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

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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 by 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 Copyright 2007 Igor Cialenco Dedication 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 ^ 3.3 Errors jθ ¡ θ0j 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 equa- tions (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 multi- plicative. 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 multi- plicative 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, C1 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 ¤ = (L) 2m is well-definite, elliptic of order s 1 and generates the scale fH (M)gs2R 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 0 1 1 X Z 2 ® 2 kukHn(M) = @ jD uj dxA ; j®j·nM 1 ® where D denotes the partial derivative of order ® = (®1; : : : ; ®d) (see [1],[9] for s s more details). We will write fH gs2R instead of fH (M)gs2R 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 fhkgk2N in the space L2(M; dx). Then for every f 2 L2(M; dx) we have the representation X1 X1 f(x) = hf; hki0hk(x) = fkhk(x) ; (x 2 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 2 N M are Fourier coefficients of f with respect to fhkgk2N. If ¹k > 0 is the eigenvalue of the operator L corresponding to the eigenvec- 1 2m s tor hk and ¸k := ¹k , then for all s ¸ 0 we have H = ff 2 L2(M; dx): 1 P 2s 2 s ¸k jfkj < 1g and for s < 0; H is the closure of L2(M; dx) in the norm k=1 ¡ 1 ¢ P 2s 2 1=2 s jjfjjs = ¸k jfkj . Hence, every element of the space H ; s 2 R, can be k=1 1 P 2s 2 identified with a sequence ffkgk2N such that ¸k jfkj < 1. Note that the space k=1 Hs, equipped with the inner product X1 2s s hf; gis = ¸k fkgk (f; g 2 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 = ¤ (L2(M; dx)); s 2 R. Note that s s s s ¤ hk = ¸khk and jjhkjjs = jj¤ hkjj = ¸k (k 2 N; s 2 R): (1.2) 2 ¡s s Moreover, the functions hk;s := ¸k hk; k 2 N, form a CONS in H for s 2 R. Similarly, the Sobolev Spaces Hs(Rd)(s 2 R) are defined. Namely, 8 9 < Z = Hs(Rd) = f 2 L (Rd): (1 + jyj2)sjf^(y)j2dy < 1 : 2 ; R R 2 2 s ^ 2 ^ with norm kfks = R(1 + jyj ) jf(y)j dy, where f denotes the Fourier Transform in R d ^ 1 R , i.e. f(y) = (2¼)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 2 R, see for instance [1], [43]. γ d For p ¸ 1 and γ 2 R, we define the space Hp (R ) as collection of all gene- ¡γ=2 d ralized functions f such that (I ¡ ¢) f 2 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 j(h; v)H j · C ¢ jjvjjV ¢ jjujjV 0 (v 2 V; u 2 H). For example the Sobolev spaces (Hl+γ(Rd); Hl(Rd); Hl¡γ(Rd)); γ > 0; l 2 R, form a normal triple. γ1 γ2 Note that Hp ½ 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 ) ½ C (R ) if pγ > d (see [9], Chapter 5). Also, we recall the interpolation theorem for spaces Hs [40], [41]. For every u 2 γ+2 θ 1¡θ s¡γ H and s 2 [γ; γ + 2], we have kuks · Ckukγ+2kukγ , where θ = 2 and C depends only on M; γ; s. jajp jbjq We will refer to inequality ab · " p + q"p¡1 , as "-inequality, where a; b 2 R; 1=p+ 1=q = 1; " > 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.
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