deﬁnition of αk = k , from the last inequality we have the following estimation

¯ ¯p X ¯ ¯ βp 2α+ 2m (β−1+2ρ) E¯u(t1, x) − u(t2, x)¯ ≤ c|t2 − t1| 2 k d . (2.17) k≥1

d The initial assumption β < 1 − 2ρ − 2m (1 + 2α) implies that the last series converges, hence the lemma is proved.

Now we are ready to prove the Theorem about regularity in t of the solution u(t, x)

m 2mρ 1 2.2.11 Theorem. Suppose that α < d − d − 2 , and assume that the assumptions (A1) and (A3) are satisﬁed. Then the solution u of equation (2.6) is Holder¨ continuous

1 1 d(1+2α) in t of order less than η := min{ 2 , 2 − ρ − 4m }.

Proof. Assumptions (A1) and (A3) imply that the equation (2.6) has a solution. Let

m 2mρ 1 β = 2τ. By initial assumption α < d − d − 2 we have that η > 0, and hence β ∈ (0, 1]. Consequently, by Lemma 2.2.10 the solution u of equation (2.6) satisﬁes the ¯ ¯p ¯ ¯ βp inequality E¯u(t1, x)−u(t2, x)¯ ≤ C|t1 −t2| 2 , where t1, t2 ∈ [0,T ], x ∈ M, p > 2.

βp Applying the Kolmogorov’s criterion (Theorem 1.1.1) with d := d, p := p, q := 2 −d, we get ¯ ¯ ³ ´2/p ¯ ¯ β − d A ¯u(t1, x) − u(t2, x)¯ ≤ Y |t1 − t2| 2 p ln . (2.18) |t1 − t2|

By the same arguments as in Theorem 2.2.7 we conclude that the process u is Holder¨

β continuous in t of any order less than 2 = η. Theorem is proved.

Taking A = ∆, and without making any geometric assumption on M, a direct consequence of Theorem 2.2.11 gives:

40 2.2.12 Corollary. Suppose that A is the Laplacian on a compact manifold with bound-

d 3 ary, and assume that γ < −dα − 2 and α < 2d − 1. Then the solution u of equation 1 3 dα d (2.6) is Holder¨ continuous in t of order less than η = min{ 2 , 4 − 2 − 2 }.

d−1 Proof. By Remark 2.1.1 ρ ≤ 4 . We apply the previous theorem, with m = 1 and m 2mρ 1 3 above mentioned ρ and conclude that for every α < d − d − 2 = 2d −1 the solution 1 1 d(1+2α) u is Holder¨ continuous in t of any order less then η = min{ 2 , 2 − ρ − 4m } = 1 3 d dα min{ 2 , 4 − 2 − 2 }.

2.2.13 Example. Let us consider the same equation as in Example 2.2.8: du(t, x) =

P − 2 uxx(t, x)dt + k 3 sin(kx)dWk(t), i.e. the stochastic heat equation with Dirichlet k≥1 2 boundary conditions, zero initial values, α = − 3 , d = 1, x ∈ M = [0, π], t ∈ [0, 1]. γ+1 1 Recall that the solution u ∈ H for every γ ≤ 6 . Since uxx is the Laplace operator, by Corollary 2.2.12 we have that the solution u is Holder¨ continuous in time variable

1 7 1 t, and the order of continuity is η = min{ 2 , 12 } − ε = 2 − ε for every ε ∈ (0, 1/2).

2.2.14 Remark. Theorem 2.2.7 implies that order of continuity of the solution u can

1 not exceed 2 . This property comes natural since the Brownian motion W is a con- 1 tinuous, nowhere differentiable process, Holder¨ continuous of order 2 − ε for every 1 0 < ε < 2 .

2.3 Equations with multiplicative noise

2.3.1 Existence and uniqueness

In this section we study the equation (2.3) in the case of multiplicative noise. For simplicity of writing we take Bku = u, but the results remain valid for Bku = g(u) where g is global Lipschitz continuous. Generally speaking, the methods used here

41 γ+m γ can be applied to general operators Bk : H → H , however we postpone these discussion for our future research and the results will be published somewhere else. Thus, keeping the same notations and boundary conditions as in the previous section, we consider the following equation

X∞ du(t) = Au(t)dt + αku(t)hk(x)dWk(t) , (2.19) k=1

α where t ∈ [0,T ], x ∈ M, a ∈ R, αk = k and α ∈ R−. First, we will establish the existence and uniqueness of the solution. For this, we make the following assumptions

m d (A4) Assume that α, γ ∈ R are such that − 2 < γ < − 2 − dα .

We will prove an auxiliary results.

2.3.1 Lemma. For every ε > 0 and γ0, γ1, γ2 ∈ R such that γ0 ∈ [γ2 − 2, γ2], γ1 ≤

γ2 − 2, there exists Mε ∈ R such that

2 2 2 kvkγ0 ≤ εkvkγ2 + Nεkvkγ1 , (2.20) for every v ∈ Hγ−2.

2 Proof. By interpolation theorem (see Section 1.1 for more details) kvks ≤ 2(1−θ ) C kvk2θ1 kvk 1 s ∈ [s , s + 2], θ = s−s1 C 1 s1+2 s1 , where 1 1 1 2 and 1 is a constant which depends only on M, s, s1. Consequently, by ε-inequality (see Section 1.1) with

1 1 ε := ε1, p := , q = we have θ1 1−θ1

− 1−θ1 2 2 θ1 2 kvks ≤ C1ε1kvks1+2 + C1ε1 kvks1 . (2.21)

42 2 Applying inequality (2.21) to kvks1 with s1 ∈ [s2, s2 + 2] we get

− 1−θ1 − 1−θ1 − 1−θ2 2 2 θ1 2 θ1 θ2 2 kvks ≤ C1ε1kvks1+2 + C1C2ε1 ε2kvks2+2 + C1C2ε1 ε2 kvks2 .

Note that s1 ≥ s2, hence kvks2+2 ≤ kvks1+2, and consequently by the above inequality we deduce

³ − 1−θ1 ´ − 1−θ1 − 1−θ2 2 θ1 2 θ1 θ2 2 kvks ≤ C1ε1 + C1C2ε1 ε2 kvks1+2 + C1C2ε1 ε2 kvks2 .

Similarly, by indication with s1 > s2 > . . . > sn, one has

1−θ ³ − 1−θ1 − n−1 ´ 2 θ1 θn−1 2 kvks ≤ C1ε1 + ... + C1 ··· Cn ε1 ··· εn−1 εn kvks1+2 1−θ − 1 − 1−θn θ1 θn 2 + C1 ··· Cn ε1 ··· ε2 kvksn . (2.22)

Note that ε1, . . . , εn are mutually independent, so we can take εs, s = 1, . . . , n 1−θ − 1−θ1 − n−1 θ1 θn−1 such that C1ε1 + ... + C1 ··· Cn ε1 ··· εn−1 εn ≤ ε. Also, we put s1 =

γ2 − 2, and continue with s2, . . . , sn until sn ≤ γ1. Finally denote by Nε := 1−θ − 1 − 1−θn θ1 θn C1 ··· Cn ε1 ··· ε2 . Lemma is proved.

γ 2.3.2 Theorem. If Assumption (A4) is satisﬁed and u0 ∈ L2(Ω, H ), then equation

γ+m γ (2.19) has a unique solution u ∈ L2((0,T ) × Ω; H ) ∩ L2(Ω; C((0,T ); H )) .

Proof. Similar to Section 2.2, the existence and uniqueness is established by Theorem 1.5.1. See also [79], Theorem 3.2, pag. 313. For the sake of completeness, let us verify conditions (iii) and (iv) of Theorem 1.5.1. In our case, (iii) becomes

X 2α 2 γ+m I1 = k khkvkγ < ∞ for all v ∈ V = H . (2.23) k≥1

43 s |s|+ε For every v1 ∈ H and every v2 ∈ H we have kv1v2ks ≤ kv1ks · kv2k|s|+ε for every ε > 0 (see for instance [78] Corollary 2.8.2). We continue (2.23)

X∞ 2α 2 2 I1 ≤ k khkkγ · kvk|γ|+ε . k=1

γ+m Since v ∈ H , kvk|γ|+ε is ﬁnite if γ + m > |γ| which holds true by initial assumption. Hence, X∞ 2α 2 I1 ≤ c k khkkγ . k=1

γ 1 d Recall that khkkγ = λk, and λk ∼ k (see notes on page 2), and we continue

X∞ 2α 2γ I1 ≤ c k k d = C. k=1

By assumption (A4) this series converges and C < ∞.

Finally let us check the condition (iv). Assume that v ∈ Hγ+m. Then,

X X 2α 2 2m 2α 2 I2 =2[Av, v] + αk ||hkv||γ = −2[Λ v, v] + k khkvkγ k≥1 k≥1 X 2 2α 2 = −2kvkγ+m + k khkvkγ . k≥1

Using the estimates for I1 established above, we continue

2 2 I2 ≤ −2kvkγ+m + Ckvk|γ|+ε1 .

44 Since |γ| ≤ γ+m, we can take ε1 > 0 such that γ+m−2 < |γ|+ε < γ+m and apply Lemma 2.3.1 to kvk2 with γ = |γ| + ε , γ = min{γ, γ + m − 2}, γ = γ + m. |γ|+ε1 0 1 1 2 From here we have

2 2 2 I2 ≤ −2kvkγ+m + Cεkvk|γ|+m + Nεkvkγ1 ,

and since γ1 ≤ γ, we continue

¡ ¢ 2 2 I2 ≤ − 2 + Cε kvk|γ|+m + Nεkvkγ .

Put δ = 2 − Cε, and take ε > 0 small enough, such that δ > 0. Thus condition (iv) from Theorem 1.5.1 is fulﬁlled. Theorem is proved.

d Note that γ < − 2 − dα is Assumption (A3), which guaranties the existence and uniqueness of the solution for equation with additive noise. It should be mentioned

1 that for d = 1, m = 1, the conditions on α and γ from Theorem 2.3.2 imply − 2 < 1 γ < −α − 2 . Consequently, α < 0. In other words, Theorem 2.3.2 ”almost” cover the results from Walsh [79] where α = 0 is considered. However, we will present here the corresponding result for α = 0 and A being the Laplacian. Although the result is known [79], our proof is different and simpler in our opinion.

d γ 2.3.3 Theorem. Suppose that −1 ≤ γ < − 2 and u0 ∈ L2(Ω, H ). Then the equation P du(t, x) = ∆u(t, x)dt+ hk(x)u(x, t)dWk(t) has a unique solution u ∈ L2((0,T )× k≥1 γ+1 γ Ω; H ) ∩ L2(Ω; C((0,T ); H )).

45 Proof. Similar to previous proof, we will apply Theorem 1.5.1. Condition (iii) gives

X X X ¯ ¯2 2 2γ ¯ ¯ I1 = khkvkγ = λj ¯hhkv, hji0¯ k≥1 j≥1 k≥1 X X 2γ 2 2γ 2 2 = λj ||vhj||0 ≤ λj ||hj||∞||v||0 . j≥1 j≥1

d−1 γ+1 2 Since γ > −1 and v ∈ H , we have ||v||0 < ∞. By Remark 2.1.1 ||hk||∞ ≤ cλk . Hence, we continue X 2 2γ+d−1 I1 ≤ c ||v||0 λj . j≥1

d Since γ < − 2 , the last series converges. Condition (iii) from Theorem 1.5.1 is veriﬁed. Now let us check coercivity condition (iv) from Theorem 1.5.1. Suppose that v ∈ √ Hγ+1. Then, by the same arguments as above and using that Λ = −∆, we get

X X 2 2 2 2γ+d−1 I2 = 2[∆v, v] + khkvkγ ≤ −2kvkγ+1 + ckvk0 λk . k≥1 k≥1

d The last series converges, since γ < − 2 . Hence

2 2 I2 ≤ −2kvkγ+1 + Ckvk0 .

Note that γ − 1 < −1 < γ < 0γ + 1. By Lemma 2.3.1 we have

2 2 2 ¡ ¢ 2 2 I2 ≤ −2kvkγ+1 + Cεkvkγ+1 + Nεkvkγ−1 ≤ − 2 + Cε kvkγ+1 + Nεkvkγ .

Take ε > 0 such that δ := 2 − Cε > 0, and conditions (iv) is veriﬁed. Theorem is proved.

46 2.3.4 Remark. Recall that for the equations with additive noise, the solution exists regardless of the space dimension and the order of the differential operator A (see Remark 2.2.2). The situation is different for the equations with multiplicative noise. It is known that the equation du = ∆udt + udW (t, x) has solution only for dimension

d d = 1. This also follows from Theorem 2.3.3. Indeed, the condition −1 ≤ γ ≤ − 2 implies d < 2. For initial equation (2.19) the restrictions are given by Assumption (A4), which also imply that the existence of the solution depends on both space dimension and order of the differential operator A. However, for any m, d ∈ N, we can choose α ¿ −1 such that Assumption (A4) is satisﬁed, and hence the solution of equation (2.19) exists.

We conclude this subsection by presenting the multiplicative counterpart of equations from Example 2.2.3.

2.3.5 Example. Assume that d = 1,M = [0, π] and let A be the one dimensional Laplace operator with zero boundary conditions. In the triple of Hilbert spaces (Hγ+1, Hγ, Hγ+1) let us consider the following SPDE

X − 2 du(t, x) = uxx(t, x)dt + k 3 sin(kx)u(t, x)dWk(t) , (2.24) k≥1

where x ∈ [0, π], t ∈ [0, 1] and Wk are independent standard Brownian motions. Note

2 1 1 that α = − 3 . By Theorem 2.3.2, for every − 2 < γ < 6 , there exists a unique strong solution of equation (2.7) in Hγ+1(0, π).

In Figure 2.2 we show realizations of the solution u of some SPDE’s with multiplicative noise (compare to Figure 2.1, where additive noise is discussed). Panel (a) corresponds to the equation du = uxxdt+hkudWk, investigated in Theorem 2.3.3. One realization of the solution of equation (2.24) is shown Panel (b). For both equations we

47 considered zero boundary conditions and the initial values u(0, x) = sin(3x) + 1, x ∈ [0, π]. To simulate the ﬁled u we used Crack-Nicolson ﬁnite difference scheme (see for instance [38], [57]). Similar to additive noise (see Figure 2.1 and Example 2.2.3), the

2 solution u looks smoother for α = − 3 than the solution of equation with corresponding noise parameter α = 0. Actually for a ﬁxed parameter α, the solutions of equations with additive and multiplicative noise, if exist, have the same order of continuity. We will show this in the next subsection.

Panel (a). α=0 Panel (b). α=−2/3

3 3

2 2

u(t,x) 1 u(t,x) 1

0 0 1 1 3 3 0.5 2 0.5 2 1 1 t 0 0 x t 0 0 x

Figure 2.2: Realization of approximated solution of stochastic heat equation with multiplicative noise

2.3.2 Regularity in space and time

Similar to additive case, to establish the regularity properties of the solution u, we will

0 prove an auxiliary result. In what follows we will assume that u0 ∈ L2(Ω, H (M)).

2.3.6 Lemma. Suppose that Assumptions (A2) and (A4) are satisﬁed and assume that γ > −m − dα. Then for any β ∈ (0, 1] and α ∈ R such that

m(1 − 2δ(β)) 1 α < − , (2.25) d 2

48 the solution u of equation (2.6) satisﬁes the following inequality

¯ ¯ ¯ ¯p E¯u(t, x) − u(t, y)¯ ≤ C|x − y|βp , (2.26) where x, y ∈ M, t ∈ [0,T ], p > 2.

Proof. Following the same arguments as in the previous section, the solution u of equation (2.19) can be represented as

t Z X Z Z u(t, x) = Gt(x, y)u0(y)dy+ Gt−s(x, y)hk(y)αku(s, y)dydWk(s) (2.27) M k≥1 0 M where x ∈ M, t ∈ [0,T ].

∞ Since hk ∈ C (M) and u0 ∈ L2(Ω × M) the ﬁrst term is a smooth function and (2.36) is satisﬁed. Hence we will study the regularity properties of the second term. Using (2.27), we get

Zt Z ¯ X ¯p p ¯ e ¯ E |u(t, x) − u(t, z)| ≤ E¯ αkG(t − s, x, y, z)hk(y)u(s, y)dydWk(s)¯ , k≥1 0 M

e where G(t − s, x, y, z) = Gt−s(x, y) − Gt−s(z, y). By BDG (1.5.3), we continue

t ¯ X Z ¯ Z p ¯ ¯ e E |u(t, x) − u(t, z)| ≤cp E¯ ¯ G(t − s, x, y, z)

k≥1 0 M (2.28) ¯2 ¯ p ¯ ¯ 2 · αkhk(y)u(s, y)dy¯ ds¯ .

49 Deﬁne ¯ ¯ ¯Z ¯2 X ¯ ³ ´ ¯ ¯ ¯ I1 = ¯ Gt−s(x, y) − Gt−s(z, y) αkhk(y)u(s, y)dy¯ ¯ ¯ k≥1 M X ¯ ¯2 ¯ e ¯ = ¯huG(t − s, x, y, z), αkhkiL2(M)¯ . k≥1

1 s s α Since λk ∼ k d , Λ (hk) = λkhk and αk = k , we continue

X ¯ ¯2 ¯ e dα ¯ I1 ≤ c ¯huG(t − s, x, y, z), Λ (hk)iL2(M)¯ . k≥1

Using the fact that Λ is a selfadjoint operator we get

X ¯ ¯2 ¯ dα e ¯ I1 ≤ c ¯hΛ (uG), hki0¯ . k≥1

Finally, by Parseval’s equality and property (1.2) of the operator Λ we have

° °2 ° dα e ° e 2 I1 ≤ c °Λ (uG)° = cku Gkdα , 0 and thus, from here and (2.28) one deduces

¯ ¯ p ¯Zt ¯ 2 ¯ ° °2 ¯ p ¯ ° e ° ¯ E |u(t, x) − u(t, z)| ≤ cE ¯ °G(t − s, x, ·, z) u(s, ·)° ds¯ . (2.29) ¯ dα ¯ 0

By the same arguments as in Theorem 2.3.2 from (2.28) we have

¯ ¯ p ¯Zt ¯ 2 ¯ ° °2 ° °2 ¯ p ¯ ° e ° ° ° ¯ E |u(t, x) − u(t, z)| ≤ cE ¯ °G(t − s, x, ·, z)° °u(s, ·)° ds¯ . (2.30) ¯ dα |dα|++ε ¯ 0

50 p p By Holder¨ inequality with p1 := p−2 , q1 := 2 , and (2.30) we get

E |u(t, x) − u(t, z)|p (2.31)   ¯ ¯ p · 1 ¯ ¯ p · 1 ¯Zt ¯ 2 q1  ¯Zt ¯ 2 p1 ¯ ¯  ¯ ¯ ¯ 2q1 ¯ ¯ e 2p1 ¯ ≤ c E ¯ ku(s, ·)k|dα|+εds¯ · ¯ kG(t − s, x, ·, z)kdα ds¯ . ¯ ¯  ¯ ¯ 0 0

By initial assumption γ + m > |dα| which implies that the ﬁrst factor in the last inequality is ﬁnite.

Now, we are going to estimate the last term on the right-hand side of (2.31), and show that it is ﬁnite too. Using the form (2.8) of the function G we get

Ã ! Zt Zt p1 X ¯ ¯2 e 2p1 ¯ e ¯ 2dα I2 := kG(t − s, x, ·, z)kdα ds = ¯hG(t − s, x, ·, z), hji0¯ λj ds 0 0 j≥1 Ã ! Zt p1 X ¯ X ¯2 ¯ lk(t−s) ¯ 2dα = ¯ (hk(x) − hk(z))e hhk, hji0¯ λj ds 0 j≥1 k≥1 Ã ! Zt p1 X ³ ´2 2lk(t−s) 2dα = hk(x) − hk(z) e λk ds . 0 k≥1

p p By Holder¨ inequality with p2 := p1 = p−2 and q2 = 2 we continue

t Ã ! p1 Z q2 X ³ ´2p2 X 2dαp2 2lk(t−s)p2 2lj (t−s)q2 I2 ≤ hk(x) − hk(z) λk e e ds , 0 k≥1 j≥1 and consequently

t ³ ´ Z ³ ´ p1 X 2p2 X q 2dαp2 lk(t−s)p2 lj (t−s)q2 2 I2 ≤ hk(x) − hk(z) λ e e ds . (2.32) k≥1 0 j≥1

51 Let us estimate the inner integral of the right-hand side of (2.32). Again by Holder¨

1 1 inequality with p3, q3 such that + = 1 we get p3 q3

t Z ³ ´ p1 X q lk(t−s)p2 lj (t−s)q2 2 I3 := e e ds (2.33) 0 j≥1 t t ³ Z ´ 1 ³ Z h i p1q3 ´ 1 p X q q ≤ elk(t−s)p2p3 ds 3 elj (t−s)q2 2 ds 3 .

0 0 j≥1

q2 p−2 If q3 := = then the last factor is ﬁnite for every p > 2 . Indeed, p1 p

t t Z p q Z h i 1 3 lj tq2 X q X X e − 1 lj (t−s)q2 2 lj (t−s)q2 I4 = e ds = e ds = . ljq2 0 j≥1 0 j≥1 j≥1

2m lj tq2 Recall that −lj ∼ j d and since lj < 0 we conclude that 1 − e < 1. Hence,

X 1 I4 ≤ c 2m , d j≥1 j

d and since m > 2 , we have that I4 is ﬁnite. Consequently, from (2.33) we get

Zt ³ ´ 1 h lktp2p3 i 1 p 1 − e p 1 lk(t−s)p2p3 3 3 − p I3 ≤ I4 e ds = I4 ≤ cp|lk| 3 , (2.34) |lk|p2p3 0

p−2 where p3 = p−4 .

52 From (2.31) using (2.32), (2.33) and (2.34) and Assumption (A2), for β > 0 one deduces

¯ ¯ ¯ ¯p E¯u(t, x) − u(t, z)¯ (2.35) ¯ ¯ p ¯X 1 ¯ 2p1 ¯ 2δ(β)p − p ¯ ≤c ¯ λ2dαp2 |x − y|2p2β l 2 l 3 ¯ p ¯ k k k ¯ k≥1 p−2 ¯ X 4mδ(β) ¯ βp ¯ p + 2αp − 2m p−4 ¯ 2 ≤cp|x − y| · ¯ k d p−2 p−2 d p−2 ¯ k≥1 ¯ ¯ p−2 2 ¯X 4mδ(β) ¯ βp ¯ p ( +2α− 2m p−4 )¯ ≤c |x − y| · ¯ k p−2 d d 4 ¯ , p ¯ ¯ k≥1

1 m(1−2δ(β)) and since, by initial assumption α < − 2 + d , we conclude that for sufﬁciently large p the last series converges. Lemma is proved.

Now, we will prove a similar result for the time variable t.

2.3.7 Lemma. Suppose that γ > −m − dα and let Assumptions (A1) and (A4) be satisﬁed. Then for every β ∈ (0, 1] such that

d(1 + 2α) β < 1 − 2ρ − (2.36) 2m the solution u of the equation (2.19) satisﬁes the following inequality

¯ ¯ ¯ ¯p βp E¯u(t1, x) − u(t2, x)¯ ≤ C|t1 − t2| 2 , (2.37)

where t1, t2 ∈ [0,T ], x ∈ M, p > 2.

53 Proof. Suppose that 0 ≤ t1 ≤ t2 ≤ T . The existence of the solution u is provided by the above assumptions and Theorem 2.3.2, and by representation (2.27) we have

Zt2 Z ¯ ¯p ¯ X ¯p ¯ ¯ ¯ ˆ ¯ J = E¯u(t1, x)−u(t2, x)¯ = E¯ G(t1, t2, s, x, y)αkhk(y)u(s, y)dydWk(s)¯ 0 k≥1 M

b where G(t1, t2, s, x, y) := Gt1−s(x, y)I(s≤t1) − Gt2−s(x, y). By similar evaluations as in Lemma 2.3.6 we deduce

¯ ¯ p · 1 ¯Zt2 ¯ 2 p1 ¯ ¯ ¯ b 2p1 ¯ J ≤ C ¯ kG(t1, t2, s, x, ·)kdα ds¯ , (2.38) ¯ ¯ 0

p where p1 = p−2 and C depends only on p and E(kukγ) (compare to (2.31)). Taking into account the deﬁnition (2.8) of the function G and properties of the norm k · kγ, we get

Zt2 Zt2 ¯ ¯ ³ X 2 ´p1 b 2p1 ¯ b ¯ 2dα J1 := kG(t1, t2, s, x, ·)kdα ds = ¯hG(t1, t2, s, x, ·), hj(·)i0¯ λj ds 0 0 j≥1 ¯ ¯2 p1 Zt2 ¯* + ¯ ¯ X ³ ´ ¯ ¯ lk(t1−s) lk(t2−s) ¯ 2dα = ¯ e I(s≤t1) − e hk(x)hk(·), hj(·) ¯ λj ds ¯ ¯ 0 k≥1 0

and since hj forms a CONS in L2(M), we continue

¯ ¯ Zt2 ¯ ³ ´ ¯p1 ¯X 2 ¯ J = ¯ elj (t1−s)I − elj (t2−s) |h (x)|2λ2dα¯ ds . 1 ¯ (s≤t1) j j ¯ 0 k≥1

54 p p By Assumption (A1) and Holder¨ inequality with p2 := p1 = p−2 and q2 = 2 we get

t2 Z ³ ´ ³ ´ 1 X 2p1 X q 2ρp1 2dαp1+ε lj (t1−s) lj (t2−s) −εq2 2 J1 ≤ c |lj| λj e I(s≤t1) − e λk ds . 0 j≥1 k≥1

P −εq2 For sufﬁciently large p and respectively large q2 the series λk converges, and k≥1 hence

Zt2 X ³ ´2p1 2ρp1 2dαp1+ε lj (t1−s) lj (t2−s) J1 ≤ c(p, ε) |lj| λj e I(s≤t1) − e ds (2.39) j≥1 0

Direct arithmetic evaluations yield to

Zt2 ³ ´2p1 lj (t1−s) lj (t2−s) p1β p1(β−1) e I(s≤t1) − e ds ≤ cp |t2 − t1| |lk| (β ∈ (0, 1)) . 0

From here and (2.39) we conclude

X p1β p1(2ρ+β−1) 2dαp1−ε J1 ≤ cp|t2 − t1| |lj| λj . (2.40) k≥1

1 2m(2ρp1+β−1) 2dαp −ε 2m d 1 Since |lk| = λk and λk ∼ k , the last series converges if d + d < p −1. Recall that p1 = , so lim p1 = 1. Hence, the series converges if β < p−2 p→∞ dα d 1 − 2ρ − m − 2m , which holds true by our initial assumption. Finally, by (2.38) and (2.40) we conclude ¯ ¯ p ¯ ¯ 2p1 βp J ≤ cp ¯J1¯ ≤ cp |t1 − t2| 2 .

Lemma is proved.

Now, we are ready to formulate the main result of this section.

55 2.3.8 Theorem. Assume that Assumption (A4) is satisﬁed and let u be the solution of the equation (2.19)

(i) if Assumption (A2) is fulﬁlled and r is the biggest integer such that there exists

β0 ∈ (0, 1] which satisﬁes the inequality 2dα + 2r + 4mδ(β0) − 2m < −d, then Λr(u) is Holder¨ continuous of order

n d dα 1 r o η := sup β | δ(β) < − − + − , β ∈ (0, 1] − ε , 4m 2m 2 2m

for every ε > 0, as long as η > 0;

m 2mρ 1 (ii) If Assumption (A1) is fulﬁlled and α < d − d − 2 , then the solution u is 1 1 d(1+2α) Holder¨ continuous in t of order less than ζ = min{ 2 , 2 − ρ − 4m }.

As long as the solutions of equations (2.2) and (2.19) exist, Lemma 2.3.6 and 2.3.7 coincide with Lemma 2.2.4 and 2.2.10 from additive noise. Under assumption that the solution of the equation 2.19 exists, which is guaranteed by Assumption (A4), the proof of Theorem 2.3.8 is the similar to the proof of Theorem 2.2.5 and 2.2.11.

2.3.9 Example. Let us consider the following SPDE

X − 2 du(t, x) = uxx(t, x)dt + k 3 sin(kx)u(t, x)dWk(t) , k≥1

where x ∈ [0, π], t ∈ [0, 1], u(t, 0) = u(t, π) = 0 and Wk are independent standard

1 1 Brownian motions. We showed in Example 2.3.5 that for every − 2 < γ < 6 , there exists a unique strong solution u in the triple of Hilbert spaces (Hγ+1, Hγ, Hγ+1).

β Recall that in this case ρ = 0, α = −2/3, δ(β) = 2 (See Example 2.1.2). By Theorem 2.3.8 we have that the solution u has one derivative in x and the ﬁrst derivative is

d dα 1 r Holder¨ continuous of order less than η := sup{β | δ(β) < − 4m − 2m + 2 − 2m , β ∈

56 β 1 1 1 1 1 (0, 1]} = sup{β | 2 < − 4 + 3 + 2 − 2 } = 6 . Respectively u is Holder¨ continuous in 1 7 1 time variable t of order less than min{ 2 , 12 } = 2 .

2.3.10 Remark. In both cases, additive and multiplicative noise, the solutions, if exists, has the same order of regularity. However, the existence and uniqueness of solution, in terms of the scale of Sobolev spaces, is more restrictive in multiplicative case.

2.4 Covariance functional and relation with some

known results

α As we already mentioned the novelty in our approach is the factor αk = k in the noise term. In this section we will show how our results are related to some known results, and also we will ﬁnd the analogous of parameter α in the known literature in terms of the asymptotic of kernel of covariance functional.

In recent paper Mytnik, Perkins, Sturm [60] investigated the equation of the form

1 p du(t, x) = ∆u(t, x)dt + u(t, x)dW (t, x) , (2.41) 2 where ∆ is Laplacian, x ∈ Rd, t ∈ [0,T ] for some ﬁnite T . W is space-time white

d noise on R+ ×R , namely W is deﬁned on a ﬁltrated probability space (Ω, F, {Ft}, P)

d and is a Gaussian martingale measure on R+ × R in sense of Walsh [79] or that introduced in Section 1.3. In [60] study the existence of mild solution of equation (2.41) and it regularity properties. The special feature of this equation is the nonlinear √ factor u. The solution u is the density for one-dimensional super-Brownian motion [63]. Of course this equation is different from those considered in our investigations.

57 Equation (2.41) is nonlinear on the whole space, in our notations M = Rd. Our equations are linear SPDE’s, but M is a bounded domain, and we allow operators of arbitrary order (not necessarily Laplacian). Here we will discuss the similarities in terms of noise W (t, x). Since we are working on different domains, some evaluations will be pure heuristic.

∞ d Denote by Cc (R+×R ) the space of compactly supported, inﬁnitely differentiable Rt R d functions on R+ × R , and put Wt(ϕ) = ϕ(s, x)W (dxds). If W (ϕ) = W∞(ϕ), 0 Rd W can be characterize by its covariance functional

h i Z∞ Z Z JΨ(ϕ, ψ) := E W (ϕ)W (ψ) = ϕ(s, x)Ψ(x, y)ψ(s, y)dxdyds , (2.42)

0 Rd Rd

∞ d 2d for ϕ, ψ ∈ Cc (R+ × R ). The function Ψ: R → R is called the correlation kernel of W . A general classes of martingale measures, spatially homogeneous noises, can be describe by (2.42) with Ψ(x, y) = Ψ(e x − y). We note that the standard space-time P e white noise hk(x)Wk(t) will correspond to the case Ψ = δ0, where δ is the Dirac k≥1 function. Suppose that the correlation is bounded by a Riesz kernel

£ ¤ |Ψ(x, y)| ≤ c |x − y|−ϑ + 1 (x, y ∈ Rd) (2.43) where ϑ > 0.

ϑ In [60] is proved that for every ξ ∈ (0, 1 − 2 ) the solution of equation (2.41) is d ξ uniformly Holder¨ continuous on compacts in [0, ∞) × R , with Holder¨ coefﬁcients 2 in time and ξ in space. Now, let us ﬁnd the correlation functional and kernel that corresponds to the equations in our research. For simplicity of writing we will consider the one-dimensional

58 P α stochastic heat equation du(t, x) = uxx(t, x)dt + k u(t, x) sin(kx)dWk(t), where k≥1 Rt Rπ P α x ∈ [0, π], t ∈ [0, 1]. In this case Wt(ϕ) = ϕ(s, x)hk(x)k dxdWk(ts). Con- k≥1 0 0 sequently,

∞ π h X Z Z α JΨ(ϕ, ψ) =E k ϕ(s, x)hk(x)dxdWk(s) k≥1 0 0 ∞ π X Z Z i α · n ψ(s, y)hn(y)dydWn(s) . n≥1 0 0

Taking into account that Wk are independent, we continue

∞ ∞ h X Z Z i 2α JΨ0 (ϕ, ψ) = E k hϕ(s, ·), hki0 dWk(s) hψ(s, ·), hki0 dWk(s) k≥1 0 0

Suppose for simplicity that the functions ϕ, ψ do not depend on time variable t. Then

X 2α JΨ0 (ϕ, ψ) = c k hϕ, hki0 hψ, hki0 = c hϕ, ψiα , k where c is a constant depending on the support of functions ϕ and ψ. Note that Rπ Rπ P 2α hϕ, ψiα = c k hk(x)hk(y)ϕ(x)ψ(y)dxdy. Hence, k≥1 0 0

X X 2α 2α Ψ0(x, y) = k hk(x)hk(y) = k sin(kx) sin(ky) . k≥1 k≥1

1 If α < − 2 , then the last series converges uniformly, and the covariance kernel is bounded. Hence, in this case there is no connection with the estimates (2.43). Suppose

1 that − 2 < α < 0. By Abel’s criterion about convergence of series, one can show that P k2α sin(kx) sin(ky) converges if x 6= y and diverges if x = y. So, the kernel Ψ k≥1 seems to have a similar asymptotic behavior to that from (2.43). To ﬁnd ϑ we note

59 √ 2α that JΨ0 (ϕ, ψ) = hΛ ϕ, ψi0 , where Λ = −∆ (see Chapter 1 for more details).

1 The counter part for the whole line will be the operator Λ = (I − ∆) 2 which can be represented as an integral operator with kernel the Bessel potential. Namely,

Z Z µZ ¶ ∞ 2 3 |x−y| −2α −α− 2 − 4t −t hΛ ϕ, ψi0 = t e dt ϕ(x)ψ(y)dydx . 0 R R

2 R ∞ 3 |x| e −α− 2 − 4t −t Hence, the correlation kernel Ψ0(x) = 0 t e dt. Note that Fourier trans- e 2 α −ϑ −1+ϑ form F [Ψ0] = (1 + |x| ) , and F [|x| ] = |x| (see for instance [77], Chapter 5). From here we conclude that −2α = 1−ϑ. Hence we can say that the parameter ϑ from

(2.43) corresponds to 1 − 2α where α is parameter involved in our research. Note that

1 α ∈ (− 2 , 0) is equivalent to ϑ ∈ (0, 1). Finally we will show that regularity properties coincide. By Theorem 2.3.8 the solution of corresponding equation is Holder¨ continu-

1 ϑ ous in space variable x with the order of continuity less than ξ = 2 − α = 1 − 2 , and 1 α 1 ϑ in time variable with order of continuity less than η = 4 − 2 = 2 − 4 , which is exactly the statement of Theorem 1.8 from [60].

2.5 Examples

2.5.1 Example. Stochastic heat equation on [0, π] with zero boundary conditions.

First we consider the classical heat equation du = uxxdt + f(u)dW (t, x), as a benchmark for testing our results. We recall that this problem was investigated by Walsh

[79]. In this case, the solution exists, is unique and Holder¨ continuous in x of order

1 1 2 − ε, and in t of order 4 − ε for every ε > 0 (see Section 2.1 for more details). Let A be the one dimensional Laplace operator on C∞(0, π). In our notations d = 1, Au = uxx, m = 1,M = [0, π]. It is well-known that the functions

60 2 hk(x) = π sin(kx), k ∈ N, are the eigenfunctions of the operator A, with corre- 2 sponding eigenvalues lk = −k . Moreover, Assumptions (A1) and (A2) are satisﬁed,

β and ρ = 0, δ(β) = 2 (see Section 2.1, Example 2.7, page 27 for the proof).

2.5.1.a. Additive noise Let us consider the heat equation with additive noise term

  P∞ du(t, x) = u (t, x)dt + kαh (x)dW (t) t ∈ (0,T ), 0 < x < π;  xx k k  k=1 u(t, 0) = u(t, π) = 0 t ∈ (0,T ); (2.44)    u(x, 0) = 0, x ∈ [0, π], where α is a real parameter.

1 Existence and Uniqueness. By Theorem 2.2.1, if γ < −α − 2 then γ+1 the equation (2.44) has a unique solution u ∈ L2((0,T ) × Ω; H (0, π)) ∩

γ γ L2(Ω; C((0,T ); H (0, π))). Note that regardless of α the solution exists in some H .

β β Regularity in x. Since δ(β) = 2m = 2 , we will apply Theorem 2.2.7. Suppose 1 1 that γ < −α − 2 and denote α < 2 . By Remark 2.2.6 we can take the operator Λ ∂ from Theorem 2.2.7 being the partial derivative operator ∂x (see also Example 2.2.8). 1 Let η = 2 − α and denote by bac the integer part of real number a (largest integer smaller than a). If η∈ / N, then the solution u has r := bηc derivatives in x and the r-th derivative is Holder¨ continuous of order ζ := η − r − ε, for every ε > 0 as long as ζ > 0. If η ∈ N, then u has r := bηc − 1 derivatives in x and the r-th derivative is Holder¨ continuous of order ζ for every ζ ∈ (0, 1).

1 Regularity in t. Suppose that α < 2 . Then, by Theorem 2.2.11, the solution u of equation (2.44) is Holder¨ continuous in time variable t with order of continuity

1 1 α η = min{ 2 , 4 − 2 } − ε, for every ε > 0 as long as η > 0.

61 1 If α = 0, we resume the results by Walsh [79]: the solution exists if γ < − 2 , and 1 1 4 −ε, 2 −ε u(t, x) ∈ Ct,x . If α decreases, the solution becomes smoother, both in time and space variable.

1 However, the maximum smoothness in time is achieved for α = − 2 . In this case, 1 the solution u has an order of continuity less than 2 and remains the same for every 1 1 α ≤ − 2 . For α = − 2 , the solution is Lipschitz continuous in x. Generally speaking, 1 (r) 1 2 −α−r there are r = b 2 − αc derivatives in x, and ux ∈ C .

2.5.1.b. Multiplicative noise.

Now, let us consider the equation

X∞ α du(t, x) = uxx(t, x)dt + k u(t, x)hk(x)dWk(t) t ∈ (0,T ), 0 < x < π , k=1 with same initial values and boundary conditions as in (2.44). By Theorem 2.3.2 this

γ+1 1 1 equation has a solution u ∈ H if − 2 < γ < − 2 − α. If the solution exists, then the regularity in time and space are the same as for additive noise, discussed above.

2 2.5.2 Example. Stochastic heat equation on torus Tπ.

2 We consider the operator A being the Laplace operator ∆ on two dimensional torus Tπ.

2 Following our previous notations d = 2, m = 1,M = Tπ. The eigenfunctions of this √ 1 ijx1 inx2 operator are hj,n(x1, x2) = π e e where x = (x1, x2) ∈ M, j, n ∈ Z, i = −1, with corresponding eigenvalues l = j2 + m2. Obviously Assumption (A1) is satisﬁed,

1 ||hk||∞ ≤ π , so ρ = 0. Let us check Assumption (A2) and ﬁnd the function δ. Denote

i(jx1+nx2) i(jy1+ny2) x = (x1, x2), y = (y1, y2). Then R := |hj,k(x)−hj,k(y)| = |e −e | = ¯ ¯ 1 ¯ ¯ 2 ¯2 − 2(cos(z1) cos(z2) + sin(z1) sin(z2))¯ , where z1 = jx1 + nx2, z2 = jy1 + ny2. ¯ ¯ 1 ¯ ¯ √ 2 ¯ z1−z2 ¯ ¯ z1−z2 ¯ β Consequently, R = 2¯1 − cos( 2 )¯ = 2¯ sin( 4 )¯ ≤ c|z1 − z2| , for every

62 ¡ β 2 2 β 2 β ∈ (0, 1]. Hence, R ≤ c |j(x1 − y1) + n(x2 − y2)| ≤ c (j + n ) 2 (x1 − y1) + β ¢ β β 2 2 2 β (x2 − y2) = c l |x − y| . Thus, δ(β) = 2 , β ∈ (0, 1]. Let us consider the corresponding equation with additive noise, i.e. du = ∆udt +

P 1 β hkdWk(t). Note that δ(β) = 2m = 2 , and hence Remark 2.2.6 holds true. By k≥1 Theorem 2.2.1 there exists a unique solution u ∈ Hγ+1 if γ < −2α − 1. Theorem 2.2.5 implies: if −2α∈ / N, then the solution u has r := b−2αc derivatives in x and the r-th derivative Dr(u) is Holder¨ continuous of order η = −2α − b2αc; if −2α ∈ N, then there exists Dr−1(u) with order of continuity τ, for every η ∈ (0, 1).

Regularity in time is given by Theorem 2.2.11. The solution u(·, x) is Holder¨

1 continuous of order η = min{ 2 , −α}, for every α ≤ 0.

2.5.3 Example. SPDE of order 2m. Let consider the equation of the form

∂2mu(t, x) X du(t, x) = dt + kαh (x)dW (t) x ∈ [0, π], t ∈ [0,T ], ∂x2m k k k≥1

2 where hk(x) = π sin(kx) and m ∈ N. We assume zero boundary conditions, and zero (or any smooth) initial values. In fact this equation is similar to the equation (2.44) from Example 1, but A = ∆m. Since the Laplace operator ∆ is a selfadjoint operator, we conclude that the functions hk are the eigenfunctions of the operator A,

2m with corresponding eigenvalues lk = −k . Thus, Assumptions (A1) and (A2) are

β satisﬁed, and ρ = 0, δ(β) = 2m . 1 Existence. Applying Theorem 2.2.1, we have: if γ < −α− 2 then there exists a unique γ+1 γ solution u ∈ L2((0,T ) × Ω; H (0, π)) ∩ L2(Ω; C((0,T ); H (0, π)))

1 m 1 Regularity in x. By Theorem 2.2.5, if α < − 2 + 2 and m − α − 2 ∈/ N, then the 1 solution u has r = bm−α− 2 c derivatives, and the last derivative is Holder¨ continuous

63 1 1 of every order less than m − α − 2 − r. Similar for m − α − 2 ∈ N. Regularity in t. If α < m−1, then the solution u is Holder¨ continuous in time variable,

1 1 1−2α with order of continuity less than min{ 2 , 2 − 4m }.

To show the impact of the parameter α, we summarize these results from the examples considered in this Chapter in the Table 2.1:

Parameters Existence Reg. in x Reg. in t

1 1 Walsh [79] d = 1, m = 1, α = 0 - 2 4 multiplicative

1 7 1 Ex. 2.1.2, 2.2.3 d = 1, m = 1 γ < 6 6 2 2 2.2.8, additive α = − 3

1 1 7 1 Ex. 2.3.5, 2.3.9 d = 1, m = 1 − 2 < γ < 6 6 2 2 multiplicative α = − 3

1 1 1 1 α Ex. 2.5.1.a d = 1, m = 1 γ < −α − 2 2 − α min{ 2 , 4 − 2 } additive

1 1 1 1 1 α Ex. 2.5.1.b d = 1, m = 1 − 2 < γ < −α − 2 2 − α min{ 2 , 4 − 2 } multiplicative

1 α Ex. 2.5.2 d = 2, m = 1 γ < −2α − 1 −2α min{ 2 , − 2 } additive

1 1 1 1−2α Ex. 2.5.3 d = 1, m > 1 γ < −2α − 1 m − α − 2 min{ 2 , 2 − 4m } additive

Table 2.1: Existence and Regularity of Solution. Summary

64 Chapter 3

Parameter estimation problems for some classes of SPDE’s

3.1 Introduction

Stochastic Partial Differential Equations (SPDE’s) are of big interest in last decades, primarily as a modelling of various phenomena from fluid mechanics [74], oceanogra- phy [64], temperature anomalies [11], [65], finance [7], [12], [14], and other domains. Every model is describe by a class of SPDE’s, with some specific properties, which are related to the corresponding phenomena, and usually this is a broad class of equations with some unspecified/unknown coefficients also called parameters. In practice we observe the process, which we believe is described by some equation, and we want to find all unknown parameters of this model by using the past data such that the equation fits and predicts, in some sense, as much as possible the real data. In mathematical terms, we want to solve a parameter estimation problem. The parameter estimation problem, generally speaking, is a particular case of inverse problem when the solution is known (observed) and an inference about the coefficients of the equation is made. Because of general setup of this problem, there are numerous methods in modern literature dedicated to this topic, covering different classes of equations. We want to emphasize that the solution of a Stochastic PDE is a random variable, and the parameter estimation problem needs to be solved by

65 some statistical methods. In contrast, the similar problem for the models described by a deterministic PDE is usually reduced to some inverse spectral problems (see for instance [68] and the references therein). In this chapter we will investigate a parameter estimation problem for evolution equations of the following form   du(t) = (A0 + θA1)u(t)dt + (Mu(t) + g(t))dW (t) , (3.1)  u(0) = u0 ,

where t ∈ [0,T ], A0, A1, M are linear operators, g a real-valued function, W is a cylindrical Brownian motion, and θ is the unknown parameter belonging to an open subset of the real line. Parameter estimation for the ordinary stochastic differential equations (SDE’s), ﬁnite dimensional counterpart of equation (3.1), has been studied by many authors under general assumptions on coefﬁcients, with very subtle results (see for instance

[25], [36], [46], [47] and references therein). Some infinite dimensional models with long time and small noise asymptotics, using system theory, have been studied by Aihara [2], Bagchi and Borkar [3], Ibragimov and Khasminskii [26]-[28] etc. Using projection-based methods, the parameter problem for SPDE (3.1) driven by additive noise was studied for the first time by Khasminskii, Rozovskii and Huebner [22] [23]. The key point is that in reality, one may not assume to observe the whole field u(t, x), so a finite dimensional approximation of the solution is used to obtain consistent estimates of unknown parameter θ. The idea is to choose a basis in the suitable Hilbert space, such that the finite dimensional projection of the solution u, i.e. Fourier coefficients w.r.t. this basis, will lead to a system of independent processes. Then, using first

66 N Fourier coefﬁcients, an explicit and simple form of the Maximum Likelihood Esti- ˆ ˆ mator (MLE) θN for θ is obtained. As one may expect, θN converges (in some sense) to the true parameter as N increases. The equation (3.1) driven by additive noise, and with A0, A1 being elliptic differential selfadjoint operators on a bounded domain in Rd, with a common system of eigenfunctions, was studied in [21], [22], [23], [66]. It ˆ was shown that if ord(A1) ≥ (ord(A0) − d)/2, then MLE θN is strongly consistent as N → ∞. We note here, that operators A0, A1 commute, since they have the same system of eigenfunctions, and hence the random ﬁeld is diagonalizable. Consequently, direct application of Galerkin type of approximations will lead to a system of ODE’s mentioned above.

The equation with none-commuting operators was considered by Lototsky and Ro- zovskii [52],[53]. Even though, applying the same technics, formally computed estimator is not an MLE, it was shown that under some technical assumptions the estimator is consistent and asymptotically normal. We would like to mention that for ordinary SDE consistent estimates are obtained if either observation time T goes to inﬁnity or the noise amplitude tends to zero (for some examples see [36]). Although, for SPDE’s within the above framework consistency is archived on any time interval [0,T ] as the dimension of the projection increases. This is due to the fact that generally speaking the probability measures generated by the solution uθ of an ordinary SDE are absolute continues for different θ, while for an SPDE with additive space-time white noise, under some conditions on the order of differential operators, the measures are singular (see for instance [56], [39]). However, the measures generated by the projection are absolute continuous for different values of the parameter, which implies the existence of MLE computed from likelihood ratio (Radon-Nikodym derivative).

67 While the SPDE’s driven by additive noise have been studied by many authors, the corresponding equations with multiplicative noise have not been covered at all (as far as we know), in particular the equation with multiplicative space-time white noise. In this work, we will address this question for several classes of equations, including space-time white noise. Despite similarity of these equations, it turns out that the measures generated by the projection of the solution with multiplicative noise, generally speaking, are not anymore absolute continuous and the likelihood ratio does not exist. For example, this is the case when W is space-time white noise, and we will discuss this in Section 3.2. Regarding this, we consider equations with some noises similar to cylindrical Brownian motion, but for which we are able to find consistent estimates of the unknown parameter, based on finite projection of the solution on some basis. For simplicity, we suppose that operators A0, A1, M commute, however the results can be extended to non-commutative ones. In Section 3.3 we set up the problem, and study existence, uniqueness and finite dimensional approximation of the solution. The estimate and its properties are discussed in Section 3.4. We find sufficient conditions on operators A0, A1, M that implies consistency, asymptotic efficiency, and normality of estimates. We conclude this chapter with some application of abstract results to general stochastic parabolic equations, presented in Section 3.5. Some numerical results are presented at the end of Section 3.5, where we show some practical applications of theoretical results, and provide some counter-examples related to sufficiency of the conditions imposed in on the original equation.

68 3.2 Special case: space-time white noise

In this section we will analyze the simplest SPDE driven by space-time white noise, and give some insides about the problems arising in parameter estimation for equations with multiplicative noise. We will follow the notations and deﬁnitions from Chapter 1.

Let us consider the stochastic one dimensional heat eqation

du(t, x) = θuxx(t, x)dt + u(t, x)dW (t, x) , (3.2)

where t ∈ [0,T ], x ∈ [0, 1], u(0) = u0 ∈ L2(0, 1) and periodic boundary conditions.

ikπx The elliptic operator Au = −uxx, with eigenfunctions hk(x) = e (k ∈ Z),

γ γ generates the scale of Sobolev spaces H = H (0, 1) (γ ∈ R). The system hk, k ∈ Z,

2 forms a CONS in L2(0, 1) and Ahk = −(λk) hk, with λk = kπ. By Theorem 1.5.1,

γ+1 if γ < −1/2 then equation (3.2) has a unique solution u ∈ L2((0,T ) × Ω; H ) ∩

γ L2(Ω; C((0,T ); H )) (see also [79]). It is assumed that the observed field u satisfies (3.2) for some unknown but fixed value θ0 of parameter θ. We suppose that the first N Fourier coefficients of u, relative to {hk}k∈Z, are known, and we would like to find a consistent estimate of θ0 as N → ∞. Denote by {uk}k∈Z the Fourier coefficients of the solution u w.r.t.

N s N hk, k ∈ Z, and by Π the projection of H , s ∈ R, on Span{hk}k=−N . Hence, N N N P u (t, x) := Π u = hk(x)uk(t). We consider the following ﬁnite-dimensional k=−N approximations of (3.2)

N N N N N dv = θ(vxx) dt + Π (v dW ) , (3.3) which to some extends corresponds to Galerkin-type approximation.

69 It should be mentioned that since operator ΠN and multiplication operator vN → vN dW N do not commute, vN from (3.3) does not coincide with projection uN of original solution u. In other words, generally speaking ΠN vN dW N 6= uN dW N . However, N N P N we will show that approximation sequence v = vk hk converges to the solution k=−N u, as N → ∞, which means that from practical point of view, we can substitute in our estimates the values of vn by Fourier coefﬁcients un, n = −N,...,N.

N Taking into account that operators Π and ∆v = vxx have a common system of eigenvalues, hence commute, we rewrite (3.3) as follows

XN XN 2 hk(x) dvk(t) = − θ λkhk(x)vk(t)dt k=−N k=−N (3.4) ³ XN XN ´ N +Π hk(x)vk hj(x)dWj(t) . k=−N j=−N

Observe that hkhm = hk+m, k, m ∈ Z, which implies that (3.4) is equivalent to a system of 2N + 1 stochastic ODEs,

³ ´ 2 dv−N = −θλ−N v−N dt + v−N dW0 + v−N+1dW−1 + ··· + v0dW−N ³ ´ 2 dv−N+1 = −θλ−N+1v−N+1dt + v−N dW1 + v−N+1dW0 + . . . v1dW−N

...... ³ ´ (3.5) 2 dv0 = −θλ0v0dt + v−N dWN + v−N+1dWN−1 + ··· + vN−1dW−N

...... ³ ´ 2 dvN = −θλN vN dt + v0dWN + v1dWN−1 + ··· + vN dW0 or, in matrix form

θ,N θ,N θ,N N dv = θA(v )dt + b(v )dWt , (3.6)

70 N 2 2 0 where A(v ) = −[λ−N v−N , . . . , λN vN ] , and    v0 v−1 . . . v−N+1 v−N 0 ... 0       v1 v0 ...... v−N+1 v−N ... 0      ......  N   b(v ) =   ,   vN vN−1 . . . v1 v0 v−1 . . . v−N      ......    0 0 ... 0 vN vN−1 . . . v0

with dim(b) = (2N + 1) × (2N + 1) and the element b00 is boldfaced. For conveniens of writing, we index the coordinates in R2N+1 from −N to N. As we mentioned above, it is natural to have some convergence of the approximation vN to the real solution u as dimension of approximation increases. To show this we prove an auxiliary result.

N 3.2.1 Lemma. If MN = (mkj)k,j=−N is a (2N + 1) × (2N + 1) real matrix such that

2 mjj = −|j| , j = −N, . . . , N, mkj = 1 if |k − j| ≤ N, and µN is the maximum eigenvalues of the matrix MN , then there exist a constant C such that µN < C for all N ∈ N.

Proof. It sufﬁces to show that hMN x, xi ≤ C for some real constant C, and for every x ∈ R2N+1 with kxk = 1, where h· , ·i and k · k are usual (Euclidian) inner product and norm in R2N+1.

Let us denote αn = mnn, n = 0, 1,...,N and f(x) = hMN x, xi. We rewrite f(x) as follows:

71 2 f(x) = αN x−N + 2x−N (x−N+1 + ··· + x0)

2 + αN−1x−N+1 + 2x−N+1(x−N+2 + ··· + x0 + x1)

......

2 + α1x−1 + 2x−1(x0 + ··· + xN−1)

2 + α0x0 + 2x0(x1 + ··· + xN ) (3.7)

2 + α1x1 + 2x1(x2 + ··· + xN )

......

2 + αN−1xN−1 + 2xN−1xN

2 + αN xN .

So, we want to maximize f under constrain kxk = 1. Using the method of Lagrange multipliers we ﬁnd that x should be a non-trivial solution of the system of linear equations Lλx = 0, where   αN + λ 1 ... 1 0 0 ... 0       0 αN−1 + λ . . . 1 1 0 ... 0       ......    Lλ =   ,    0 0 . . . α0 + λ 1 1 ... 1       ......    0 0 ... 0 0 0 . . . αN + λ

and λ is the multiplier parameter. Obviously, Lλx = 0 has a non-zero solution, iff

αj +λ = 0 for some j. Suppose αN +λ = 0, since otherwise, we get the same system, but of smaller dimension. Let a = max{|x−N |, |xN |}. Note that since kxk = 1 we

72 a have a ≤ 1. By direct arithmetic evaluations, one can deduce that |x±j| ≤ n+j , j = −N + 1,...,N + 1. Hence, by (3.7), we get f(x) ≤ C.

Now we can prove the following result

N 3.2.2 Theorem. For every γ < −1/2, the sequence {v }n≥1 converges weakly in Hγ+1 to the weak solution of the equation (3.9).

Proof. Similar to deterministic case, we will use weak sequential compactness of Hilbert spaces (see for instance [9]). The main part of the proof is to establish that the

θ,N 2 1 sequence kv k γ is uniformly bounded. Since γ < − , it sufﬁces to show L2(Ω×[0,T ];H ) 2 RT N θ,N 2 P 2 that E kv k0dt is uniformly bounded in N. By Ito’sˆ formula for f(x) = xk 0 k=−N we deduce dY = AY dt ,

h i0 2 2 where Y = E(v−N ),..., E(vN ) and

  1 − 2θλ2 ... 1 ... 0  −N       ......    A =  2  .  1 ... 1 − 2θλ0 ... 1       ......    2 0 ... 1 ... 1 − 2θλN

By Lemma 3.2.1 the maximal eigenvalue of the matrix A is uniformly (in N) bounded

θ,N from above. Consequently, by Theorem 4.2 from [16] the norm kv k0 is also uniformly bounded in N. Hence, there exists a subsequence of vθ,N that converges weakly

γ+1 γ+1 in L2(Ω×[0,T ]; H ). Let us denote this limit by v˜. By (3.3), for every ϕ ∈ H we Rt Rt θ,N N θ,N N θ,N N have (v , ϕ) = (u0 , ϕ) + θ [vxx , ϕ] + (Π (v dW (s), ϕ)). Taking N → ∞ 0 0

73 Rt Rt we deduce (˜v, ϕ) = (u0, ϕ) + θ [˜vxx, ϕ] + (˜vdW (s), ϕ)). Since the last equality 0 0 holds for every ϕ ∈ Hγ+1, we get v˜ = u. Theorem is proved.

Let us denote by Pθ,N the measure in X = C([0,T ]; ΠN (H0)), generated by the solution vθ,N (s), 0 ≤ s ≤ T of equation (3.6), i.e. Pθ,N (a) = P(vθ,N ∈ a) for all a ∈ BT , where BT is the Borel σ-algebra on X .

By Theorem 1.2.2 (see also Theorem 7.6.4 in [47]) the measures Pθ,N and Pθ0,N are mutually absolutely continuous if the algebraic system of equations

b(x)y = (θ − θ0)A(x) (3.8) has a bounded solution (in y ∈ R2N+1) for each t ∈ [0,T ], x ∈ R2N+1. It turns out that this algebraic equation with apparently nice structure does not have an obvious solution. Direct evaluations show that for N = 1, 2, 3, the system (3.8) does not have solution for some values of x, which means that we can not apply the general results about absolute continuity of measures. On the other hand, we can not conclude that these measures are singular for different parameters θ. Some examples and generalization of Theorem 7.6.4 from [47] are discussed in [36]. In particular, it is shown that the corresponding measures can be singular as well as absolute continuous, if the system (3.8) does not have a finite solution. Unfortunately, these results can not be applied to our system of SDE’s, nor the proofs can be adapted (by best of our knowledge). We want to mention, that simple structure of matrix b is due to the unique property of eigenfunctions, hkhm = hk+m. In general, b will be a full matrix, without any specific properties, which makes impossible to find a solution of equation (3.8). ˆ We estimated naively the MLE θN of θ, by maximizing the likelihood ratio

θ,N dP (vN,θ0 ) dPθ0,N , and used it for some numerical simulations. The results suggest us that

74 the measures look like to be singular, which consequently implies that some different approach needs to be considered for the equations with multiplicative space-time white noise.

3.3 General case. Approximation of the solution

In this section we will set up the parameter estimation problem, discuss the existence and uniqueness of the solution of the corresponding equations, and consider the ﬁnite dimensional approximation of the solution.

Let H be a separable Hilbert space with the inner product (·, ·)0, and the corresponding norm k · k0. Suppose Λ is a linear operator on H such that kΛuk0 ≥ ckuk0 for every u from the domain of Λ. Then the operators Λγ, γ ∈ R, are well deﬁned, and generate the spaces Hγ as follows: the domain of Λγ coincides with Hγ for every

γ γ γ ≥ 0; H is the completion of H with respect to the norm k · kγ := kΛ · k0,

γ for all γ < 0 (see Chapter 1 here, or [41]). The set {H }γ∈R is called a Hilbert scale. The spaces (Hγ+m, Hγ, Hγm) form a normal triple with canonical bilinear form

γ−m γ+m γ−m γ+m γ r r−γ hu1, u2i = (Λ u1, Λ u2)0, where u1 ∈ H , u2 ∈ H ; Λ (H ) = H for every γ, r ∈ R.

γ+m γ γ−m Let (Ω, F, {Ft}, P) be a stochastic basis. In the triple (H , H , H ), for some γ ∈ R, m ∈ N, we consider the following SPDE

 ¡ ¢ n  P du(t) = (A0 + θA1)u(t) + f(t) dt + (Mju(t))dWj(t) j=1 (3.9)   u(0) = u0 ,

75 where A0, A1, Ms are linear operators, f is a given vector-valued function, Wj are independent standard Brownian motions, n ∈ N or n = ∞, t ∈ [0,T ], and θ is a scalar parameter subject to estimation belonging to an open set Θ ⊂ R. The ﬁrst question we will address is the existence and uniqueness of the solution of equation (3.9). In connection with this, throughout in what follows, we will assume that:

(H1) there exists m > 0 such that the linear operator Aθ := A0 + θA1, is bounded from Hγ+m to Hγ−m for every γ ∈ R;

γ+m γ (H2) Mj (j = 1, . . . , n) are linear bounded operators acting from H into H ;

γ (H3) u0 ∈ H ;

γ−m (H4) f ∈ L2(Ω × (0,T ); H );

∞ (H5) The operators Λ, A1, A0, Mj have the same system of eigenfunctions {hk}k=1,

∞ γ the system {hk}k=1 is a complete system in H, and hk ∈ ∩γ≥0H .

Condition (H5) is equivalent to say that operators A1, A0, M commute, which consequently implies that eigenfunctions and eigenvalues of these operators does not depend on the parameter θ. We want to note that this is the case in many applications. Usually the operator −Aθ is a differential, selfadjoint, positive deﬁned operator

γ of order 2m deﬁned on the scale of Sobolev spaces {H (M)}γ∈R, where M is a d-

∞ 1 dimensional compact orientable C manifold. In this case Λ = (−Aθ) 2m and with

γj Mj = Λ the assumption (H5) holds true. We will discuss this example in details later on (see Section 3.5).

Most of the results remain true in more general situations, in particular when operators A1, A0, M do not commute, and will be published elsewhere.

76 1 0 j Denote by lk, lk, λk and µk the eigenvalues of the operators A1, A0, Λ and Mj respectively, which correspond to the eigenfunction hk for every k ∈ N. Assume that:

r j (H6) lk < 0, µk 6= 0 for r = 0, 1, j = 1, . . . , n, k ∈ N, and there exist ε > 0,M ∈ R n −2m P s 2 −2m 1 0 such that λk (µk) ≤ ε − 2λk (θlk + lk) ≤ M for every θ ∈ Θ, k ∈ N. s=1

r Suppose lk, are enumerated in order of magnitude

r r r l1 ≥ l2 · · · ≥ lk ≥ ..., where r = 0, 1.

3.3.1 Theorem. Under Assumptions (H1)-(H6), there exists a unique weak solution u

γ γ+m of equation (3.9), the solution belongs to L2(Ω,C((0,T ); H ))∩L2(Ω×[0,T ]; H ) and   µ ¶ ZT 2  2  E sup ku(t)kγ + E ku(t)kγ+mdt 0

Proof. This theorem follows from Theorem 1.5.1, a general result from the theory of SPDEs (see also [70], Theorem 3.1.4). Conditions (i)-(iii) and (v) of Theorem 1.5.1 come naturally from the structure of equation (3.9) and follow immediately from Assumptions (H1)-(H4).

77 Condition (iv) represents strong parabolicity or coercivity assumption. It sufﬁces

0 1 to check this condition for the vectors hk. Put lk(θ) = lk + θlk, then using above assumptions we have

Xn Xn 2 j 2 2 2[Aθhk, hk] + kMjhkkγ = 2lk(θ)[hk, hk] + (µk) khkkγ j=1 j=1 (3.11) Xn ³ Xn ´ 2 j 2 2 j 2 −2m 2 ≤ 2lk(θ)khkkγ + (µk) khkkγ = 2lk(θ) + (µk) λk khkkγ+m . j=1 j=1

n −2m P j 2 Denote by δ = − max{λk (2lk(θ) + (µk) )}. By Assumption (H6) we have that k j=1 δ > 0, thus the condition (v) is veriﬁed, and the Theorem is proved.

To make an inference about parameter θ we will project the solution u on some ﬁnite-dimensional subspace of Hγ+m, and essentially reduce the SPDE to a system of SODE’s. As we mentioned, a natural approach is to project on the subspace generated by the eigenfunctions hk, and use the Fourier coefﬁcients of the solution u with respect

∞ to {hk}k=1. Suppose that the observed field u satisfies (3.9) for some unknown but fixed value

θ0 of the parameter θ. Denote by {uk}k∈N the Fourier coefﬁcients of the solution u

N γ N w.r.t. {hk}k∈N, and by Π the projection of H , γ ∈ R, on Span{hk}k=1. Hence, N N N P u (t, x) := Π u = uk(t)hk(x). k=1 We consider the following ﬁnite-dimensional approximation of (3.9)

N N du (t, x) = Π ((A0 + θA1)u(t, x) + f(t)) dt Xn (3.12) N + Π (Mju(t, x))dWj(t) . j=1

78 γ+m 3.3.2 Remark. By Theorem 3.3.1, u ∈ L2(Ω × [0,T ]; H ), which implies that

ZT N 2 lim E ||u − u||γ+mdt = 0 . N→∞ 0

Denote by fk the Fourier coefficients of the function f. From (3.12) and by assumption (H6), we deduce that Fourier coefficient uk satisfies the following equation

³ ´ Xn 0 1 j duk(t) = (lk + θlk)uk(t) + fk(t) dt + µkuk(t)dWj(t) , (3.13) j=1

with uk(0) = (u0, hk)γ, and k ∈ N.

3.4 The estimate and its properties

In this section we will ﬁnd a Maximum Likelihood Estimate (MLE) of the parameter θ by using the approximation (3.12) of the solution to the original equation.

Let us fix a number θ0 ∈ Θ, that represents the true value of the parameter θ subject to estimation. In what follows we will suppose that the first k Fourier coefficients of the

ﬁeld u are observed, and we will estimate θ0 from them. For simplicity of writing we will drop the index θ0 where there is no room for ambiguity. Namely, we will write uk

γ+m instead of uθ0,k. Let BT be the Borel σ-algebra on the space C([0,T ], H ). Denote

θ by Pk the measure on BT generated by the solution uθ,k of equation (3.13). Note that n P j 1 the equation (1.5) from Theorem 1.2.2 in this case becomes µky = (θ − θ0)lk, j=1

79 θ θ0 which has a ﬁnite solution for every k ∈ N, and hence the measures Pk and Pk are mutually absolute continuous, with Radon-Nikodym derivative

T θ n Z 1 0 1 dPk lk(θ − θ0) duk lklk(θ − θ0) T (uk) = exp − θ0 Pn Pn dPk j 2 uk j 2 0 (µk) (µk) j=1 j=1 (3.14) ZT l1(θ − θ ) f (s)ds (l1)2(θ2 − θ2) T o − k 0 k − k 0 . Pn Pn j 2 uk(s) j 2 (µk) 0 2 (µk) j=1 j=1

ˆN The maximum likelihood estimate θ of θ0 is obtained by maximizing the Radon- Nykodim derivative (likelihood ratio) (3.14) with respect to the parameter of interest θ ∈ Θ. Direct computations yield

T T Z 0 Z ˆ 1 duk lk 1 fk(s) θk = 1 − 1 − 1 ds . (3.15) lkT uk lk lkT uk(s) 0 0

Pn duk 1 j 2 By Ito’sˆ Lemma, we ﬁnd that d ln(uk) = − (µ ) dt, and hence from (3.15) uk 2 k j=1 we get ZT Xn 0 ˆ 1 uk(T ) 1 j 2 lk 1 fk(s) θk = 1 ln + 1 (µk) − 1 − 1 ds . (3.16) lkT uk(0) 2lk lk lkT uk(s) j=1 0 Also, from (3.15) and taking into account (3.13) we have the following representation for the estimate 1 Xn θˆ = θ + µj W (T ) , (3.17) k 0 l1T k j k j=1 ˆ which will play the key role in our future investigations. Note that by (3.15) θk is an unbiased estimate of θ0.

80 In addition to Assumptions (H1)-(H6), we will assume the following:

Pn j 2 (µk) j=1 (H7) There exists the limit lim 1 2 = µ < ∞. k→∞ (lk)

We want to mention that assumption (H7) does not follow from (H6) neither converse

m 0 2m 1 2 m is true. For example, take lk = −λk , lk = −λk , µk = λk , n = 1. Then (H6) is

0 1 2m 2m−1 satisﬁed, while (H7) does not hold. If we put lk = 0, lk = −λk , µk = λk , n = 1, m > 2, then (H7) holds, and µ = 0, but (H6) is not satisﬁed. Both assumptions are related to the order of operators A0, A1, Mj, when all of them are some pseudo- differential operators. The relationship between orders of operators will be discussed in details in the next section.

Denote by L the set of all real-valued, nonnegative functions w deﬁned on R, which are symmetric, w(0) = 0 and monotone for x > 0. These functions are called loss functions.

3.4.1 Theorem. Under Assumptions (H1)-(H6) the following holds true:

ˆ (i) lim θk = θ0, P-a.s., for every k ∈ N (consistency in time); T →∞

Suppose in addition that (H7) is satisﬁed. Then

ˆ (ii) if µ = 0, then lim θk = θ0, P-a.s., for every T > 0 (consistency in k). More- k→∞ ˆ over, in this case, MLE θk is asymptotic efﬁcient, that is for any loss function

w ∈ L    √   1   |lk| T ˆ  lim Ew s (θk − θ0) = Ew(ξ) , k→∞ Pn  j 2  (µk) j=1

were ξ a Gaussian random variable with zero mean and unit variance;

81 ˆ (iii) if µ 6= 0, then lim θk = θ0 + ξ(µ), P-a.s., for every T > 0, where ξ(µ) ∼ k→∞ µ N (0, T ).

Proof. Property (i) follows from (3.17) and the fact that lim W (t) = 0. t→∞ t j µk If µ = 0 then lim 1 = 0 for every j = 1, . . . , n. Thus, from equality (3.17) we k→∞ lk ˆ conclude that lim θ = θ0. Also, by (3.17) we note that k→∞ √ 1 |lk| T ˆ ξk := s (θk − θ0) ∼ ξ , (3.18) Pn j 2 (µk) j=1

where ξ ∼ N (0, 1). Hence E(w(ξk)) = E(w(ξ)), that consequently implies lim E(w(ξk)) = E(w(ξ)), and (ii) is proven. k→∞ Property (iii) follows from (3.18) and Assumption (H7).

ˆ 3.4.2 Remark. By (3.18) one gets at once that θk is also asymptotic efﬁcient in T . We live this property apart since the main goal is to get a consistent estimate on every small time interval [0,T ].

ˆ By the above theorem, we can use every individual θk as an estimate for θ0, and by increasing T to get a good estimate of the true parameter, regardless of Assump- tion (H7). However in practice we would like to ﬁnd θ0 with any precision in a small interval of time. As we can see from (ii), Theorem 3.4.1, this is possible by increas-

θ ing the number of Fourier coefﬁcients. This is due to the fact that the measures Pu, that correspond to the solution of the original equation (3.9), are singular for different values of the parameter θ. Comparing (ii) and (iii), we also note that for µ = 0 we get consistent estimates,s while for µ 6= 0, the limit estimate is biased. The speed Pn j 2 (µk) j=1 √ ˆ of convergence is of order 1 . Although for µ = 0, θk is a consistent esti- |lk| T mate of the true values as k → ∞, we want to mention that in practice the estimates

82 will be evaluated by formula (3.16), that involves the values of uk and fk. Note that

γ uk and fk being Fourier coefﬁcients of some functions in H will have at least polynomial decay to zero. This implies that for large k we will not be able to observe and compute uk, fk reliably. However, in many applications f ≡ 0, and in this case to apply (3.16) we need to know only the values of log uk(T ) . In this situation, uk(0) equation (3.13) becomes the well-known geometrical Brownian motion, with solution n n P j 2 P j uk(t) = uk(0) exp{(θ0lk(θ) − (µk) /2)t + µkWj(t)}. Obviously, j=1 j=1

u (T ) ³ Xn ´ Xn log k = θ l (θ) − (µj )2/2 T + µj W (T ) (3.19) u (0) 0 k k k j k j=1 j=1

uk(T ) which implies that log has the same magnitude as lk for a reasonable time interval uk(0) [0,T ] , and thus the estimator given by (3.16) is computable for sufﬁciently large k. As will see later on, it sufﬁces to take k being around ten, to get an approximation of order 10−4. On the other hand, a reasonable and natural approach is to consider a weighted ˆ average of the estimates θk. For example, if n = 1, and the sign of µk varies with k, ˆ using (3.17) we observe that the estimates θk will approach the true value by oscillating around it (depending on the sign of W and µk). Hence, some proper chosen weighted average of θk will converge faster to the true value. For the rest of this section we will discuss how to choose the weights such that to have at least the same rate of ˆ convergence of new estimates comparative to θk.

∞ Suppose {βk}1 ⊂ R+, and put

N P ˆ βkθk θˆ = k=1 . (3.20) (N) PN βk k=1

83 ˆ ˆ ˆ Since E(θk) = θ0, by (3.20) obviously we have that E(θ(N)) = θ0, i.e. θ(N) is an unbiased estimator of θ0. From (3.16), we get

N ³ T ´ N n N P R P β P j P β l0 βk uk(T ) fk(s) k (µ )2 k k l1 ln u (0) − u (s) ds l1 k l1 k k k k=1 k j=1 k θˆ = k=1 0 + − k=1 , (3.21) (N) PN PN PN T βk 2 βk βk k=1 k=1 k=1 which will be the main formula used in computational tasks. Similarly, by (3.17) we deduce

N n P P j βk ηkWj(T ) k=1 j=1 θˆ = θ + , (3.22) (N) 0 PN T βk k=1

j j 1 where ηk = µk/lk for j = 1, . . . , n and k = 1,...,N. It turns out, to preserve consistency and normality, it sufﬁces to make the following assumption:

PN (H8) βk ≥ 0, k ∈ N and lim βk = ∞. N→∞ k=1 3.4.3 Theorem. If Assumptions (H1)-(H8) are fulﬁlled then:

ˆ (i) lim θ(N) = θ0, P-a.e., for every N ∈ N (consistency in time) and MLE is T →∞ asymptotic efﬁcient as T → ∞;

ˆ (ii) if µ = 0, then lim θ(N) = θ0, P-a.e., for every T > 0 (consistency in N), N→∞ ˆ and MLE θ(N) is asymptotic efﬁcient;

ˆ (iii) if µ 6= 0, then lim θ(N) = θ0 + ξ(µ), P-a.s., for every T > 0, where ξ(µ) ∼ N→∞ µ N (0, T ).

84 Proof. Similar to Theorem 3.3.1, (i) follows from (3.22) and the fact that lim W (t) = t→∞ t 0. By Stolz-Cesaro Theorem (discrete version of L’Hospital’s rule, see for instance [33], p.35 or [67], p.17), under Assumption (H8) we have

N P j βkηk k=1 j lim = lim ηk , (3.23) N→∞ PN k→∞ βk k=1 for every j = 1, . . . , n.

j If µ = 0, then lim ηk = 0 for every j = 1, . . . , n, hence by (3.23) and (3.24) it k→∞ ˆ follows that lim θ(N) = θ0. N→∞ Using (3.22) one can show that

PN √ βk T k=1 ˆ s (θ(N) − θ0) ∼ N (0, 1) (3.24) PN Pn j 2 βk (ηk) k=1 j=1 and similar to Theorem 3.3.1 the rest of the proof follows.

In practice, ideally we want to choose the weights βk such that the rate of conver- ˆ gence of θ(N) → θ0 to be at least the same (or faster) than the rate of convergence of ˆ θk → θ0 and also, as we mentioned before, βk should offset somehow the fast decay of the Fourier coefﬁcients uk. It should be mentioned that Assumption (H8) does not

−2 guarantee the same rate of convergence. For example, if ηk = k and βk = k, then PN βkηk k=1 δ ε lim PN = ∞. However, if βk = k with δ > 1, or βk = exp(k ) with ε > 0, then k→∞ ηN βk k=1 the rate of convergence is preserved. These heuristic discourses lead to the conclusion that βk should grow fast enough. On the other hand, we have to keep in mind that

85 βk ﬁnally should be computable on PC. Thus, a reasonable choose for βk would be

δ βk = exp(k ) with δ =∈ {0.5, 1, 2, 3} and depends on how many Fourier coefﬁcients we are going to use in our computations. Finally, we will prove the following technical lemma about rate of convergence.

∞ 3.4.4 Lemma. Suppose {an}n=1 is an increasing sequence of positive numbers such ¯ ¯ (2) ¯ ∆ (an) ¯ (k) that lim an = ∞, ¯ (1) ¯ < M, where ∆ is the k-th ﬁnite difference. Then, n→∞ ∆ (an)

N P −1 an exp(an) n=1 ∼ a−1 . PN N exp(an) n=1

Proof. We will apply several times the Stolz-Cesaro Theorem to the sequence N N P −1 P aN an bn/ bn, where bn := exp(an), and show that this sequence has a ﬁnite n=1 n=1 limit.

PN PN NP−1 a a−1b a bn − a bn N n n N an N−1 an A = lim n=1 = lim n=1 n=1 N N→∞ P N→∞ bN bn n=1 N−1 P −1 (aN − aN−1) an bn = 1 + lim n=1 N→∞ bN

NP−1 NP−2 (a − a ) bn − (a − a ) bn N N−1 an N−1 N−2 an = 1 + lim n=1 n=1 N→∞ b − b N N−1 (3.25) NP−2 bn (aN − 2aN−1 + aN−2) (a − a )b an = 1 + lim N N−1 N−1 + lim n=1 . N→∞ aN−1(bN − bN−1) N→∞ bN − bN−1

86 We claim that both limits in the last expression are zero. Recall that bn = exp(an). Thus,

aN − aN−1 bN−1 aN − aN−1 1 A1 := lim · = lim · , N→∞ aN−1 bN − bN−1 N→∞ aN−1 exp(aN − aN−1) − 1

and since the sequence an is increasing, by Taylor series expansion for exponential functions, from the last equality we get

1 A1 ≤ lim = 0 , N→∞ aN−1

hence A1 = 0. Similarly,

NP−2 NP−2 (a − 2a + a ) bn ∆(2)(a ) bn N N−1 N−2 an N an n=1 n=1 A2 = lim = lim (1) . N→∞ bN − bN−1 N→∞ ∆ (aN ) exp(aN ) ¯ ¯ (2) ¯ ∆ (aN ) ¯ By the initial assumption, ¯ (1) ¯ < M, that implies ∆ (aN )

NP−2 bn an n=1 A2 ≤ M lim , N→∞ exp(aN ) and again by the Stolz-Cesaro Theorem, we continue

exp(aN−2) 1 A2 ≤ M lim ≤ M lim = 0 . N→∞ aN−2(exp(aN ) − exp(aN−1) N→∞ aN−2

Finally by (3.25) we have A = 1, and lemma is proved.

87 PN −1 an bn n=1 −1 −1 Similarly to Lemma 3.4.4, one can prove that PN ∼ aN , if an has polyno- bn n=1 mial growth, and bn grows faster than an.

ˆ 3.4.5 Remark. Observe that θk involves only the k-th Fourier coefﬁcient, while evalu- ˆ ˆ ating θ(N) we use all uk with k = 1,...,N. One should expect that using θ(N) we will get more precise estimates since more information is used. However, we get actually ˆ the same precision in our estimates by applying either simple estimates θk or weighted ˆ average estimates θ(N), since the rates of convergence are the same. Generally speak-

uk ing, the ﬁltration Ft (see Section 1.2) generated by the solution uk is smaller then the

ﬁltration Ft generated by the Brownian Motions Wj(t), and could be different for different k’s. In our case, each uk is a geometrical Brownian motion driven by the same processes Wj(t)(j = 1, . . . , n), hence are given by the same function with different

uk parameters but with same noise. This impplies that the ﬁltrations Ft are the same for all k ∈ N. In other words, each Fourier coefﬁcient uk contains the same amount of ˆ ˆ information about θ0, that explains why the precisions of θk and θ(N) are the same.

3.5 Applications to stochastic parabolic

differential equations

In this section, we will present some applications of Theorems 3.4.1 and 3.4.3 to equation (3.9) with Aθ, Mj being some pseudo-differential operators. We will begin with an abstract case, and continue with some particular examples, including some numerical results for the initial parameter estimation problem. If otherwise it is not mentioned all notations from Section 3.3 will be preserved.

88 Let M be a d-dimensional compact, orientable, C∞ manifold with a smooth

s positive measure dx, {H (M)}s∈R be the scale of Sobolev spaces on M (see for instance[1], [76]), and let (Ω, F, {Ft}, P) be a stochastic basis.

Suppose that Aθ = A0 + θA1, is a differential, elliptic operator of order 2m, formally self-adjoint, and the operator Aθ is lower semi-bounded for every θ ∈ Θ (uniformly in θ). The latter means that there exists a positive number ε > 0 such that

γ+m −Aθ > εI, for every θ ∈ Θ, where I is the identity operator in H .

It is well-known that the operator Aθ can be extended to a closed, self-adjoint operator on L2(M), the spectrum of this operator is discrete, consisting of eigenvalues of ﬁnite multiplicity (for more details see the discussion in Chapter 1 and references mentioned therein). Denote by {hk,θ} the eigenfunctions of Aθ, and suppose that

∞ {hk,θ}k=1 forms an orthonormal system in L2(M). By Hilbert-Schmidt theory, this

γ ∞ system is complete in L2(M), and hk,θ ∈ H (M) ∩ C (M) for every γ ∈ R. Gen- erally speaking, the eigenfunctions hk,θ depend on θ, however, for sake of simplicity we will rule this out, and write hk. From general theory of elliptic operators, under

γ+m γ−m above assumptions, the operators Aθ : H → H are linear bounded operators for every γ ∈ R. Thus Assumption (H1) from Section 3.3 is satisﬁed.

1 We set Λθ := (εI − Aθ) 2m . For every θ ∈ Θ, the operator Λθ generates the Hilbert

s s scale {Hθ(M)}s∈R. The spaces {Hθ(M)}s∈R are equivalent for all θ ∈ Θ (see for

s1 s2 instance [76]), namely Hθ (M) = Hθ (M) as sets of functions, and topologies are equivalent. Thus without loss of generality, we will consider the operator Λ = Λθ for some ﬁxed value of the parameter θ. Now we are under the setup of general results from section 3.3.

pj Assume that Mj = bjΛ , where pj ≤ m, bj ∈ R and j = 1, . . . , n. Then the operators Mj are pseudo-differential operators of order less than m, and hence are

89 linear bounded operators acting from Hγ+m in Hγ for every γ ∈ R. Thus Assumption (H2) is satisﬁed.

We assume that initial condition u0 satisﬁes (H3) and free term f satisﬁes (H4).

Finally, we suppose that the operators A0 and A1 have the same system of eigenfunc-

tions {hk}k∈N, so (H5) is satisﬁed too.

Let mi := ord(Ai), i = 0, 1, where by ord(A) we denote the order of differen-

tial operator A. From the above assumptions obviously 2m = max{m0, m1}, and

ord(Mj) = pj ≤ m. From spectral theory of elliptic operators it follows that the asymptotics of the

j eigenvalues lk,i, µk and λk are given by

p mi j j 1 d d d lk,i ∼ k , µk ∼ k , λk ∼ k , (3.26)

1 0 lkθ+lk where i = 0, 1 and j = 1, . . . , n. From here we get that lim 2m is finite for every k→∞ λ j pj θ ∈ Θ. Obviously, µk = bjλk 6= 0 for every j = 1 . . . , n, k ∈ N. Hence, to fulfill condition (H6) it is sufficient to assume the following:

Pn 1 0 2 2(pj −m) 2(lkτ+lk) (H6.1) There exists ε > 0 such that bj λk ≤ ε − λ2m , where k ∈ N and j=1 k τ = sup(Θ).

In the nutshell, Assumption (H6) requires

max{ord(Mj)} ≤ 2 max{ord(A1), ord(A0)} , j

plus the coercivity condition on the coefﬁcients. Also, (H6.1) can be replaced with the following, more restrictive but easier to check, condition

Pn 1 0 2 2(lkτ+lk) (H6.2) bj < − λ2m , where τ = sup(Θ), k ∈ N. j=1 k

90 Under above assumptions, the equation (3.9) has a unique solution in the space

γ γ+m L2(Ω,C((0,T ); H )) ∩ L2(Ω × [0,T ]; H ), and estimate (3.10) holds true. Finally, we will discuss Assumption (H7). For the differential operators considered in this section, this condition becomes

(H7.1) p := max{p1, . . . , pn} ≤ m1.

The equivalence of (H7) and (H7.1) follows directly from (3.26). Since all assumptions (H1)-(H7) are satisﬁed, we conclude

3.5.1 Proposition. Under above assumptions on differential operators Aθ, Mj, the ˆ estimates θk are unbiased and consistent in both time and dimension of the projection (number of Fourier coefﬁcients), namely Theorem 3.4.1 holds true. If in addition we assume (H8), then Theorem 3.4.3 about weighted average follows. Moreover, if p <

m1, then µ = 0, and hence the MLE’s from Theorems 3.4.1 and 3.4.3 are asymptotic efﬁcient.

3.6 Examples and some numerical results

In this section we will proceed to some particular examples, each followed by numerical results.

3.6.1 Example. Stochastic heat equation. Let us consider the following SPDE    du(t, x) = θ∆u(t, x)dt + u(t, x)dW (t), t ∈ (0,T ], x ∈ (0, 1) ,  u(0, x) = u (x) x ∈ (0, 1) , (3.27)  0    u(t, 0) = u(t, 1) = 0 t ∈ [0,T ] ,

91 where ∆u = uxx,W is a standard Brownian motion, and u0 ∈ L2(0,T ). For simplicity of writing we consider T = 1, but all results remain true for an arbitrary time √ T . Using previous notations we put Λ := −∆ acting in L2(0, 1) with zero boundary √ conditions. It is well-known that hk(x) = 2 sin(kπx), k ∈ N, are the eigenfunctions of the operator Λ, with corresponding eigenvalues λk = πk, the operator Λ satis- ﬁes all conditions mentioned in Section 3.3 and generates the scale of Sobolev spaces

s {H (0, 1)}s∈R (compare to Example 2.2.3 - 2.3.5). The operators corresponding to the original evolution equation (3.9) are as follows:

1 2 A1 = ∆ and lk = −(kπ) , k ∈ N;

A0 = 0, f ≡ 0;

M = I, n = 1 and µk = 1, k ∈ N. Equation (3.26) makes sense only for θ > 0. Indeed, Assumption (H6.1) leads to the following inequality 1 ≤ ε + 2θ(kπ)2 for every k ∈ N, which obviously is satisﬁed only for θ > 0. Thus, the existence and uniqueness of the solution is guaranteed in Hs(0, 1), s ∈ R if θ > 0.

The MLE’s (see (3.16) and (3.21)) in this case have the following form µ ¶ ˆ 1 uk(1) 1 θk = − 2 log + , (kπ) uk(0) 2 PN ³ ´ βk uk(1) 1 2 log + (3.28) (kπ) uk(0) 2 θˆ = − k=1 , (N) PN βk k=1

where uk are the Fourier coefﬁcients of the observed solution of equation (3.26) w.r.t.

CONS {hk}k∈N, and βk’s satisfy (H8).

92 Since p = 0 < m1 = 2, we have that Assumptions (H7.1) is satisﬁed, and hence Theorems 3.4.1 and 3.4.3 hold true. Moreover, µ = 0, which consequently implies that estimates (3.28) are consistent, and asymptotically efﬁcient and unbiased.

We simulated the Fourier coefﬁcients uk, using Matlab7, taking the true parameter ˆ θ0 = 1. Applying formulas (3.28), we evaluated the estimates θk (simple estimates), ˆ θ(N) with βk = k (weighted polynomial estimates), and βk = exp(k) (weighted exponential estimates).

In Figure 3.1 we present the graph for all three estimators and the true value of the parameter, as functions of the number of Fourier coefﬁcients.

1.01

θ (true value) 1.008 0 θ (simple estimates) k θ (weighted polynomial est), β =k (N) k 1.006 θ (weighted exponential est), β =exp(k) (N) k

1.004 estimated θ

1.002

0 5 10 15 20 25 30 Fourier Coefficient

Figure 3.1: Estimated Parameter. Example 3.6.1

93 ˆ ˆ We note that θk and θ(N) with exponential weights converge to the true parameter, basically with the same speed of convergence. However, as mentioned before, the polynomial weighted estimate converge slower. ˆ In the Table 3.1 we present the corresponding errors, i.e. |θ0 − θ|, and as we can see, for k = 10 we are within four digits precision, and taking into account the ﬁrst 20 Fourier coefﬁcients we get an error of magnitude less then 10−5. We see that indeed

k \ Error Simple Polynomial Exponential 1 0.00961 0.0096 0.0096 5 0.00038 0.0008 0.0006 10 0.00009 0.0002 0.0001 15 4.3−5 0.0001 4.7 × 10−5 20 2.4 × 10−5 6.7 × 10−5 2.5 × 10−5 30 1.6 × 10−5 3.0 × 10−5 1.1 × 10−5

Table 3.1: Errors of estimated parameter the rate of convergence is equal to k−2.

The numerical results are similar if the noise is driven by several independent Brownian motions, i.e. for n > 1.

3.6.2 Example. Now we will consider an example where the unknown parameter is not a factor of the leading differential operator. In our notations this means ord(A0) > ord(A1). Suppose that u satisﬁes the following SPDE

Xn 1 du = (u − θu)dt + (1 − ∆)qj udW (t) , x ∈ [0, 1], t ∈ [0, 1], (3.29) xx j2 j j=1 with periodic boundary conditions, u(t, 0) = u(t, 1), with initial condition u(0, x) =

∂ u0(x) ∈ L2(0, 1) and qj < 0. Denote by ∆ the Laplace operator ∂x2 on L2([0, 1]) √ with periodic boundary conditions. Let us consider Λ = 1 − ∆, then Λ generates

94 s the Hilbert scale {H }s∈R. The orthonormal system of eigenfunctions of this operator √ is given by: h0(x) = 1, h2k(x) = 2 cos(2πkx), h2k−1 = sin(2πkx) for k ∈ N.

2 qj Obviously, the operators A1 = −I, A0 = ∆, Mj = 1/j (1 − ∆) and Λ have a common system of eigenfunctions. Also, note that the corresponding eigenvalues p 1 0 0 2 0 2 j satisfy the relations lk = −1, l2k = l2k−1 = −(2kπ) , λk = 1 − (lk) and µk =

2qj 2 λk /j . We see that conditions (H1)-(H5) and (H7.1) are satisﬁed, and now, the objective

2 is to show that (H6.2) holds. With bj = 1/j , (H6.2) becomes

Xn 1 −τ + l0 < −2 k . j2 1 − l0 j=1 k

∞ P 1 π2 Without sake of completeness we take τ = 1, and since j2 = 6 < 2, we con- j=1 clude that (H6.2) is veriﬁed. Hence, by Proposition 3.5.1 (see also Theorems 3.4.1 and representation (3.16)) there exists a unique solution of equation (3.29), and the estimators

n u (0) 1 X (1 − (2kπ)2)qj θˆ = log k − + (2kπ)2 (k ∈ N) k u (1) 2 j2 k j=1 are consistent and asymptotic efﬁcient. Similarly, one may apply the Theorem 3.4.3 and get the results about the weighted average MLE’s.

Now we will present some numerical results for this example. We take qj = −1 and n = 100. Figure 3.2 exhibits the estimated parameters evaluated by three different methods, using the ﬁrst 30 Fourier coefﬁcients. The solid horizontal line corresponds to the true value of the parameter θ0 = 1. As we mentioned before, Simple Estimates ˆ ˆ θk and Weighted Exponential Estimates θ(N) for k, N ≥ 10 are almost the same, while

95 ˆ polynomial weighted average converge slower to the true value. The error |θ−θ0|, k = 25,..., 45 are presented in Figure 3.3.

1.001 θ (true value) 0 θ (simple estimates) 1.0008 k θ (weighted polynomial est), β =k (N) k θ (weighted exponential est), β =exp(k) 1.0006 (N) k

estimated 1.0004 θ

1.0002

0 5 10 15 20 25 30 Fourier Coefficient

Figure 3.2: Estimated Parameter. Example 3.6.2.

3.6.3 Example. We will conclude this section with some hypothetical examples (by indicating only the eigenvalues but not the operators themselves), that show different rates of converges of the estimates, and also, some counter-examples related to conditions (H6) and (H7). In what follows we assume (H1)-(H5) and (H8) hold true.

3.6.3.a. Let A0 = 0, A1 = ∆ with zero boundary conditions. Similar to Example 1, √ 1 2 k lk = −(kπ) . Suppose that M is the operator with eigenvalues µk = (−1) k. It is clear that (H6) and (H7) are satisﬁed and µ = 0. Thus, we can apply Theorems 3.4.1 and 3.4.3. The results are presented in Figure 3.4. We observe that in this case, the weighted exponential estimates perform better than simple or polynomial averaged ones. This is due to the fact that the sign of µk alternates.

96 −6 10

−7 10 Simple Estimates Weighted Polynomial Est Weighted Exponential Est

−8

Error (log scale) 10

−9 10 30 35 40 45 50 Fourier Coefficient

ˆ Figure 3.3: Errors |θ − θ0| vs number of Fourier coefﬁcients. Example 3.6.2

0 2 1 3.6.3.b. Let lk = −(kπ) , lk = 1, µk = kπ, then Assumption (H6) is satisﬁed. Since ³ ´ 2 2 µk (kπ) lim 1 = lim 1 = ∞, we have that (H7) is violated. This means that there k→∞ lk k→∞ exists a unique solution, but we can not apply the above results about MLE’s. The numerical results, formally applied to this problem are shown on Figure 3.5. As we can see, all estimators diverge.

0 1 4 3.5 3.6.3.c. Now we take lk = 1, lk = −(kπ) , µk = (kπ) , then one may check that Assumption (H7) is satisﬁed, while (H6) does not hold. Note that (H6) is related only to existence and uniqueness and not to the convergence of the estimators. It is known that coercivity condition (H6) is a sufﬁcient and in some sense a necessary condition for the existence and uniqueness of the solution for parabolic SPDE. It turns out that numerical results also show that if only (H6) is violated, the estimators do not converge to the true values of the parameter θ0 = 0. The simulation are presented in Figure 3.6. We see that the error is more than .1 for k=100.

97 1.015 θ (true value) 0 θ (simple estimates) k 1.01 θ (weighted polynomial est), β =k (N) k θ (weighted exponential est), β =exp(k) (N) k 1.005

1 estimated θ 0.995

0.99

0.985 5 10 15 20 25 Fourier Coefficient

Figure 3.4: Estimators vs number of Fourier coefﬁcients. Example 3.6.3.a

50 θ (true value) 0 θ (simple estimates) 45 k θ (weighted polynomial est), β =k (N) k θ (weighted exponential est), β =exp(k) 40 (N) k

25 estimated θ 20

0 5 10 15 20 25 Fourier Coefficient

Figure 3.5: Estimators vs number of Fourier coefﬁcients. Example 3.6.3.b

98 1.1

0.9

0.8 estimated θ θ (true value) 0 0.7 θ (simple estimates) k θ (weighted polynomial est), β =k (N) k 0.6 θ (weighted exponential est), β =exp(k) (N) k

0 10 20 30 40 50 60 70 80 Fourier Coefficient

Figure 3.6: Estimators vs number of Fourier coefﬁcients. Example 3.6.3.c

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