BEN-GURION UNIVERSITY OF THE NEGEV THE FACULTY OF NATURAL SCIENCES DEPARTMENT OF MATHEMATICS
GENERALIZED WHITE NOISE SPACE ANALYSIS AND STOCHASTIC INTEGRATION WITH RESPECT TO A CLASS OF GAUSSIAN STATIONARY INCREMENT PROCESSES
THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE MASTER OF SCIENCES DEGREE
ALON KIPNIS
UNDER THE SUPERVISION OF: PROFESSOR DANIEL ALPAY
MAY 2012 ABSTRACT
We extend the ideas in the basis of Hida’s white noise space into the case where the fundamental stochastic process has a non-white spectrum. In particular, we show that a Skorokhod-Hitsuda integral with respect to this process, which obeys the Wick-Itˆo calculus rules, can be naturally defined in this new setting. We use the spectral representation of the process to define a Fourier integral operator on L2(R). The Bochner-Minlos theorem is then applied to a characteristic functional on the Schwartz space of rapidly decreasing functions defined in terms of this operator, to obtain a probability measure on the topological dual of the Schwartz space, the space of tempered distributions. In the probability space thus obtained we define the counterpart of the S-transform, and use it to define the stochastic integral and prove an associated Itˆoformula. We demonstrate an application of our stochastic integration approach to formulate and solve an optimal stochastic control problem. CONTENTS
1. Introduction ...... 4
Part I Preliminaries 6
2. Background ...... 7 2.1 Countably-Normed spaces...... 7 2.2 Countably-Hilbert spaces...... 8 2.3 Nuclear spaces...... 9 2.4 Gelfand triples...... 11 2.5 The space of Schwartz Functions and its Dual...... 11 2.6 Abstract Gaussian Hilbert Spaces...... 12
3. Hida’s White Noise space ...... 14 3.1 The Wiener-Hermite Chaos Expansion...... 15 3.2 Spaces of stochastic test functions and generalized functions...... 16 3.3 The S-Transform...... 18 3.4 Stochastic Integration in the White Noise Space...... 18
Part II Results 22
4. Introduction ...... 23
5. The m Noise Space ...... 27 5.1 Definition...... 27
5.2 The process Bm ...... 31
5.3 The Sm transform...... 34
6. Stochastic integration in Wm ...... 40 6.1 Itˆo’sformula...... 44 Contents 3
6.2 Relation to other white-noise extensions of Wick-Itˆointegral...... 47
7. Application in optimal control ...... 50 7.1 Solution...... 53 7.2 Simulation...... 54
8. Conclusions ...... 56 1. INTRODUCTION
In many mathematical models of real world systems some parameter values may fluc- tuate and vary in time or space in such a way that seems random to us. One way of dealing with this randomness is to replace the values of these parameters by some kind of average and hope that this will give a good approximation to the original one. One problem of this approach is that even if we assume that the model obtained by averaging is reasonable, we might still want to know how do the small fluctuations of the parameters actually affect on the solution. In addition, it may also be that the actual fluctuations of the parameter values affect the solution in such a way that the averaged model is not even near to be a reliable description of what is actually happening. This has motivated the development of stochastic integration theory and associated stochastic calculus by K. Itˆoin the 40’s of the previous century [30]. This stochastic integration theory is based on the Brownian Motion, and can be defined with respect to any continuous semi-local Martingale. A few other stochastic integrals have been presented since Itˆo,for various other classes of stochastic processes. The Skorokhod-Hitsuda integral was initially introduced in [44] and [26] as a non adapted version of the Itˆointegral, which satisfies similar calculus rules as that of Itˆo.The Stratonovich integral satisfies the regular calculus rules as that of the Riemann integral, but is regularly defined for a narrower class of integrand [45, 46]. In this work we are interested in stochastic integration with respect to processes that are not necessarily semi Martingales. These processes are useful in, e.g., modeling of time- dependent phenomena in signal processing, in information theory, in telecommunication and in a host of applications. Some attempts have been made to extend the definition of Itˆo’sintegral to general stationary increments processes, a special attention was given to the case of the fractional Brownian motion [14, 38, 28,7].
White noise is regarded as a zero mean stationary stochastic process which is independent at different times. The covariance function of such process must vanish anywhere outside zero, but in order for such a process to have a physical meaning as a random signal, the variance of the process must be unbounded, and hence such process does not exist 1. Introduction 5 in the ordinary sense. The increment process of the Brownian motion, normalized by the length of the increment, seems to obtain the properties of the white noise described above as the length of the increment approaches zero. This is the reason why white noise is informally regarded as the time derivative of the Brownian motion which has nowhere differentiable sample path with probability one. Due to the above features, white noise it is often used in system model equations as an idealization of random noise arises in nature, such as the roar of a jet engine, or the noise disturbing the transmission of a communication system. As of today, there are several approaches to stochastic integration. An intuitive approach is to define such an integral directly with white noise as part of the integrand. This re- quires the building of a rigorous mathematical theory of white noise, such as the one introduced by T. Hida in 1975 [22]. His idea was to realize non-linear functions on a Hilbert space as functions of white noise. During the last three decades the theory of white noise has evolved into an infinite dimensional distribution theory.
In this work we consider Gaussian stationary increment processes and extend Hida’s white noise space theory to a wide family of such processes. In particular we introduce stochastic integration theory with respect to these processes based on the Skorokhod- Hitsuda integral, which can be useful in modeling systems in which the underlying noise has a non-white spectrum, namely a colored noise. Our main tool is a version of the S transform adapted to our new setting. The S transform is an elementary transformation in the white noise space which allows a rather simple definition of the Itˆointegral and other important results in the white noise space settings. The fact that this transfor- mation can be naturally extended suggests that our new integration theory is a natural extension of the Itˆointegral. In the present thesis, I show that a Wick-Itˆostochastic integral, with respect to any sta- tionary increment Gaussian process, can be naturally defined using an associated family of Fourier integral operators and some ideas taken from Hida’s white noise space theory. In other words, this is an extension of Hida’s white noise space theory to the case of non- white noises. In particular, this integration theory generalizes many works in stochastic integration with respect to fractional Brownian motion done in the recent years [10].
The white noise space theory is an elegant example of the combination of many de- velopments in functional analysis to the study of stochastic dynamics in probability. The first part of this dissertation is devoted to review Hida’s white noise space theory and the relevant notions in functional analysis. New results are included in the second part. Part I
PRELIMINARIES 2. BACKGROUND
In this chapter we describe the background concerning the concepts needed in the con- struction of the white noise space and its extension in this work. The main references for this chapter are [19] for countably-normed spaces; [20] for countably-Hilbert spaces, nuclear spaces and Gel’fand triples; [42] for The space of Schwartz functions and [31] for abstract Gaussian Hilbert spaces.
2.1 Countably-Normed spaces
Definition 2.1.1. Two norms k · k1 and k · k2, defined in a vector space V , will be called comparable if there is constant C such that the inequality
kvk1 ≤ Ckvk2
holds for all v ∈ V .
In the above definition the norm k · k2 is considered to be stronger then k · k1 in the sense
that every Cauchy sequence with respect to the k · k2 is also a Cauchy sequence with
respect to k · k1.
Definition 2.1.2. Two norms k · k1 and k · k2, defined in a vector space V , will be called
compatible if every Cauchy sequence {vn}n≥1 ⊂ V in both norms that converges to the zero element with respect to one of the norms, also converges to the zero element with respect to the second.
If two comparable and compatible norms ,k · k1 and k · k2 such that k · k1 ≤ Ck · k2, are
defined in a space V , then the completion V1 and V2 of V with respect to these norms may be considered to have the following relationship:
V1 ⊃ V2 ⊃ V.
If the two norms k · k1 and k · k2 are compatible but not comparable, we can introduce a 2. Background 8
third norm k · k3 defined by
kvk3 = max {kvk1, kvk2} .
It is easy to verify that k · k3 is comparable and compatible with the other two norms.
Thus, given any family {k · kn}n≥1 of norms on V we may assume they satisfy the relation
k · k1 ≤ C2k · k2 ≤ ..., and the completions of V with respect to each of the norms satisfy
V1 ⊃ V2 ⊃ ... ⊃ Vn ⊃ ... ⊃ V.
For a vector space V with a countable system of norms {k · kn}n≥1 we can define a topology by the following system of neighborhoods of zero,
Uk, = {v : kvk1 < , kvk2 < , ..., kvkk < } , k ∈ N, > 0.
We note that the topology defined in this way coincides with the topology in V defined by the metric ∞ X kvkn kvk = 2−n . 1 + kvk n=1 n Definition 2.1.3. A vector space V in which a topology is defined by a countable familiy of compatible norms is called a countably normed space.
2.2 Countably-Hilbert spaces
Suppose we are given a countably normed space V in which the topology is defined by p a countable set of inner product norms kvkn = (v, v)n. The space V will be called a countably Hilbert space if it is complete relative to the stated countably-normed topology.
Note that also in this case, any system of scalar products (·, ·)n, n = 1, 2, ... in V can be 0 replaced by a new system of scalar products (·, ·)n which does not alter the topology in V , by setting n 0 X (v, w)n = (v, w)k , v, w ∈ V. k=1 This new system has the property that
0 0 (v, v)1 ≤ (v, v)2 ≤ ..., (2.1) 2. Background 9
so without losing generality we can further assume that given a countably Hilbert space V and a system of norms in it, condition (2.1) is satisfied.
Let Vn denote the completion of the space V relative to the norm k · kn. So each Vn is a Hilbert space and from the completeness of V it follows that
∞ \ V = Vn. n=1
It is not hard to show that the dual space V 0 of a countably Hilbert space V equals
∞ 0 [ 0 V = Vn. n=1
2.3 Nuclear spaces
Let V be a countably-Hilbert space associated with an increasing sequence {k · kn}n = p (·, ·) of Hilbert norms. Denote Vn the completion of V with respect to the norm k · kn. In each of these spaces the set of elements of V is an everywhere dense set. By hypothesis, if m ≤ n then (v, v)m ≤ (v, v)n ∀v ∈ V . From this it follows that the function maps en element v ∈ V from Vn to Vm (i.e. the same element v considered in two different spaces) is a continuous function of an everywhere dense set in Vm, so it can be extended to a n p n p continuous linear operator Tm : Vn −→ Vm. Note that Tm = TmTn if m ≤ n ≤ p.
Definition 2.3.1. A countably Hilbert space V is called nuclear, if for any m there is n an n such that the operator Tm is nuclear, i.e. has the form
∞ n X Tmv = λk (v, vk)n wk, v ∈ Vn, k=1
where {vk} and {wk} are orthonormal systems of vectors in the spaces Vn and Vm respec- P∞ tively, λk > 0 and k=1 λk < ∞.
Every nuclear space is perfect, i.e. every bounded closed set in a nuclear space V is compact. From this follows the following properties of a nuclear space V :
1. V is separable.
2. V is complete relative to weak convergence.
3. Both in V and its dual V 0, strong and weak convergence coincide. 2. Background 10
4. V 0 is perfect relative to the topology of weak and strong convergence.
It follows that the σ-field on V 0 generated by the three topologies(weak, strong and inductive limit) are all the same. This σ-field is regarded as the Borel field of V and denoted B (V ). Two of the most important properties of nuclear spaces are given below.
Theorem 2.3.2 (Bochner-Minlos [20]). Let V be a real nuclear space. Let C : V −→ C be a complex valued function on V satisfying:
1. C is continuous.
2. C(0) = 1.
3. C is a positive function in the sense that for any
z1, ..., zn ∈ C and v1, ..., vn ∈ V ,
n X zizjC (vi − vj) ≥ 0. i,j=1
Then there exist a unique probability measure P on V 0 such that Z C(v) = eihx,vidP (x), v ∈ V. V 0
Note that the above theorem is not true for a general real separable infinite dimensional − 1 kvk2 Hilbert space H. To see this, take C(v) = e 2 H and let {e1, e2, ...} be an orthonormal basis for H. Then Z i(x,e ) − 1 e n dP (x) = e 2 . H
But for every x ∈ H,(x, en) −→ 0 as n −→ ∞. Another important fact about nuclear spaces is the following abstract kernel theorem.
Theorem 2.3.3 (Schwartz’s kernel theorem [20]). Let V be a nuclear space associated with an increasing sequence {k · kn} of norms and let Vn be the completion of V with respect to k · kn. Suppose F : V × V → C is a bilinear continuous functional. Then there 0 exist n, p ≥ 1, and a Hilbert-Schmidt operator A from Vn into Vp such that
F (u, v) = hAu, vi, u, v ∈ V. 2. Background 11
2.4 Gelfand triples
Let V be a nuclear space densely imbedded into some Hilbert space H, relative to the norm of H. We may identify H with its dual space H0 using Riesz representation theorem: 0 each h ∈ H is identified as the element φh in H defined by
φh(x) = hx, hiH , x ∈ H.
Each h ∈ H can be further identified with an element of V 0, by
hh, viV 0,V = hh, viH , v ∈ V.
Thus H is densely embedded in V 0 with respect to the weak topology of V 0. We get the triple V ⊂ H ⊂ V 0, which is called a Gelfand triple. H is dense in V 0 with the weak topology of V 0.
Suppose that V is associated with a sequence {k · kn} of norms and let Vn be the com- pletion of V with respect to k · kn. By setting H = V1, we get the continuous inclusion
0 0 0 V ⊂ ... ⊂ Vn+1 ⊂ Vn ⊂ ... ⊂ V1 ⊂ ... ⊂ Vn ⊂ Vn+1 ⊂ ... ⊂ V ,
0 and in particular V ⊂ V1 ⊂ V is a Gelfand triple.
2.5 The space of Schwartz Functions and its Dual
Definition 2.5.1. The Schwartz space SR of rapidly decreasing functions consists of all functions s ∈ C∞ (R) such that for every α, β ∈ N,
dβs lim | 1 + |x|2α (x)| = 0. |x|→∞ dxβ
The family of norms
Z β 1/2 α d s kskα,β = |x β (x)|dx , α, β ∈ N, R dx
makes into a countably normed space. The space and its topological dual 0 , SR SR SR the space of tempered distributions, play a central role in White Noise Space theory. We will list some of their important properties. 2. Background 12
1. If s ∈ SR and P is a polynomial, then the mapping
s −→ P (D)s,
n dns where D (s) = dxn , is a continuous map of SR into itself. In addition,
P\(D)s = P s,b and Pc s = P (−D)s.b
2. The Fourier transform is a continuous linear mapping of SR into itself [42, Theorem 7.4].
3. is a nuclear spaces, thus so is 0 [20, section 3.6]. SR SR 4. We have the Gelfand triple [41].
0 SR ⊂ L2 (R) ⊂ SR.
2.6 Abstract Gaussian Hilbert Spaces
As can be deduced by its name, a Gaussian Hilbert space is a notion combining probability theory and Hilbert space theory. A Gaussian linear space is a linear space of random variables defined in some probability space (Ω, F,P ) with central normal distribution.
The inner product in L2 (Ω, F,P ) assigned to a Gaussian linear space turns it into a pre-Hilbert. A Gaussian Hilbert space is a complete Gaussian linear space, i.e. a closed linear space of L2 (Ω, F,P ) consisting of zero mean Gaussian variables.
Let H be a Gaussian Hilbert space on (Ω, F,P ). Since variables in H belong to Lp for every finite p ≥ 0, H¨older’sinequality shows that any finite product of variables in H
belongs to L2 (Ω, F,P ). This allows us to consider subspaces of L2 (Ω, F,P ) consists of polynomials in the elements of H. We define
n ⊥ H , P n(H) ∩ P n−1(H) , where P n(H) is the closure in L2 (Ω, F,P ) of the linear space generated by polynomials in the elements of H of degree ≤ n. It follows that the spaces Hn, n ≥ 0, are mutually orthogonal, and if we consider the 2. Background 13
sub-sigma field F(H) generated by the elements of H in L2 (Ω, F,P ), we get
∞ M n H = L2 (Ω, F,P ) , (2.2) n=0 which means that each elements X(ω) ∈ L2 (Ω, F,P ) has the representation
∞ X n X(ω) = Xn(ω),Xn(ω) ∈ H . n=0
This decomposition of L2 (Ω, F,P ) is called the Wiener chaos decomposition. The Wick product of two elements X(ω) ∈ Hn and Y (ω) ∈ Hm can be defined by
X Y = πm+n(XY ),
n where πn is the orthogonal projection of L2 (Ω, F,P ) on H , and the definition may be extended to any two element of L2 (Ω, F,P ) in view of their chaos representation. In the sequel we will investigate a particular example of the Wiener chaos decomposition in the white noise space, given in terms of the Hermite polynomials and the Hermite functions. 3. HIDA’S WHITE NOISE SPACE
In 1975, T. Hida [21] defined white noise in rigorous mathematical terms as generalized functions on an infinite dimensional space. This approach has been extensively studied in the last decades and we will review it here briefly. We refer to [24], [27] and [35] for more details.
The starting point for the construction of the white noise space is the positive function − 1 ksk 2 L2(R) e defined on the nuclear space SR. Applying the Bochner-Minlos theorem 2.3.2 to it we obtain a probability measure P on 0 such that SR Z − 1 ksk2 hs0,si 0 2 L2(R) e = e dP (s ), s ∈ SR. (3.1) S 0 R In accordance with the notation common in probability theory, we set
0 Ω := SR
and denote by ω the elements of Ω. The Borel sigma algebra is denoted by B. The probability space (Ω, B( 0 ),P ) is called a white noise space. The measure P is SR called the standard Gaussian measure on 0 or the white noise measure. We also set SR
B 0 L2(Ω) , L2 (Ω, (SR),P ) .
Taking s = s1 with ∈ R in (3.1) and expanding both sides into a power series we may
conclude that for each s ∈ SR, hω, si is a normally distributed random variable with zero 2 mean and variance ksk . The isometric map s −→ hω, si from into L2(Ω) can be L2(R) SR
extended to any function f ∈ L2(R) by taking a sequence {sn} in SR such that sn → f in L2(R) and setting
hω, fi lim hω, sni , n→∞
in L2(Ω). It follows that H , {hω, fi, f ∈ L2(R)} 3. Hida’s White Noise space 15
is a Gaussian Hilbert space isomorphic to L2(R). The Brownian motion can be defined to be the continuous version of the process
B(t) , B(t, ω) , hω, 1[0,t]i, t ≥ 0. (3.2)
3.1 The Wiener-Hermite Chaos Expansion
It turns out to be convenient to express the Wiener chaos (2.2) for the space L2(Ω) in
terms of Hermite polynomials and Hermite functions. The nth (probabilistic) Hermite polynomial is defined by
n n 1 x2 d − 1 x2 h (x) = (−1) e 2 e 2 , n = 1, 2, .... n dxn
The Hermite functions are defined by
1 2 − 2 x √ − 1 e ηn(x) , π 4 hn−1 2x , n = 0, 1, 2, .... p(n − 1)! and constitutes an orthonormal basis for L2(R). We denote by J the set of multi-indices over N, which can be viewed as the set of infinite sequences α = (α1, α2, ...), αi ∈ N for
which αi = 0 for all i large enough. Define
∞ Y Hα(ω) , hαi (hω, ηii) . i=0
The family {Hα}α∈J constitutes an orthogonal basis for L2(Ω). In addition, if α =
(α1, α2...) then we have
H2 = kH k2 = α! α !α ! ··· . E α α L2(Ω) , 1 2
It follows that every X ∈ L2(Ω) can be decomposed as
X X(ω) = cαHα(ω), cα ∈ R, α∈J
and we have 2 X 2 E X(·) = α!cα. α∈J 3. Hida’s White Noise space 16
For example, the Brownian motion has the Wiener-Hermite expansion
∞ ∞ ∞ X Z t X Z t X Z t B(t) = hω, 1[0,t]i = hω, ηj(u)du ηji = ηj(u)duhω, ηji = ηj(u)duH(j), j=1 0 n=1 0 n=1 0 (3.3) where (j) = (0, 0, ..., 1, ..) with 1 on the entry number j.
The Wick product on L2(Ω) is defined through
Hα Hβ , Hα+β.
Wick powers, Wick polynomials and Wick versions of analytic functions can be defined as well. For example, the Wick exponential is defined by
∞ X Xn eX (ω) . , n! n=0
For a Gaussian X(ω) = hω, fi ∈ H, with f ∈ L2(Ω), it can be shown that X X− 1 X2 1 2 e = e 2 E[ ] = exp hω, fi − kfk . 2 L2(R)
We note that the definition of the Wick product is independent of the particular choice
of basis elements {Hα} [27, App. D].
3.2 Spaces of stochastic test functions and generalized functions
P Let X(ω) = α∈J cαHα(ω). If X 2 α!cα < ∞ (3.4) α∈J P then X ∈ L2(Ω). Moreover, if Y (ω) = α∈J bαHα(ω) then X E [X(·)Y (·)] = bαcαα!. α
The main idea in the following is to replace condition 3.4 by various other conditions, and thus obtain a family of stochastic distributions and test functions.
The Kondratiev spaces of stochastic test functions Sρ for 0 ≥ ρ ≥ 1 consist of those
X φ(ω) = cαHα(ω) ∈ L2(Ω) α∈J 3. Hida’s White Noise space 17
such that
2 X 2 1+ρ Y kαj kφkρ,k , cα (α!) (2j) < ∞ (3.5) α∈J j for all k ∈ N.
The corresponding Kondratiev space of stochastic distributions S−ρ consist of all formal expansions X Φ = bαHα(ω) α∈J such that
2 X 2 1−ρ Y −qαj kΦk−ρ,−k , bα (α!) (2j) < ∞ (3.6) α∈J j
for some q ∈ N. The topologies of Sρ and S−ρ are defined by the corresponding families
of seminorms defined in 3.5 and 3.6 respectively. Note that the duality between Sρ and
S−ρ is well defined by the action
X hΦ, φ, i , bαcαα!, Φ ∈ S−ρ, φ ∈ Sρ, (3.7) α∈J
since for q large enough ! ! 1−ρ 1+ρ X X Y qαj /2 Y −qαj /2 |bαcα|α! = |bαcα| (α!) 2 (α!) 2 (2j) (2j) α α j j !!1/2 !!1/2 X 2 1−ρ Y −qαj X 2 1+ρ Y qαj ≤ bα (α!) (2j) bα (α!) (2j) < ∞. α j α j
For general 0 ≤ ρ ≤ 1 we have
S1 ⊂ Sρ ⊂ Sρ ⊂ L2(Ω) ⊂ S−0 ⊂ S−ρ ⊂ S−1.
The Gelfand triples
S1 ⊂ L2(Ω) ⊂ S−1 and S0 ⊂ L2(Ω) ⊂ S−0
are used in stochastic analysis as the analog of the triple ⊂ L ( ) ⊂ 0 that SR 2 R SR is commonly used in differential equations. Indeed, it turns out that these spaces of stochastic distributions host many of the solution for stochastic differential equations. For example, consider the Wiener-Itˆoexpansion of the Brownian motion (3.3) and take 3. Hida’s White Noise space 18 its formal time derivative. The resulting sum
∞ X ηj(t)H(j) (3.8) n=1 does not satisfy condition (3.4) hence cannot be a member of L2(Ω), but it is not hard to prove that this sum belongs to Hida’s space of stochastic distributions S−0.
3.3 The S-Transform
We will now introduce a transformation on (L2) which its extension plays a central role in our work. This transformation was introduced in [18] and [34].
For X ∈ L2(Ω) we define the S-transform of X to be Z hω,si − 1 ksk2 h·,si 2 L2(R) (SX)(s) , X(ω)e dP (ω) = e E X(·)e , s ∈ SR. Ω
Due to the translation invariance of the Gaussian measure P [23], it follows that Z SX(s) = X(ω + s)dP (ω). Ω P In terms of the chaos expansion, we can express the S-transform of X = α∈J aαHα as
X α (SX)(s) = aα (s, η) , α∈J where (s, η)α Q∞ (s, η )αi . This allows us to formally extend the definition of the , i=1 i L2(R) S-transform to any element of S−1. In addition, we conclude that
S (X Y )(s) = (SX)(s) · (SY )(s)
for any X,Y ∈ S−1. The importance of the S-transform follows from the fact that it is injective [35, Propo- sition 5.10], hence one can specify a generalized function by its S-transform.
3.4 Stochastic Integration in the White Noise Space
The White Noise distribution theory allows a convenient framework for various levels of stochastic integrals, each generalizes the other. The following integrals are said to be of 3. Hida’s White Noise space 19
Wick-Itˆotype, since they all satisfy the Wick-Itˆocalculus rules that follows from Itˆo’s formula.
3.4.1 Wiener Integral
The Wiener integral of a function f ∈ L2(R) is defined by the isometric map f −→ hω, fi which identifies L2(R) with the Gaussian Hilbert space {hω, fi, f ∈ L2(Ω)} ⊂ L2 (Ω). Recall the definition of the Brownian motion {B(t)} in the white noise space (3.2) to justify the notation
Z Z d Z d fdB(t) := f hω, 1ti = hω, ” f 1t”i = hω, fi. R R dt R dt The Weiner integral carries a deterministic function into a Gaussian random variable. It is merely an isometric embedding of an abstract Hilbert space in its corresponding Gaussian Hilbert space.
3.4.2 ItˆoIntegral
The Itˆointegral is defined for non-anticipating stochastic processes, i.e. a stochastic
process {X(t)} defined on L2(Ω) such that for any t, the random variable X(t) is mea-
surable with respect to Ft, which is the sigma-field generated by {B(s), s ≤ t}. The most important properties of the Itˆointegral are given below. W Let {X(t)} be a non-anticipating stochastic process on L2(Ω, t≥0 Ft,P ). Denote
Z t It(X) , x(s)dB(s), 0
where the left hand side is the Itˆointegral of {X(t)}. We have [32, Chapter 3]
1.
I0(X) = 0, a.s. P.
2.
E [It(X)|Fs] = Is(X), a.s. P.
3. Z t 2 2 E (It(X)) = E X(s) ds , 0
2 4. Itˆo’srule: Let g : R −→ R be a function of class C and let X(t) = X0 + 3. Hida’s White Noise space 20
R t 0 f(s)dB(s). Then,
Z t 1 Z t g(X(t)) = g(X(0)) + g0(X(s))f(s)dB(s) + g00(X(s))(f(s))2ds, a.s. P. 0 2 0 (3.9)
We note that the Itˆointegral can be extended to be defined with respect to the class of semi-local martingales [32].
3.4.3 Hitsuda-Skorokhod integral
Hitsuda [26] and Skorokhod [44] introduced an integral which is not restricted to inte- grands of the class of non-anticipating process, but which reduces to the Itˆointegral if the integrand happens to be non-anticipating, as was proved in [39]. In the white noise space framework the Hitsuda-Skorokhod integral can be defined by the relation Z ∞ Z ∞ X(t)δB(t) = X(t) B˙ (t)dt, (3.10) −∞ −∞ where B˙ (t) is the singular white noise, a stochastic distribution defined by the sum (3.8).
The integral at the right hand side should be interpreted as an S−1 valued Pettis/Bochner integral (see for example [25] for Pettis integrability). Relation (3.10) presents a natural definition for the Hitsuda-Skorokhod integral in the white noise setting. Even more natural is an equivalent definition for it in terms of the S-transform.
Definition 3.4.1. Let {X(t), t ∈ R} be an L2(Ω) valued stochastic process such that
S (X)(s) ∈ L1(R) for any s ∈ SR, and such that for any Borel set E the function R d E S (X(t)) (s) dt (s, 1t) dt is the S-transform of a unique element in L2(Ω). Then {X(t), t ∈ R} is called Hitsuda-Skorokhod (S-transform) integrable and we define
Z Z −1 d X(t)δB(t) , S S (X(t)) (s) (s, 1t) dt . E E dt
It can be shown that the last definition coincides with the definition of Hitsuda and
Skorokhod for their integral, and it can be extended to S−1 valued processes as well. (see [24], [35, 13.3] and especially [5] for more on the S-transform approach to stochastic integration). Note that the S-transform definition for stochastic integration of L2(Ω) processes does not involve Wick product nor stochastic distributions, and that for any s ∈ SR the function (s, 1t) is absolutely continuous with respect to the Lebesgue measure as can be seen by (6.2). In view of this, the S-transform integral can be defined in
only in terms of expectation and the inner product in L2(R). This distinction suggests 3. Hida’s White Noise space 21 the possibility to naturally extend the Hitsuda-Skorokhod integral to a setting where the underlying noise is not necessarily white. As we shall see in the Results part of this work, it will be required to replace L2(R) by another Hilbert space, one which is determined by the spectrum of the noise. Part II
RESULTS 4. INTRODUCTION
In this work we take an approach which is based on an extension of the S-transform in order to develop stochastic integration for the family of centered Gaussian processes with covariance function of the form
Z eiξt − 1 e−iξs − 1 Km(t, s) = m(ξ)dξ R ξ ξ where m is a positive measurable even function subject to
Z m(u) 2 dξ < ∞. (4.1) R ξ + 1
Note that Km(t, s) can also be written as
Km(t, s) = r(t) + r(s) − r(t − s),
where Z 1 − cos(tξ) r(t) = 2 m(ξ)dξ. R ξ This family includes in particular the fractional Brownian motion, which corresponds (up to a multiplicative constant) to m(ξ) = |ξ|1−2H , where H ∈ (0, 1). We note that complex-valued functions of the form
K(t, s) = r(t) + r(s) − r(t − s) − r(0),
where r is a continuous function, have been studied in particular by von Neumann, Schoenberg and Krein. Such a function is positive definite if and only if r can be written in the form Z iξt iξt dσ(ξ) r(t) = r0 + iγt + e − 1 − 2 2 , R ξ + 1 ξ where σ is an increasing right continuous function subject to R dσ(ξ) < ∞. See [36], [33], R ξ2+1 and [2] for more information on these kernels. 4. Introduction 24
As in [2], our starting point is the (in general unbounded) operator Tm on the Lebesgue
space of complex-valued functions L2(R) defined by √ Tdmf(ξ) = m(ξ)fb(ξ), (4.2)
with domain Z 2 D(Tm) = f ∈ L2(R); m(ξ)|fb(ξ)| dξ < ∞ , R where f(ξ) = √1 R e−iξtf(t)dt denotes the Fourier transform of f. Clearly, the Schwartz b 2π R space SR of real smooth rapidly decreasing functions belong to the domain of Tm. The indicator functions 1[0,t], t ≥ 0, 1t = −1[t,0] t ≤ 0,
also belong to D(Tm). In [2], and with some restrictions on m, a centered Gaussian process B with covariance function K (t, s) = (T 1 ,T 1 ) was constructed in m m m t m s L2(R) Hida’s white noise space. In the present work we chose a different path. We define the characteristic functional 2 − kTmsk Cm(s) = e 2 , s ∈ S . (4.3)
It has been proved in [3] that Cm is continuous from SR into R. Restricting Cm to real-valued functions and using the Bochner-Minlos theorem 2.3.2, we obtain an analog
of the white noise space in which the process Bm is built in a natural way. Stochastic calculus with respect to this process is then developed using an S-transform approach.
The S-transform of an element X(ω) of the white noise space is defined by
h·,si − 1 ksk SX(s) = E X(·)e e 2 L2(R) .
An S-transform approach to stochastic integration in the white noise setting can be found in [24], [35, Section 13.3] and especially in [5]. The main idea is to define the Hitsuida- Skorohod integral of a stochastic process X(t) with respect to the Brownian motion B(t) over a Borel set E, by
Z Z −1 X(t)δB(t) , S S (X(t)) (s)s(t)dt . E E
Namely, the integral of X(t) over the Borel set E is the unique stochastic process Φ(t) 4. Introduction 25
such that for any t ≥ 0 and s ∈ S , Z (SΦ(t)) (s) = S (X(t)) (s)s(t)dt. E
Since s(t) = d (s, 1 ) , it suggests to extend the last definition of the integral by dt t L2(R) replacing the inner product in L2(R) by a different one. In the present work, this inner
product is determined by the spectrum of the process through the operator Tm. 1−2H 1 We note that when m(ξ) = |ξ| , and H ∈ ( 2 , 1), the operator Tm reduces, up to a 2 multiplicative constant, to the operator MH defined in [16] and in [7]. The set Lφ pre-
sented in [12, Eq. 2.2], is closely related to the domain of Tm, and the functional Cm was used with the Bochner-Minlos theorem in [8, (3.5), p. 49]. In view of this, our work gen- eralized the stochastic calculus for fractional Brownian motion presented in these works to the aforementioned family of Gaussian processes.
Note moreover that the function φ from the last references defines the kernel associated ∗ to the operator TmTm via Schwartz’ kernel theorem, with m = MH . In the general case, ∗ the kernel associated to the operator TmTm is not a function. This last remark is the source for some of the difficulties arises in extending Wick-Itˆointegration for fractional 1 1 1 Brownian motion such as the distinction between the cases H < 2 , H > 2 and H = 2 .
There are two main results in this work. The first is the construction of a probability space in which a stationary increment process with spectral density m is naturally de- fined. This result, being a concrete example of Kolmogorov’s extension theorem on the existence of a Gaussian process with a given spectral density, is interesting in its own right. The second main result deals with developing stochastic integration with respect to the fundamental process in this space. We take an approach based on the analog of the S-transform in our setting, and show that this stochastic integral coincides with the one already defined in [1] but in the framework of Hida’s white noise space.
The results section consists of four chapters besides the introduction. In Chapter 5 we construct an analog of Hida’s white noise space using the characteristic function Cm,
define and study the fundamental stationary increment process Bm and the analog of the S-transform in this space. In Chapter6 we define a Wick-Itˆotype stochastic integral
with respect to Bm, and prove an associated Itˆoformula. We explain the relation of this integral to other works on white noise based stochastic integrals. In Chapter 7 we use our stochastic integration approach to formulate and solve a stochastic optimal control 4. Introduction 26 problem. The last chapter contains the main conclusion and some issues for further discussion. 5. THE M NOISE SPACE
5.1 Definition
We set to be the space of real-valued Schwartz functions, and Ω = 0 . We denote SR SR by B the associated Borel sigma algebra. Throughout this paper, we denote by h·, ·i the duality between 0 and , and by (·, ·) the inner product in L ( ). In case there is no SR SR 2 R danger of confusion, the L2(R) norm will be denoted as k · k.
Theorem 5.1.1. There exists a unique probability measure µm on (Ω, B) such that
2 kTmsk Z − 2 ihω,si e = e dµm(ω), s ∈ SR, Ω
Proof: The function Cm(s) is positive definite on SR since
1 1 C (s − s ) = exp − kT s k2 × exp {(T s ,T s )} × exp − kT s k2 , m 1 2 2 m 1 m 1 m 2 2 m 2 where now the middle term is positive since an exponent of a positive function is still positive. Moreover, the operator Tm is continuous from S (and hence from SR) into L2(R). This was proved in [3], and we repeat the argument for completeness. As in [3] we set K = R m(u) du and s](u) = s(−u). We have for s ∈ : R 1+u2 S Z 2 2 kTmsk = |sb(u)| m(u)du R Z m(u) = |(1 + u2)s(u)|2 du b 2 R 1 + u Z Z ≤ K |s ? s]|(ξ)dξ + |s0 ? (s])0|(ξ)dξ R R Z 2 Z 2! ≤ K |s(ξ)|dξ + |s0(ξ)|dξ , R R
where we have denoted convolution by ?. It follows that Cm is a continuous map from
SR into R, and the existence of µm follows from the Bochner-Minlos theorem. 5. The m Noise Space 28
The triplet (Ω, B, µm) will be used as our probability space.
Proposition 5.1.2. Let s ∈ SR. Then:
2 2 E[hω, si ] = kTmsk . (5.1)
Proof: We have
1 2 Z − kTmsk ihω,si e 2 = e dµm(ω). (5.2) Ω Expanding both sides of (5.2) in power series for s we obtain
Z E [hω, si] = hω, sidµm(ω) = 0. (5.3) Ω and Z 2 2 2 E hω, si = hω, si dµm(ω) = kTmsk . (5.4) S 0
We now want to extend the isometry (5.1) to any function in the domain of Tm. This extension involves two separate steps: First, an approximation procedure, and next com- plexification. For the approximation step we introduce an inner product defined by the
operator Tm. For f and g in D(Tm) we define the inner product Z ˆ ∗ (f, g)m , fgˆ m(ξ)dξ. R
Note that D(Tm) is consist of those functions f in L2(R) that satisfy
2 kfkm = (f, f)m < ∞.
We define the space LSm and Lm to be the closure of S and D(Tm) in the norm k · km, respectively.
Proposition 5.1.3. We have
Lm = LSm.
Proof: Let f ∈ Dm be orthogonal to any s ∈ S in the norm k · km, i.e. Z ∗ 0 = (s, f)m = sbfb m(ξ)dξ, ∀s ∈ S . R 5. The m Noise Space 29
It follows that fb∗m = 0 almost everywhere since it defines the zero distribution on S . But that also means Z fbfb∗m(ξ)dξ = 0 R
so f is zero in Lm.
Theorem 5.1.4. The isometry (5.1) extends to any f ∈ L2(R) where f is real-valued
and in the domain of Tm.
Proof: We first note that, for f in the domain of Tm we have
Tmf = Tmf. (5.5)
Indeed, since m is even and real we have
] √ ] √ ] Tdmf = m(fb) = ( mfb) = Tdmf = Tdmf.
Let now f be real-valued and in D(Tm) ⊂ Lm. It follows from Proposition 5.1.3 that there exists a sequence (sn)n∈N of elements in S such that
lim ksn − fkm = 0. (5.6) n→∞
In view of (5.5), and since f is real-valued we have
lim ksn − fkm = lim kTmsn − TmfkL ( ) = lim kTmsn − TmfkL ( ) = 0. (5.7) n→∞ n→∞ 2 R n→∞ 2 R
Together with (5.6) this last equation leads to
lim kTm(Re sn) − TmfkL ( ) = 0. (5.8) n→∞ 2 R
In particular (Tm(Re sn))n∈N is a Cauchy sequence in L2(R). By (5.1), (hω, Re sni)n∈N
is a Cauchy sequence in Wm. We denote by hω, fi its limit. It is easily checked that the limit does not depend on the given sequence for which (5.6) holds.
We denote by DR(Tm) the elements in the domain of Tm which are real-valued.
Let f, g ∈ DR(Tm). The polarization identity applied to
[hω, fi2] = kT fk2 , f ∈ D (T ). (5.9) E m L2(R) R m 5. The m Noise Space 30
leads to [hω, fihω, gi] = Re (T f, T g) . E m m L2(R)
In view of (5.5), Tmf and Tmg are real and so we have:
Proposition 5.1.5. Let f, g ∈ DR(Tm). It holds that
E [hω, fihω, gi] = (Tmf, Tmg) . (5.10)
Proposition 5.1.6. {hω, fi, f ∈ DR(Tm)} is a Gaussian process in the sense that for Pn any f1, ..., fn ∈ DR(Tm) and a1, ..., an ∈ R, the random variable i=1 aihω, fii has a normal distribution.
Proof: By (5.2), for λ ∈ R we have,
Pn Z Pn iλ aihω,fii iλ aihω,fii E[e i=1 ] = e i=1 dµm(ω) Ω Z Pn ihω,λ aifii (5.11) = e i=1 dµm(ω) Ω − 1 λ2k Pn a T f k2 = e 2 i=1 i m i .
In particular, we have that for any ξ1, ..., ξn ∈ DR (Tm) such that Tmξ1, ..., Tmξn are n orthonormal in L2 (R) and for any φ ∈ L2(R )
Z n 1 Y 1 2 − 2 xi E [φ (hω, ξ1i, ...hω, ξ1i)] = n φ(x1, ..., xn) e dx1 · ... · dxn. (5.12) (2π) 2 n R i=1
Definition 5.1.7. We set G to be the σ-field generated by the Gaussian elements
{hω, fi, f ∈ DR (Tm)} , and denote
Wm , L2 (Ω, G, µm) .
Note that G may be significantly smaller than B, the Borel σ-field of Ω. For example, if
m ≡ 0, then Tm is the zero operator and G = {∅, Ω, 0, Ω\{0}}. We will see in the following section that the time derivative, in the sense of distributions,
of the fundamental stochastic process Bm in the space Wm has spectral density m(ξ). It
is therefore justified to refer Wm as the m-noise space. 5. The m Noise Space 31
In the case m (ξ) ≡ 1, Tm is the identity over L2 (R) and µm is the white noise measure used for example in [24, (1.4), p. 3]. Moreover, by Theorem 1.9 p. 7 there, G equals the Borel sigma algebra and so the 1-noise space coincides with Hida’s white noise space.
5.2 The process Bm
We now define the fundamental stationary increment process Bm : R −→ Wm via
Bm(t) , Bm(t, ω) , hω, 1ti.
This process plays the role of the Brownian motion for the stochastic integral and the
Itˆoformula in the space Wm. Note that this is the same definition as the Brownian mo- tion in white noise space (3.2), the difference being the probability measure assigned to Ω.
Theorem 5.2.1. Bm has the following properties:
1. Bm is a centered Gaussian random process.
2. For t, s ∈ R, the covariance of Bm(t) and Bm(s) is
Z eiξt − 1 e−iξs − 1 Km(t, s) = m(ξ)dξ = (Tm1t,Tm1s) . (5.13) R ξ ξ
3. The process Bm has a continuous version under the condition
Z m(ξ) dξ < ∞ (5.14) R 1 + |ξ|
Proof: (1) follows from (5.11) and (5.3). To prove (2), we see that by (5.10) we have
E [Bm(t)Bm(s)] = E [hω, 1tihω, 1si]
= Re (Tm1t,Tm1s)
= (Tm1t,Tm1s) , since this last expression is real. To prove (3) we use similar arguments to [3, Theorem 10.2]. For t, s ∈ R, Z 2 2 1 − cos ((t − s)ξ) E (Bm(t) − Bm(s)) = E h·, 1[s,t]i = 2 2 m(ξ)dξ, R ξ 5. The m Noise Space 32
where the last equality follows by vanishing imaginary part of (5.13). We now compute
Z 1 Z 1 tξ 2 1 − cos(tξ) 2 2 sin 2 2 2 m(ξ)dξ = 2 t 2 2 m(ξ)dξ 0 ξ 0 ξ t 2 ≤ C1t (C1 > 0 independent of t).
Using the mean-value theorem for the function ξ → cos(tξ) we have
1 − cos(tξ) = tξ sin(tθt), θt ∈ [0, ξ].
Thus,
Z ∞ 1 − cos(tξ) Z ∞ m(ξ) Z ∞ m(ξ) 2 m(ξ)dξ = t sin(tθt) dξ ≤ t dξ ≤ C2t, 1 ξ 1 ξ 1 ξ
where we have used (5.14) for the last move. Since Bm(t)−Bm(s) is zero mean Gaussian, we obtain
4 2 2 E (Bm(t) − Bm(s)) = C3E (Bm(t) − Bm(s)) ≤ C4 (t − s) .
˘ Thus Bm satisfies Kolmogorov-Centsov test for the existence of a continuous version [32, Theorem 2.8].
We bring here two interesting examples for specific choices of the spectral density m and
the corresponding process Bm.
Example 5.2.2 (Band-limited noise). Consider the spectral density
m1 (ξ) = 1[−∆,∆], ∆ ≥ 0.
The corresponding process Bm1 has the covariance function
1 Z ∆ 1 − cos(tξ) − cos(sξ) + cos(ξ(t − s)) √ Km1 (t, s) = 2 dξ, t, s ∈ R. 2π −∆ ξ
The time derivative of this process also belongs to Wm, and is a stationary Gaussian process with covariance
2 Z ∆ ∂ 1 i(t−s)ξ 2 sin (∆(t − s)) Km1 (t, s) = √ e dξ = . (5.15) ∂t∂s 2π −∆ t − s
This process can be obtained in physical models by passing a white noise through a low- 5. The m Noise Space 33
Fig. 5.1: Various sample paths for the stationary increment process of Example 5.2.2 (left) and Example 5.2.3 (right).
pass filter with cut-off frequency ∆. We see from the covariance function (5.15) that each π time sample t0 ∈ R is positively correlated with time samples in the interval t0, t0 + 2∆ , π 3π negatively correlated with time samples in the interval t0 + 2∆ , t0 + 2∆ and so on with decreasing magnitude of correlation. This behavior may describes well price fluctuation of some financial asset.
Example 5.2.3 (Band limited fractional noise). We can combine the spectral density m1 from the previous example with the spectral density |ξ|1−2H of the fractional noise with Hurst parameter H ∈ (0.5, 1) to obtain a process with covariance function
1 Z ∆ 1 − cos(tξ) − cos(sξ) + cos ((t − s)ξ) √ 1−2H Km2 (t, s) = 2 |ξ| , t, s ∈ R. 2π −∆ ξ
This process shares both properties of long range dependency of the fractional Brown- ian motion with Hurst parameter H as well as the ripples of the filtered noise for its
time derivative. As the bandwidth ∆ approaches infinity, the covariance function Km2 uniformly converges (up to a multiplicative constant) to the covariance of the fractional Brownian motion.
Our next goal is to define stochastic integration with respect to the process Bm in the space Wm. The definition of the Wiener integral with respect to Bm for f ∈ D (Tm) is straightforward in view of the Hilbert spaces isomorphism (5.4) and given by
Z τ f(t)dBm(t) , hω, 1τ fi. (5.16) 0 5. The m Noise Space 34
Note that since Z Z 2 2 2 m(ξ) m(ξ)|fb(ξ)| dξ ≤ sup(1 + ξ )|fb(ξ)| 2 dξ, R ξ∈R R 1 + ξ
a sufficient condition for a function f ∈ L2 (R) to be in the domain of Tm is
sup(1 + ξ2)|fb(ξ)|2 ≤ ∞. ξ∈R
This is satisfied in particular if f is differentiable with derivative in L2 (R). Recall that in the white noise space one may defines the Skorokhod-Hitsuda stochastic
integral of Xt on the interval [a, b] as
Z b Z b ˙ XtdB(t) = Xt Bmdt a a
˙ where Bm denotes the time derivative of the Brownian motion and denotes the Wick product [27]. The chaos decomposition of the white noise space is used in order to define ˙ the Wick product and appropriate spaces of stochastic distributions where Bm lives.
Chaos decomposition for Wm can be obtained by a similar procedure to the one explained in 3.1 for the fractional Brownian motion. A space of stochastic distributions that con- ˙ tains Bm and is closed under the Wick product can similarly be defined. A somewhat alternative approach, which uses only the expectation and the Lebesgue integral on the real line, is achieved by using the S-transform [5]. As we shall see below,
an analogue of the S-transform can be naturally defined in the space Wm, thus allows us
to introduce Skorokhod-Hitsuda integral for Wm valued processes which is based on this transform.
5.3 The Sm transform
We now define the analog of the S transform in the space Wm and study its properties.
For s ∈ SR we define the analog of the Wick exponential in the space Wm:
hω,si hω,si− 1 kT sk2 e , e 2 m
Note that this definition is not yet related to the Wick product which has not yet been defined in Wm. 5. The m Noise Space 35
Definition 5.3.1. The Sm transform of Φ ∈ Wm is defined by Z hω,si hω,si (SmΦ)(s) , e Φ(ω)dµm(ω) = E e Φ(ω) , s ∈ SR. Ω
Theorem 5.3.2. Let Φ, Ψ ∈ Wm. If (SmΦ) (s) = (SmΨ) (s) for all s ∈ S , then Φ = Ψ.
Proof: We follow the same arguments as in [5, Theorem 2.2] with some small changes.
By linearity of the Sm transform, it is enough to prove
(∀s ∈ S , (SmΦ) (s) = 0) ⇒ Φ = 0.
Let {ξ } ⊂ be a countable dense set in L ( ) and denote by G the σ-field n n∈N SR 2 R n generated by {hω, ξ i, ..., hω, ξ i}. We may choose {ξ } such that {T ξ } are 1 n n n∈N m n n∈N orthonormal. For every n ∈ N, E [Φ|Gn] = φn (hω, ξ1i, ..., hω, ξni) for some measurable n function φn : R −→ R such that Z Z − 1 x0x EΦ = ··· φn(x)e 2 dx < ∞,
Rn
where x0 denotes the transpose of x; see for instance [9, Proposition 2.7, p. 7]. Thus, for n t = (t1, ..., tn) ∈ R , using (5.12) we obtain
Z Pn Z Pn hω, tkξki hω, tkξki 0 = e k=1 Φ(ω)dµm = e k=1 E [Φ|Gn] dµm(ω) Ω Ω 1 Pn 2 2 Z Pn − t kTmξkk tkhω,ξki = e 2 k=1 k e k=1 φn (hω, ξ1i, ..., hω, ξni) dµm(ω) Ω 1 Pn 2 2 1 Z Z Pn 1 Pn 2 − 2 k=1 tkkTmξkk k=1 tkxk − 2 k=1 xk = e n ··· e φn (x1, ..., xn) e dx1...dxn (2π) 2 Rn Z Z − 1 (x−t)0(x−t) = ··· φn (x) e 2 dx.
Rn
Since the last expression is a convolution integral of φn with a positive eigne vector of
the Fourier transform, by properties of the Fourier transform we get that φn = 0 for all n ∈ . Since S G = G we have Φ = 0. N n∈N n Definition 5.3.3. A stochastic exponential is a random variable of the form
hω,fi e , f ∈ DR (Tm) .
We denote by E the family of linear combinations of stochastic exponentials. 5. The m Noise Space 36
hω,fi − 1 kT fk2 hω,fi Since e = e 2 m e , the following claim is a direct consequence of Theorem 5.3.2.
Proposition 5.3.4. E is dense in Wm.
Definition 5.3.5. A stochastic polynomial is a random variable of the form
p (hω, f1i, ..., hω, f2i) , f1, ..., fn ∈ DR (Tm) . for some polynomial p in n variables. We denote the set of stochastic polynomials by P.
Corollary 5.3.6. The set of stochastic polynomials is dense in Wm.
Proof: We first note that the stochastic polynomials indeed belong to Wm because the random variables hω, fi are Gaussian and hence have moments of any order.
Let Φ ∈ Wm such that E [Φp] = 0 for each p ∈ P. Then any f ∈ DR(Tm),
" ∞ # ∞ X hω, fin X [hω, finΦ(ω)] E ehω,fiΦ(ω) = E Φ(ω) = E = 0, (5.17) n! n! n=0 n=0 where interchanging of summation is justified by Fubini’s theorem since
∞ n ∞ X hω, fi X 1 q 2 | Φ(ω) | ≤ [hω, fi2n] Φ(ω) E n! (n!) E E n=0 n=0 ∞ s X (2n − 1)!! nq 2 ≤ k T f k Φ(ω) (n!)2 m E n=0 ∞ n X 2 nq 2 ≤ k T f k Φ(ω) n! m E n=0
2 q 2kTmfk 2 = e · E Φ(ω) < ∞.
(We have used the Cauchy-Schwarz inequality and the moments of a Gaussian distribu- tion). hω,fi We have shown that E e Φ(ω) = 0 for any f ∈ DR(Tm) so by Theorem 5.3.2 we obtain Φ = 0 in Wm.
Lemma 5.3.7. Let f, g ∈ DR(Tm). Then
E[ehω,fi ehω,gi] = e(Tmf,Tmg). 5. The m Noise Space 37
Proof: hω,fi − 1 kT fk2 hω,fi E[e ] = e 2 m E[e ] = 1, (5.18)
since E[ehω,fi] is the moment generating function of the Gaussian random variable hω, fi 2 with variance kTmfk evaluated at 1. Thus we get hω,fi hω,gi (T f,T g) hω,f+gi (T f,T g) E[e e ] = e m m E[e ] = e m m .
The following formula is useful in calculating the Sm transform of the multiplication of two random variables, and can be easily proved using Lemma 5.3.7.