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-Ito

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

Ito formula.

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 Ito’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Contents 3

6.2 Relation to other white-noise extensions of Wick-Ito integral . . . . . . . 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. Ito in 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 Ito, 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 Ito integral, which satisfies similar calculus rules

as that of Ito. 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

Ito’s integral 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 Ito integral 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 Ito integral.

In the present thesis, I show that a Wick-Ito stochastic 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 ‖ · ‖1 and ‖ · ‖2, defined in a vector space V , will be called

comparable if there is constant C such that the inequality

‖v‖1 ≤ C‖v‖2

holds for all v ∈ V .

In the above definition the norm ‖ · ‖2 is considered to be stronger then ‖ · ‖1 in the sense

that every Cauchy sequence with respect to the ‖ · ‖2 is also a Cauchy sequence with

respect to ‖ · ‖1.

Definition 2.1.2. Two norms ‖ · ‖1 and ‖ · ‖2, defined in a vector space V , will be called

compatible if every Cauchy sequence vnn≥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 ,‖ · ‖1 and ‖ · ‖2 such that ‖ · ‖1 ≤ C‖ · ‖2, 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 ‖ · ‖1 and ‖ · ‖2 are compatible but not comparable, we can introduce a

2. Background 8

third norm ‖ · ‖3 defined by

‖v‖3 = max ‖v‖1, ‖v‖2 .

It is easy to verify that ‖ · ‖3 is comparable and compatible with the other two norms.

Thus, given any family ‖ · ‖nn≥1 of norms on V we may assume they satisfy the relation

‖ · ‖1 ≤ C2‖ · ‖2 ≤ ...,

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 ‖ · ‖nn≥1 we can define a

topology by the following system of neighborhoods of zero,

Uk,ε = v : ‖v‖1 < ε, ‖v‖2 < ε, ..., ‖v‖k < ε , k ∈ N, ε > 0.

We note that the topology defined in this way coincides with the topology in V defined

by the metric

‖v‖ =∞∑n=1

2−n‖v‖n

1 + ‖v‖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

a countable set of inner product norms ‖v‖n =√

(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

replaced by a new system of scalar products (·, ·)′n which does not alter the topology in

V , by setting

(v, w)′n =n∑k=1

(v, w)k , v, w ∈ V.

This new system has the property that

(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 ‖ · ‖n. So each Vn is a

Hilbert space and from the completeness of V it follows that

V =∞⋂n=1

Vn.

It is not hard to show that the dual space V ′ of a countably Hilbert space V equals

V ′ =∞⋃n=1

V ′n.

2.3 Nuclear spaces

Let V be a countably-Hilbert space associated with an increasing sequence ‖ · ‖nn =√(·, ·) of Hilbert norms. Denote Vn the completion of V with respect to the norm ‖ · ‖n.

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

continuous linear operator T nm : Vn −→ Vm. Note that T pm = T nmTpn if m ≤ n ≤ p.

Definition 2.3.1. A countably Hilbert space V is called nuclear, if for any m there is

an n such that the operator T nm is nuclear, i.e. has the form

T nmv =∞∑k=1

λk (v, vk)nwk, v ∈ Vn,

where vk and wk are orthonormal systems of vectors in the spaces Vn and Vm respec-

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 ′, strong and weak convergence coincide.

2. Background 10

4. V ′ is perfect relative to the topology of weak and strong convergence.

It follows that the σ-field on V ′ 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 −→ Cbe 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∑i,j=1

zizjC (vi − vj) ≥ 0.

Then there exist a unique probability measure P on V ′ such that

C(v) =

∫V ′ei〈x,v〉dP (x), v ∈ V.

Note that the above theorem is not true for a general real separable infinite dimensional

Hilbert space H. To see this, take C(v) = e−12‖v‖2

H and let e1, e2, ... be an orthonormal

basis for H. Then ∫H

ei(x,en)dP (x) = e−12 .

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 ‖ · ‖n of norms and let Vn be the completion of V with

respect to ‖ · ‖n. Suppose F : V × V → C is a bilinear continuous functional. Then there

exist n, p ≥ 1, and a Hilbert-Schmidt operator A from Vn into V ′p such that

F (u, v) = 〈Au, v〉, 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 H ′ using Riesz representation theorem:

each h ∈ H is identified as the element φh in H ′ defined by

φh(x) = 〈x, h〉H , x ∈ H.

Each h ∈ H can be further identified with an element of V ′, by

〈h, v〉V ′,V = 〈h, v〉H , v ∈ V.

Thus H is densely embedded in V ′ with respect to the weak topology of V ′. We get the

triple

V ⊂ H ⊂ V ′,

which is called a Gelfand triple. H is dense in V ′ with the weak topology of V ′.

Suppose that V is associated with a sequence ‖ · ‖n of norms and let Vn be the com-

pletion of V with respect to ‖ · ‖n. By setting H = V1, we get the continuous inclusion

V ⊂ ... ⊂ Vn+1 ⊂ Vn ⊂ ... ⊂ V1 ⊂ ... ⊂ V ′n ⊂ V ′n+1 ⊂ ... ⊂ V ′,

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,

lim|x|→∞

|(1 + |x|2

)α dβsdxβ

(x)| = 0.

The family of norms

‖s‖α,β =

(∫R|xα d

βs

dxβ(x)|dx

)1/2

, α, β ∈ N,

makes SR into a countably normed space. The space SR and its topological dual S ′R,

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,

where Dn(s) = dnsdxn

, is a continuous map of SR into itself. In addition,

P (D)s = P s, and P s = P (−D)s.

2. The Fourier transform is a continuous linear mapping of SR into itself [42, Theorem

7.4].

3. SR is a nuclear spaces, thus so is S ′R [20, section 3.6].

4. We have the Gelfand triple [41].

SR ⊂ L2 (R) ⊂ S ′R.

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, Holder’s inequality 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

Hn , 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

∞⊕n=0

Hn = L2 (Ω,F, P ) , (2.2)

which means that each elements X(ω) ∈ L2 (Ω,F, P ) has the representation

X(ω) =∞∑n=0

Xn(ω), Xn(ω) ∈ Hn.

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 ),

where πn is the orthogonal projection of L2 (Ω,F, P ) on Hn, 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

e−12‖s‖L2(R) defined on the nuclear space SR. Applying the Bochner-Minlos theorem 2.3.2

to it we obtain a probability measure P on S ′R such that

e− 1

2‖s‖2

L2(R) =

∫S ′R

e〈s′,s〉dP (s′), s ∈ SR. (3.1)

In accordance with the notation common in probability theory, we set

Ω := S ′R

and denote by ω the elements of Ω. The Borel sigma algebra is denoted by B.

The probability space (Ω,B(S ′R), P ) is called a white noise space. The measure P is

called the standard Gaussian measure on S ′R or the white noise measure. We also set

L2(Ω) , L2 (Ω,B(S ′R), P ) .

Taking s = εs1 with ε ∈ R in (3.1) and expanding both sides into a power series we may

conclude that for each s ∈ SR, 〈ω, s〉 is a normally distributed random variable with zero

mean and variance ‖s‖2L2(R). The isometric map s −→ 〈ω, s〉 from SR into L2(Ω) can be

extended to any function f ∈ L2(R) by taking a sequence sn in SR such that sn → f

in L2(R) and setting

〈ω, f〉 , limn→∞〈ω, sn〉

in L2(Ω). It follows that

H , 〈ω, f〉, 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) , 〈ω,1[0,t]〉, 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

hn(x) = (−1)ne12x2 d

n

dxn

(e−

12x2), n = 1, 2, ....

The Hermite functions are defined by

ηn(x) , π−14

e−12x2√

(n− 1)!hn−1

(√2x), n = 0, 1, 2, ....

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

Hα(ω) ,∞∏i=0

hαi (〈ω, ηi〉) .

The family Hαα∈J constitutes an orthogonal basis for L2(Ω). In addition, if α =

(α1, α2...) then we have

E[H2α

]= ‖Hα‖2

L2(Ω) = α! , α1!α2! · · · .

It follows that every X ∈ L2(Ω) can be decomposed as

X(ω) =∑α∈J

cαHα(ω), cα ∈ R,

and we have

E[X(·)2

]=∑α∈J

α!c2α.

3. Hida’s White Noise space 16

For example, the Brownian motion has the Wiener-Hermite expansion

B(t) = 〈ω,1[0,t]〉 = 〈ω,∞∑j=1

∫ t

0

ηj(u)du ηj〉 =∞∑n=1

∫ t

0

ηj(u)du〈ω, ηj〉 =∞∑n=1

∫ t

0

ηj(u)duHε(j),

(3.3)

where ε(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

eX(ω) ,∞∑n=0

Xn

n!.

For a Gaussian X(ω) = 〈ω, f〉 ∈ H, with f ∈ L2(Ω), it can be shown that

eX = eX−12E[X2] = exp

〈ω, f〉 − 1

2‖f‖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

Let X(ω) =∑

α∈J cαHα(ω). If ∑α∈J

α!c2α <∞ (3.4)

then X ∈ L2(Ω). Moreover, if Y (ω) =∑

α∈J bαHα(ω) then

E [X(·)Y (·)] =∑α

bαcαα!.

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

φ(ω) =∑α∈J

cαHα(ω) ∈ L2(Ω)

3. Hida’s White Noise space 17

such that

‖φ‖2ρ,k ,

∑α∈J

c2α (α!)1+ρ

∏j

(2j)kαj <∞ (3.5)

for all k ∈ N.

The corresponding Kondratiev space of stochastic distributions S−ρ consist of all formal

expansions

Φ =∑α∈J

bαHα(ω)

such that

‖Φ‖2−ρ,−k ,

∑α∈J

b2α (α!)1−ρ

∏j

(2j)−qαj <∞ (3.6)

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

〈Φ, φ, 〉 ,∑α∈J

bαcαα!, Φ ∈ S−ρ, φ ∈ Sρ, (3.7)

since for q large enough

∑α

|bαcα|α! =∑α

|bαcα| (α!)1−ρ2 (α!)

1+ρ2

(∏j

(2j)qαj/2)(∏

j

(2j)−qαj/2)

≤

(∑α

b2α (α!)1−ρ

(∏j

(2j)−qαj

))1/2(∑α

b2α (α!)1+ρ

(∏j

(2j)qαj

))1/2

<∞.

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 SR ⊂ L2(R) ⊂ S ′R that

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-Ito expansion of the Brownian motion (3.3) and take

3. Hida’s White Noise space 18

its formal time derivative. The resulting sum

∞∑n=1

ηj(t)Hε(j) (3.8)

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

(SX) (s) ,∫

Ω

X(ω)e〈ω,s〉dP (ω) = e− 1

2‖s‖2

L2(R)E[X(·)e〈·,s〉

], s ∈ SR.

Due to the translation invariance of the Gaussian measure P [23], it follows that

SX(s) =

∫Ω

X(ω + s)dP (ω).

In terms of the chaos expansion, we can express the S-transform of X =∑

α∈J aαHα as

(SX)(s) =∑α∈J

aα (s, η)α ,

where (s, η)α ,∏∞

i=1 (s, ηi)αiL2(R). This allows us to formally extend the definition of the

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-Ito type, since they all satisfy the Wick-Ito calculus rules that follows from Ito’s

formula.

3.4.1 Wiener Integral

The Wiener integral of a function f ∈ L2(R) is defined by the isometric map f −→ 〈ω, f〉which identifies L2(R) with the Gaussian Hilbert space 〈ω, f〉, 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∫RfdB(t) :=

∫Rfd

dt〈ω,1t〉 = 〈ω, ”

∫Rfd

dt1t”〉 = 〈ω, f〉.

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 Ito Integral

The Ito integral 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 Ito integral are given below.

Let X(t) be a non-anticipating stochastic process on L2(Ω,∨t≥0 Ft, P ). Denote

It(X) ,∫ t

0

x(s)dB(s),

where the left hand side is the Ito integral 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.

E[(It(X))2] = E

[∫ t

0

X(s)2ds

],

4. Ito’s rule: Let g : R −→ R be a function of class C2 and let X(t) = X0 +

3. Hida’s White Noise space 20

∫ t0f(s)dB(s). Then,

g(X(t)) = g(X(0)) +

∫ t

0

g′(X(s))f(s)dB(s) +1

2

∫ t

0

g′′(X(s))(f(s))2ds, a.s. P.

(3.9)

We note that the Ito integral 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 Ito integral 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 ∫ ∞−∞

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∫ES (X(t)) (s) d

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∫E

X(t)δB(t) , S−1

(∫E

S (X(t)) (s)d

dt(s,1t) 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

Km(t, s) =

∫R

eiξt − 1

ξ

e−iξs − 1

ξm(ξ)dξ

where m is a positive measurable even function subject to∫R

m(u)

ξ2 + 1dξ <∞. (4.1)

Note that Km(t, s) can also be written as

Km(t, s) = r(t) + r(s)− r(t− s),

where

r(t) =

∫R

1− cos(tξ)

ξ2m(ξ)dξ.

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

r(t) = r0 + iγt+

∫R

eiξt − 1− iξt

ξ2 + 1

dσ(ξ)

ξ2,

where σ is an increasing right continuous function subject to∫Rdσ(ξ)ξ2+1

<∞. See [36], [33],

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

Tmf(ξ) =√m(ξ)f(ξ), (4.2)

with domain

D(Tm) =

f ∈ L2(R) ;

∫Rm(ξ)|f(ξ)|2dξ <∞

,

where f(ξ) = 1√2π

∫R e−iξtf(t)dt denotes the Fourier transform of f . Clearly, the Schwartz

space SR of real smooth rapidly decreasing functions belong to the domain of Tm. The

indicator functions

1t =

1[0,t], t ≥ 0,

−1[t,0] t ≤ 0,

also belong to D(Tm). In [2], and with some restrictions on m, a centered Gaussian

process Bm with covariance function Km(t, s) = (Tm1t, Tm1s)L2(R) was constructed in

Hida’s white noise space. In the present work we chose a different path. We define the

characteristic functional

Cm(s) = e−‖Tms‖2

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

SX(s) = E[X(·)e〈·,s〉

]e−

12‖s‖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∫E

X(t)δB(t) , S−1

(∫E

S (X(t)) (s)s(t)dt

).

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 ,

(SΦ(t)) (s) =

∫E

S (X(t)) (s)s(t)dt.

Since s(t) = ddt

(s,1t)L2(R), it suggests to extend the last definition of the integral by

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.

We note that when m(ξ) = |ξ|1−2H , and H ∈ (12, 1), the operator Tm reduces, up to a

multiplicative constant, to the operator MH defined in [16] and in [7]. The set L2φ 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 T ∗mTm via Schwartz’ kernel theorem, with m = MH . In the general case,

the kernel associated to the operator T ∗mTm is not a function. This last remark is the

source for some of the difficulties arises in extending Wick-Ito integration for fractional

Brownian motion such as the distinction between the cases H < 12, H > 1

2and H = 1

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 Chapter 6 we define a Wick-Ito type stochastic integral

with respect to Bm, and prove an associated Ito formula. 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 SR to be the space of real-valued Schwartz functions, and Ω = S ′R. We denote

by B the associated Borel sigma algebra. Throughout this paper, we denote by 〈·, ·〉 the

duality between S ′R and SR, and by (·, ·) the inner product in L2(R). In case there is no

danger of confusion, the L2(R) norm will be denoted as ‖ · ‖.

Theorem 5.1.1. There exists a unique probability measure µm on (Ω,B) such that

e−‖Tms‖2

2 =

∫Ω

ei〈ω,s〉dµm(ω), s ∈ SR,

Proof: The function Cm(s) is positive definite on SR since

Cm(s1 − s2) = exp

−1

2‖Tms1‖2

× exp (Tms1, Tms2) × exp

−1

2‖Tms2‖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 =∫Rm(u)1+u2

du and s](u) = s(−u). We have for s ∈ S :

‖Tms‖2 =

∫R|s(u)|2m(u)du

=

∫R|(1 + u2)s(u)|2 m(u)

1 + u2du

≤ K

(∫R|s ? s]|(ξ)dξ +

∫R|s′ ? (s])′|(ξ)dξ

)≤ K

((∫R|s(ξ)|dξ

)2

+

(∫R|s′(ξ)|dξ

)2),

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:

E[〈ω, s〉2] = ‖Tms‖2. (5.1)

Proof: We have

e−12‖Tms‖2=

∫Ω

ei〈ω,s〉dµm(ω). (5.2)

Expanding both sides of (5.2) in power series for εs we obtain

E [〈ω, s〉] =

∫Ω

〈ω, s〉dµm(ω) = 0. (5.3)

and

E[〈ω, s〉2

]=

∫S ′〈ω, s〉2dµm(ω) = ‖Tms‖2. (5.4)

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

(f, g)m ,∫Rf g∗m(ξ)dξ.

Note that D(Tm) is consist of those functions f in L2(R) that satisfy

‖f‖2m = (f, f)m <∞.

We define the space LSm and Lm to be the closure of S and D(Tm) in the norm ‖ · ‖m,

respectively.

Proposition 5.1.3. We have

Lm = LSm.

Proof: Let f ∈ Dm be orthogonal to any s ∈ S in the norm ‖ · ‖m, i.e.

0 = (s, f)m =

∫Rsf ∗m(ξ)dξ, ∀s ∈ S .

5. The m Noise Space 29

It follows that f ∗m = 0 almost everywhere since it defines the zero distribution on S .

But that also means ∫Rf f ∗m(ξ)dξ = 0

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

Tmf =√m(f)] = (

√mf)] =

(Tmf

)]= Tmf.

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

limn→∞

‖sn − f‖m = 0. (5.6)

In view of (5.5), and since f is real-valued we have

limn→∞

‖sn − f‖m = limn→∞

‖Tmsn − Tmf‖L2(R) = limn→∞

‖Tmsn − Tmf‖L2(R) = 0. (5.7)

Together with (5.6) this last equation leads to

limn→∞

‖Tm(Re sn)− Tmf‖L2(R) = 0. (5.8)

In particular (Tm(Re sn))n∈N is a Cauchy sequence in L2(R). By (5.1), (〈ω,Re sn〉)n∈Nis a Cauchy sequence in Wm. We denote by 〈ω, f〉 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

E[〈ω, f〉2] = ‖Tmf‖2L2(R), f ∈ DR(Tm). (5.9)

5. The m Noise Space 30

leads to

E [〈ω, f〉〈ω, g〉] = Re (Tmf, Tmg)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 [〈ω, f〉〈ω, g〉] = (Tmf, Tmg) . (5.10)

Proposition 5.1.6. 〈ω, f〉, f ∈ DR(Tm) is a Gaussian process in the sense that for

any f1, ..., fn ∈ DR(Tm) and a1, ..., an ∈ R, the random variable∑n

i=1 ai〈ω, fi〉 has a

normal distribution.

Proof: By (5.2), for λ ∈ R we have,

E[eiλ∑ni=1 ai〈ω,fi〉] =

∫Ω

eiλ∑ni=1 ai〈ω,fi〉dµm(ω)

=

∫Ω

ei〈ω,λ∑ni=1 aifi〉dµm(ω)

= e−12λ2‖

∑ni=1 aiTmfi‖2 .

(5.11)

In particular, we have that for any ξ1, ..., ξn ∈ DR (Tm) such that Tmξ1, ..., Tmξn are

orthonormal in L2 (R) and for any φ ∈ L2(Rn)

E [φ (〈ω, ξ1〉, ...〈ω, ξ1〉)] =1

(2π)n2

∫Rnφ(x1, ..., xn)

n∏i=1

e−12xi

2

dx1 · ... · dxn. (5.12)

Definition 5.1.7. We set G to be the σ-field generated by the Gaussian elements

〈ω, f〉, 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) , 〈ω,1t〉.

This process plays the role of the Brownian motion for the stochastic integral and the

Ito formula 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

Km(t, s) =

∫R

eiξt − 1

ξ

e−iξs − 1

ξm(ξ)dξ = (Tm1t, Tm1s) . (5.13)

3. The process Bm has a continuous version under the condition∫R

m(ξ)

1 + |ξ|dξ <∞ (5.14)

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 [〈ω,1t〉〈ω,1s〉]

= 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,

E[(Bm(t)−Bm(s))2] = E

[〈·,1[s,t]〉2

]= 2

∫R

1− cos ((t− s)ξ)ξ2

m(ξ)dξ,

5. The m Noise Space 32

where the last equality follows by vanishing imaginary part of (5.13). We now compute

2

∫ 1

0

1− cos(tξ)

ξ2m(ξ)dξ = 2

∫ 1

0

t22 sin

(tξ2

)2

ξ2t2m(ξ)dξ

≤ C1t2 (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, ∫ ∞1

1− cos(tξ)

ξ2m(ξ)dξ = t

∫ ∞1

sin(tθt)m(ξ)

ξdξ ≤ t

∫ ∞1

m(ξ)

ξdξ ≤ C2t,

where we have used (5.14) for the last move. Since Bm(t)−Bm(s) is zero mean Gaussian,

we obtain

E[(Bm(t)−Bm(s))4] = C3E

[(Bm(t)−Bm(s))2] ≤ C4 (t− s)2 .

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

Km1(t, s) =1√2π

∫ ∆

−∆

1− cos(tξ)− cos(sξ) + cos(ξ(t− s))ξ2

dξ, t, s ∈ R.

The time derivative of this process also belongs to Wm, and is a stationary Gaussian

process with covariance

∂2

∂t∂sKm1 (t, s) =

1√2π

∫ ∆

−∆

ei(t−s)ξdξ =2 sin (∆(t− s))

t− s. (5.15)

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) andExample 5.2.3 (right).

pass filter with cut-off frequency ∆. We see from the covariance function (5.15) that each

time sample t0 ∈ R is positively correlated with time samples in the interval[t0, t0 + π

2∆

),

negatively correlated with time samples in the interval(t0 + π

2∆, t0 + 3π

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

Km2(t, s) =1√2π

∫ ∆

−∆

1− cos(tξ)− cos(sξ) + cos ((t− s)ξ)ξ2

|ξ|1−2H , t, s ∈ R.

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∫ τ

0

f(t)dBm(t) , 〈ω,1τf〉. (5.16)

5. The m Noise Space 34

Note that since ∫Rm(ξ)|f(ξ)|2dξ ≤ sup

ξ∈R(1 + ξ2)|f(ξ)|2

∫R

m(ξ)

1 + ξ2dξ,

a sufficient condition for a function f ∈ L2 (R) to be in the domain of Tm is

supξ∈R

(1 + ξ2)|f(ξ)|2 ≤ ∞.

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∫ b

a

XtdB(t) =

∫ b

a

Xt Bmdt

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:

e〈ω,s〉 , e〈ω,s〉−12‖Tms‖2

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

(SmΦ)(s) ,∫

Ω

e〈ω,s〉Φ(ω)dµm(ω) = E[e〈ω,s〉Φ(ω)

], s ∈ SR.

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 ξnn∈N ⊂ SR be a countable dense set in L2(R) and denote by Gn the σ-field

generated by 〈ω, ξ1〉, ..., 〈ω, ξn〉. We may choose ξnn∈N such that Tmξnn∈N are

orthonormal. For every n ∈ N, E [Φ|Gn] = φn (〈ω, ξ1〉, ..., 〈ω, ξn〉) for some measurable

function φn : Rn −→ R such that

EΦ =

∫· · ·∫

Rn

φn(x)e−12x′xdx <∞,

where x′ denotes the transpose of x; see for instance [9, Proposition 2.7, p. 7]. Thus, for

t = (t1, ..., tn) ∈ Rn, using (5.12) we obtain

0 =

∫Ω

e〈ω,∑nk=1 tkξk〉Φ(ω)dµm =

∫Ω

e〈ω,∑nk=1 tkξk〉E [Φ|Gn] dµm(ω)

= e−12

∑nk=1 t

2k‖Tmξk‖

2

∫Ω

e∑nk=1 tk〈ω,ξk〉φn (〈ω, ξ1〉, ..., 〈ω, ξn〉) dµm(ω)

= e−12

∑nk=1 t

2k‖Tmξk‖

2 1

(2π)n2

∫· · ·∫

Rn

e∑nk=1 tkxkφn (x1, ..., xn) e−

12

∑nk=1 x

2kdx1...dxn

=

∫· · ·∫

Rn

φn (x) e−12

(x−t)′(x−t)dx.

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 ∈ N. Since⋃n∈N Gn = G we have Φ = 0.

Definition 5.3.3. A stochastic exponential is a random variable of the form

e〈ω,f〉, f ∈ DR (Tm) .

We denote by E the family of linear combinations of stochastic exponentials.

5. The m Noise Space 36

Since e〈ω,f〉 = e−12‖Tmf‖2e〈ω,f〉, 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 (〈ω, f1〉, ..., 〈ω, f2〉) , 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 〈ω, f〉 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),

E[e〈ω,f〉Φ(ω)

]= E

[∞∑n=0

〈ω, f〉n

n!Φ(ω)

]=∞∑n=0

E [〈ω, f〉nΦ(ω)]

n!= 0, (5.17)

where interchanging of summation is justified by Fubini’s theorem since

∞∑n=0

E[| 〈ω, f〉

n

n!Φ(ω) |

]≤

∞∑n=0

1

(n!)

√E [〈ω, f〉2n]E

[Φ(ω)2]

≤∞∑n=0

√(2n− 1)!!

(n!)2‖ Tmf ‖n

√E[Φ(ω)2]

≤∞∑n=0

2n

n!‖ Tmf ‖n

√E[Φ(ω)2]

= e2‖Tmf‖2 ·√E[Φ(ω)2] <∞.

(We have used the Cauchy-Schwarz inequality and the moments of a Gaussian distribu-

tion).

We have shown that E[e〈ω,f〉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[e〈ω,f〉 e〈ω,g〉] = e(Tmf,Tmg).

5. The m Noise Space 37

Proof:

E[e〈ω,f〉] = e−12‖Tmf‖2E[e〈ω,f〉] = 1, (5.18)

since E[e〈ω,f〉] is the moment generating function of the Gaussian random variable 〈ω, f〉with variance ‖Tmf‖2 evaluated at 1.

Thus we get

E[e〈ω,f〉 e〈ω,g〉] = e(Tmf,Tmg)E[e〈ω,f+g〉] = e(Tmf,Tmg).

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.

Sm(e〈ω,f〉 e〈ω,g〉

)= e(Tms,Tmf)e(Tms,Tmf)e(Tms,Tmg), f, g ∈ DR(Tm). (5.19)

Proposition 5.3.8. Let Φn be a sequence in Wm that converges in Wm to Φ. Then

for any s ∈ SR the sequence of real numbers Sm (Φn) (s) converges to Sm (Φ) (s).

Proof: For any s ∈ SR,

|SmΦn(s)− SmΦ(s)| = |E[e〈ω,s〉(Φn − Φ)

]| ≤

√E[(e〈ω,s〉)

2]·√E[(Φn − Φ)2].

By direct calculation E[(e〈ω,s〉

)2]

= e‖Tms‖2

and since E[(Φn − Φ)2] −→ 0, the claim

follows.

In the statement of Theorem 5.3.9 recall that Tm is a continuous operator from SR into

L2(R) and so its adjoint is a continuous operator from L2(R) into S ′R.

Theorem 5.3.9. For Φ ∈Wm and s ∈ SR,

SmΦ(s) =

∫Ω

Φ(ω + T ∗mTms)dµm(ω).

Proof: Assume first that Φ (ω) = e〈ω,s1〉 where s1 ∈ SR. We have by Lemma 5.3.7

that ∫Ω

Φ (ω + T ∗mTms) dµm (ω) =

∫Ω

e〈ω,s1〉e〈T∗mTms,s1〉dµm (ω)

= e〈T∗mTms,s1〉

∫Ω

e〈ω,s1〉dµm (ω)

= e(Tms,Tms1) · 1

= SmΦ(s).

5. The m Noise Space 38

The result may be extended by linearity and by Proposition 5.3.8 to any Φ ∈ E, which

by Propositions 5.3.4 is a dense subset of Wm.

We can find the Sm transform of powers of 〈ω, f〉 for f ∈ DR(Tm) by the formula for

Hermite polynomials.

Corollary 5.3.10. For f ∈ DR(Tm) and s ∈ SR, we have that

(Tms, Tmf)n = n!

bn/2c∑k=0

(−1

2

)k (Sm〈ω, f〉n−2k

)(s) ‖Tmf‖2k

k!(n− 2k)!, (5.20)

in particular

(Sm〈ω, f〉)(s) = (Tmf, Tms) (5.21)

and

(Sm〈ω, f〉2)(s) = (Tmf, Tms)2 + ‖Tms‖2 (5.22)

Proof: From Lemma 5.3.7 we get that

(Sme〈ω,f〉)(s) = e(Tms,Tmf),

then,

e−12‖Tmf‖2Sm

(∞∑k=0

〈ω, f〉k

k!

)(s) =

∞∑k=0

(Tms, Tmf)k

k!(5.23)

By the linearity of the Sm transform and Fubini’s theorem, and replacing f by tf with

t ∈ R we compare powers of t at both sides to get (5.20).

This last corollary can be also formulated in terms of the Hermite polynomials. Recall

that the nth Hermite polynomial with parameter t ∈ R is defined by

h[t]n (x) , n!

bn/2c∑k=0

(−1

2

)kxn−2k · t2k

k!(n− 2k)!(5.24)

(see for instance [35, p. 33]). For f ∈ D(Tm) we define

hn (〈ω, f〉) , h[‖Tmf‖]n (〈ω, f〉) = n!

bn/2c∑k=0

(−1

2

)k 〈ω, f〉n−2k · ‖Tmf‖2k

k!(n− 2k)!, (5.25)

and we also set h0 = 1.

5. The m Noise Space 39

So by (5.20) we have that (Smhn (〈ω, f〉)

)(s) = (Tms, Tmf)n . (5.26)

Using Equation 5.20 and Lemma 5.3.7, one can easily verify the following result:

Proposition 5.3.11. Let f ∈ DR(Tm). It holds that:

e〈ω,f〉 =∞∑k=0

hk (〈ω, f〉)k!

(5.27)

It is possible to define a Wick product in Wm using the Sm transform.

Definition 5.3.12. Let Φ,Ψ ∈ Wm. The Wick product of Φ and Ψ is the element

Φ Ψ ∈Wm that satisfies

ST (Φ Ψ)(s) = (STΦ)(s)(STΨ)(s), ∀s ∈ SR,

if it exists.

As this definition suggests, in general the Wick product is not stable in Wm.

From (5.26), the Wick product of Hermite polynomials satisfies

hn (〈ω, f〉) hk (〈ω, f〉) = hn+k (〈ω, f〉) , n, k ∈ N, f ∈ DR(Tm).

6. STOCHASTIC INTEGRATION IN WM

We now use the Sm-transform to define a Wick-Ito type stochastic integral which can be

seen as a version of Hitsuda-Skorokhod integral in Wm, and prove an Ito formula for this

integral. In the next section we also show that for particular choices of m, our definition

of the stochastic integral coincides with previously defined Wick-Ito stochastic integrals

for fractional Brownian motion; see [13, 8]. We set

Bs(t) = Sm (Bm(t)) (s).

By (5.21) we see that

Bs(t) = (Tms, Tm1t) =

∫Rm(ξ)s(ξ)

eiξt − 1

ξdξ. (6.1)

This function is absolutely continuous with respect to Lebesgue measure and its derivative

is

(Bs(t))′ =

∫Rm(ξ)s(ξ)eiξtdξ. (6.2)

We note that when Tm is a bounded operator from L2(R) into itself we have by a result

of Lebesgue (see [43, p. 410]), (Bs(t))′ = (T ∗mTms)(t) (a.e.).

Definition 6.0.13. Let M ∈ R be a Borel set and let X : M −→Wm be a stochastic pro-

cess. The process X will be called integrable over M if for any s ∈ SR, (SmXt) (s)Bs(t)′

is integrable on M , and if there is a Φ ∈Wm such that

SmΦ(s) =

∫M

(SmXt) (s)Bs(t)′dt.

for any s ∈ SR. If X is integrable, Φ is uniquely determined by Theorem 5.3.2 and we

denote it by∫MXtdBm (t).

If T = IdL2(R), this definition coincides with the Hitsuda-Skorokhod integral [24, Chapter

8]. See also Section 6.

6. Stochastic integration in Wm 41

Note that since

|Bs(t)′| ≤

∫Rm(ξ)|s(ξ)|dξ ≤ sup

ξ|(1 + ξ2)s(ξ)|

∫R

m(ξ)

1 + ξ2dξ <∞,

for any s ∈ S there exists a constant Ks such that

|∫M

SmXt(s)Bs(t)′dt| ≤ Ks

∫M

|E[Xte

〈ω,s〉] |dt≤ KsE

[e2〈ω, s〉

] ∫M

E[X2t ]dt.

Thus a sufficient condition for the integrability of SmXt(s)Bs(t)′ is∫ME[X2

t ]dt <∞.

Theorem 6.0.14. Any non-random f ∈ DR(Tm) is integrable and we have,∫ τ

0

f(t)dBm(t) = 〈ω,1[0,τ ]f〉. (6.3)

Proof: By virtue of (5.21) and the definition of the stochastic integral, we need to show

that ∫ τ

0

f(t)Bs(t)′dt =

(Tms, Tm1[0,τ ]f

).

Using formula (6.2) and Fubini’s theorem, we have:∫ τ

0

f(t)Bs(t)′dt =

∫ τ

0

f(t)

(∫Rm(ξ)s(ξ)eiξtdξ

)dt

=

∫Rm(ξ)s(ξ)

(∫ τ

0

f(t)eitξdt

)dξ

=

∫Rm(ξ)s(ξ)

(f1[0,τ ](ξ)

)dξ

=(Tms, Tm1[0,τ ]f

).

Proposition 6.0.15. The stochastic integral has the following properties:

1. For 0 ≤ a < b ∈ R,

Bm (b)−Bm (a) =

∫ b

a

dBm(t)

6. Stochastic integration in Wm 42

2. Let X : M −→Wm be an integrable process. Then∫M

XtdBm(t) =

∫R

1MXtdBm(t).

3. Let X : M −→Wm an integrable process. Then

E[∫

M

XtdBm(t)

]= Sm

(∫M

XtdBm(t)

)(0) = 0

.

4. The Wick product and the stochastic integral can be interchanged: Let X : R −→Wm an integrable process and assume that for Y ∈Wm, Y Xt is integrable. Then,

Y ∫RXtdBm(t) =

∫RY XtdBm(t)

Proof: The proof of the first three items is easy and we omit it. The last item is proved

in the following way:

Sm(Y ∫RXtdBm(t))(s) = (SmY )(s)

∫M

(SmXt)(s)dBm

=

∫M

(SmY )(s)(SmXt)(s)dBm

= Sm(

∫RY XtdBm(t))(s).

Example 6.0.16. For τ ≥ 0, we have by equation (5.22),∫ τ

0

(Tms, Tm1t)d

dt(Tms, Tm1t) dt =

1

2(Tms, Tm1τ )

2

=1

2Sm(〈ω,1t〉 − ‖Tm1t‖2

)(s).

Then Bm is integrable on the interval [0, τ ], and we have∫ τ

0

Bm(t)dBm(t) =1

2Bm(τ)2 − 1

2‖Tm1τ‖2.

This reduces to the well known result if m (ξ) ≡ 1 and Tm is then the identity operator.

6. Stochastic integration in Wm 43

Example 6.0.17. Let hn be defined by (5.25). A similar argument to the one in (6.2)

will show that any f such that for any f1t ∈ D(Tm) we have that the function t 7→(Tms, Tmf1t) is differentiable with time derivative

d

dt(Tms, Tmf1t) = f(t)

∫Rm(ξ)s(ξ)e−itξdξ = f(t)Bs(t)

′.

By a similar argument to Theorem 6.0.14 we have

1

n+ 1Sm

(hn+1 (〈ω,1τf〉) (t)

)(s) =

1

n+ 1(Tms, Tm1τf)

=

∫ τ

0

(Tms, Tmf1t)n f(t)Bs(t)

′dt

= Sm

(∫ τ

0

f(t)hn (〈ω,1tf〉) dBm (t)

)(s).

Thus, ∫ τ

0

f(t)hn (〈ω,1tf〉) dBm (t) =1

n+ 1hn+1 (〈ω,1τf〉) . (6.4)

It follows from (6.4) that for any polynomial p and f with 1tf ∈ D(Tm) the process

p(〈ω,1tf〉) is integrable. This result can be easily extended to the process

t 7→ e〈ω,1tf〉, 1tf ∈ D(Tm),

and we also obtain:

Corollary 6.0.18. ∫ τ

0

f(t)e〈ω,1tf〉dBm(t) = e〈ω,1tf〉 − 1.

Example 6.0.19. Let f ∈ DR(Tm). Using (5.19) we can obtain

Sm

(e〈ω,f〉

∫ τ

0

e〈ω,1t〉dBm(t)

)(s) = Sm

(e〈ω,f〉 e〈ω,1τ 〉 − e〈ω,f〉

)(s)

= e(Tms,Tmf)(e(Tms,Tm1τ )e(Tmf,Tm1τ ) − 1

).

On the other hand,

Sm

(∫ τ

0

e〈ω,f〉 e〈ω,1t〉dBm(t)

)(s) = e(Tms,Tmf)

∫ τ

0

e(Tms,Tm1t)e(Tmf,Tm1t)d

dt(Tms, Tm1t) dt

= e(Tms,Tmf)(e(Tms,Tm1τ )e(Tmf,Tm1τ ) − 1

)−∫ τ

0

e(Tms,Tm1t)e(Tmf,Tm1t)d

dt(Tmf, Tm1t) dt.

6. Stochastic integration in Wm 44

So in general for an integrable stochastic process X and a random variable Y we have

the undesirable result

Y

∫ τ

0

XtdBm(t) 6=∫ τ

0

Y XtdBm(t).

6.1 Ito’s formula

In this section we prove an Ito’s formula. We begin by proving an extension of the

classical Girsanov theorem to our setting.

Theorem 6.1.1. Let f ∈ D(Tm), and let µ be the measure defined by µ(A) = E[e〈ω,f〉1A].

The process

Bm(t) , Bm(t)− (Tmf, Tm1t),

is Gaussian and satisfies

Eµ[Bm(t)Bm(s)] = (Tm1t, Tm1s).

Proof: We will first prove that for all t ≥ 0, Bm(t) is a Gaussian random variable relative

to the measure µ by considering its moment generating function Eµ[eλBm(t)

], λ ∈ R,

Eµ[eλBm(t)

]= E

[e〈ω,f〉−

12‖Tmf‖2eλ〈ω,1t〉−λ(Tmf,Tm1t)

]= e−λ(Tmf,Tm1t) e−

12‖Tmf‖2E

[e〈ω,f+λ1t〉

].

(6.5)

Since 〈ω, f + λ1t〉 is a zero mean Gaussian random variable with variance

‖Tm (f + λ1t) ‖2 = ‖Tmf‖2 + λ2‖Tm1t‖2 + 2λ (Tmf, Tm1t) ,

its moment generating function evaluated at 1 is given by

E[e〈ω,Tmf+λ1t〉

]= e

12‖Tmf‖2e

12λ2‖Tm1t‖2eλ(Tmf,Tm1t), (6.6)

and we conclude from (6.5) that

Eµ[eλBm(t)

]= e

12λ2‖Tm1t‖2 . (6.7)

Thus for all t ≥ 0, Bm(t) is a zero mean Gaussian random variable on (Ω,G, µ). Similar

arguments will show that any linear combination of time samples is a Gaussian variable,

6. Stochastic integration in Wm 45

and so Bm(t), t ≥ 0 is a Gaussian process. Finally, by the polarization formula,

Eµ[Bm(t)Bm(s)] = (Tm1t, Tm1s).

We now interpret integrals of the type∫ τ

0Φ(t)dt, where for every t ∈ [0, τ ], Φ(t) ∈ Wm,

as Pettis integrals, that is as

E[(∫ τ

0

Φ(t)dt

)Ψ

]=

∫ τ

0

E[Φ(t)Ψ]dt, ∀Ψ ∈Wm,

under the hypothesis that the function t 7→ E[Φ(t)Ψ] belongs to L1([0, τ ], dt) for every

Ψ ∈ Wm. See [25, pp. 77-78]. We note that if X is moreover pathwise integrable and

such that the pathwise integral belongs to Wm, then∫ τ

0

E[|Xt|]dt <∞,

and we can apply Fubini’s theorem to show that the Pettis integral coincides with the

pathwise integral. It is also clear from the definition of the Pettis integral that it com-

mutes with the Sm transform.

We introduce the conditions

E[|F (t,Xt)|e〈ω,s〉

]< ∞ (6.8)

E[|∂F∂t

(t,Xt)|e〈ω,s〉]

< ∞ (6.9)

E[|∂F∂x

(t,Xt)|e〈ω,s〉]

< ∞ (6.10)

for F ∈ C1,2 ([0,∞) ,R).

We shall now develop an Ito formula for a class of stochastic processes of the form,

Xt (ω) =

∫ τ

0

f(t)dBm(t) = 〈ω,1τf〉, τ ≥ 0, 1τf ∈ D (Tm) . (6.11)

Theorem 6.1.2. Let F ∈ C1,2 ([0,∞) ,R) satisfying (6.8)-(6.10), and assume that the

function ‖Tm1tf‖2 is absolutely continuous with respect to the Lebesgue measure as a

function of t. Then we have,

6. Stochastic integration in Wm 46

F (τ,Xτ )− F (0, 0) =

∫ τ

0

∂

∂tF (t,Xt)dt+

∫ τ

0

f(t)∂

∂xF (t,Xt) dBm (t) (6.12)

+1

2

∫ τ

0

d

dt‖Tm1tf‖2 ∂

2

∂x2F (t,Xt)dt

in Wm.

This proof is based on the proof for Ito’s formula for the S-transform approach to Hitsuda-

Skorokhod integration in the standard white noise space found in [35, Section 13.5].

Proof: Let s ∈ SR and f ∈ D (Tm). It follows from Theorem 6.1.1 that for every

t ∈ [0, τ ], Xt(ω) = 〈ω,1tf〉 is normally distributed under the measure

µs(A) , E[1A exp

〈ω, s〉 − 1

2‖Tms‖2

]= E

[1Ae

〈ω,s〉] ,with mean

(Tms, Tm1tf)

and variance

‖Tm1tf‖2.

Thus,

(SmF (t,Xt)) (s) = E[e〈ω,s〉F (t,Xt)

](6.13)

=

∫RF (t, u+ (Tm1tf, Tms)) ρ

(‖Tm1tf‖2, u

)du,

where ρ(w, u) = 1√2πw

e−u2

2w and satisfies,

∂

∂wρ =

1

2

∂2

∂u2ρ. (6.14)

Integrating by part we obtain:∫RF (t, u)

∂2

∂u2ρ(w, u)du =

∫R

∂2

∂u2F (t, u)ρ(w, u)du. (6.15)

In view of (6.8)-(6.10) we may differentiate under the integral sign by (6.13), (6.14) and

6. Stochastic integration in Wm 47

(6.15) and obtain for 0 ≤ t ≤ τ ,

d

dtSm (F (t,Xt)) (s) =

∫R

∂

∂tF (t, u+ (Tm1tf, Tms)) ρ

(‖Tm1tf‖2, u

)du

+

∫R

∂

∂xF (t, u+ (Tm1tf, Tms))

d

dt(Tm1tf, Tms) ρ

(‖Tm1tf‖2, u

)du

+

∫RF (t, u+ (Tm1tf, Tms))

d

dt‖Tm1tf‖2 ∂

∂tρ(‖Tm1tf‖2

)du

= Sm

(∂

∂tF (t,Xt)

)(s) +

d

dt(Tms, Tm1tf) Sm

(∂

∂xF (t,Xt)

)(s)

+1

2

d

dt‖Tm1tf‖2 · Sm

(∂2

∂x2F (t,Xt)

)(s).

Hence,

Sm (F (τ,Xτ )− F (0, 0)) (s) =

∫ τ

0

Sm

(∂

∂tF (t,Xt)

)(s)dt (6.16)

+

∫ τ

0

d

dt(Tms, Tm1tf) Sm

(∂

∂xF (t,Xt)

)(s)dt

+1

2

∫ τ

0

d

dt‖Tm1tf‖2 · Sm

(∂2

∂x2F (t,Xt)

)(s)dt.

By the definition of the stochastic integral,

Sm

(∫ τ

0

f(t)∂

∂xF (t,Xt) dBm(t)

)(s)

=

∫ τ

0

Sm

(∂

∂xF (t,Xt)

)(s)f(t)Bs(t)dt,

which in view of Example 6.0.17 equals∫ τ

0

d

dt(Tms, Tm1tf) Sm

(∂

∂xF (t,Xt)

)(s)dt.

Thus we may now use Fubini’s theorem to interchange the Sm-transform and the pathwise

integral, and obtain that the Sm-transform of the right hand side of (6.12) is exactly the

right hand side of (6.16) and the theorem is proved.

6.2 Relation to other white-noise extensions of Wick-Ito integral

Recall that the white noise space corresponds to m(ξ) ≡ 1, so denoting it W1 is consistent

with our notation, and S1 is the classical S-transform of the white noise space. We define

6. Stochastic integration in Wm 48

a map Tm : Wm −→W1 by describing its action on the dense set of stochastic polynomials

in Wm:

Tm〈ω, f〉n = 〈ω, Tmf〉n, f ∈ D (Tm) .

Note that since the range of Tm is contained in D(T1) = D(idL2(R)) = L2(R), this map is

well defined. It is easy to see that Tm is an isometry of Hilbert spaces. By continuity we

obtain that

Tme〈ω,f〉 = e〈ω,Tmf〉, f ∈ D (Tm) ,

hence (S1Tme

〈ω,f〉)

(Tms) = e(Tms,Tmf) =(Sme

〈ω,f〉) (s).

So this relation between S1 and Sm is extended such that for any Φ ∈Wm,(S1TmΦ

)(Tms) = (SmΦ) (s).

Let X : [0, τ ] −→ Wm be a stochastic process. We have defined its Ito integral as the

unique element Φ ∈Wm (if exists) having Sm-transform

(SmΦ) (s) =

∫ τ

0

(Xt) (s)d

dt(Tms, Tm1t) (s)dt.

This suggests that if we define in the white noise the process Bm as 〈ω, Tm1t〉 and

stochastic integral with respect to Bm as the unique element Φ ∈ W1(if exists) having

S1-transform

(S1Φ) (s) =

∫ τ

0

(Xt) (s)d

dt(s, Tm1t)) (s)dt, (6.17)

both definitions coincides in the sense that

Tm

∫ τ

0

XtdBm(t) =

∫ τ

0

TmXtdBm(t). (6.18)

Recall that the fractional brownian motion can be obtained in our setting by taking

m (ξ) = 12|ξ|1−2H , H ∈ (0, 1), which results in Tm = MH , where MH is defined in [16]. In

the white noise, the fractional Brownian motion can be defined by the continuous version

of the process 〈ω,MH1t〉t≥0.

An approach that is based on the definition described in (6.17) for the fractional Brownian

motion was given in [5]. Due to Theorem 3.4 there, under appropriate conditions our

definition of the Hitsuda-Skorokhod integral in the case of Tm = MH coincides in the

sense of (6.18) with the Hitsuda-Skorokhod integral defined there. Stochastic integration

6. Stochastic integration in Wm 49

in the white noise setting for the family of stochastic processes considered in this paper

can be found in [1], and its equivalence to the integral described here can be obtained by

a similar argument to that of Theorem 3.4 in [5].

7. APPLICATION IN OPTIMAL CONTROL

The study of continuous time systems driven by noisy input has been focused since its

beginning primarily on the case of white noise and Brownian motion. This is mainly

thanks to the fact that the Brownian motion arises as the limit distribution of many

discrete time models and physical processes, what makes it a reasonable model for ran-

domness in many applications.

Motivated from some applications in hydrology, telecommunications, queueing theory

and mathematical finance, there has been a recent interest in input noises without in-

dependent increments. The fractional Brownian motion and the fractional noise are of

particular interest and we refer to [15], [11], [37] and the book [6] for recent development.

Note that the Ito formula (6.12) can be rewritten in differential form as

dF (Xt, t) =∂

∂tF (t,Xt)dt+ f(t)

∂

∂xF (Xt, t)dBm(t) +

1

2

d

dt‖Tm1tf‖2 ∂

2

∂x2F (Xt, t)dt, (7.1)

or in its Sm transform differential form as

d

dtSm (F (t,Xt)) (s) =

∂

∂tSm (F (t,Xt)) (s) + f(t)Sm

(∂

∂xF (t,Xt)

)(s)B′s(t)

+1

2

d

dt‖Tm1tf‖2Sm

(∂2

∂x2F (t,Xt)

)(s),

(7.2)

which is easily obtained form the proof of Theorem 6.1.2 and Example 6.0.17.

The fact that the Ito formula (6.12) for stochastic integrals in Wm takes a similar struc-

ture as the classical one (3.9) and as some formulas developed for the fractional Brownian

motion [12, Eq. 4.1], suggests that some results in stochastic analysis based on the Brow-

nian motion and the fractional Brownian motion can be extended to our more general

setting. In this chapter we bring a simple example for such a result in the field of stochas-

tic optimal control. The same problem was formulated and solved in [29] for the case of

fractional Brownian motion. We show that the generalization to our setting is straight-

forward.

7. Application in optimal control 51

We fix m such that∫ m(ξ)

1+ξ2dξ < ∞, and consider the case where the control dynamics

is given by the following Wick-Ito type stochastic differential equation(SDE), where the

state is scalar valued:dX(t) = (a(t)X(t) + b(t)u(t)) dt+ c(t)X(t)dBm(t),

X(0) = x0 ∈ R (given and deterministic).(7.3)

Here a(t) and b(t) := (b1(t), ..., bp(t)) are measurable and essentially bounded determin-

istic functions in t ≥ 0, dBm(t) is the Ito-type differential of Bm(t), t→ c(t), t ∈ [0, T ] ,

belongs to D(Tm) and ‖Tm (1tc(·)) ‖2 is an absolutely continuous function in t.

Stochastic differential equations of the type (7.3) perturbed by white noise is widely

common in the description of price development over time of a risky financial assets [40,

Eq. 1.5.1]. In this case, a(t) is denoted the drift, c(t) is called the volatility and we can

further interpret u(t) as some type of controlled influence on the price.

For every initial state and control u(t), 0 < t ≤ τ,, we define an associated cost functional

J (x0,u(·)) , E[∫ τ

0

(q(t)X(t)2 + u∗(t)R(t)u(t)

)dt+ gX(τ)2

], (7.4)

where q(t) and each entry in R(t) are given essentially bounded deterministic functions

in t, and g is a given deterministic scalar. We reduce ourselves to the case where the

control u(t), 0 < t ≤ τ, is taken to be of Markovian linear feedback type, namely,

u(t) = k(t)X(t),

where k(t) := (k1(t), ..., kp(t))∗ is an essentially bounded deterministic function of t.

Under these assumptions, (7.3) reduces to the following SDE:dX(t) = (a(t) + b(t)k(t))X(t)dt+ c(t)X(t)dBm(t),

X(0) = x0 ∈ R.(7.5)

The cost functional (7.4) may now be associated directly with the initial state and the

7. Application in optimal control 52

feedback gain k(t),

J (x0,k(·)) , E[∫ τ

0

(q(t) + k∗(t)R(t)k(t))X(t)2dt+ gX(τ)2

]. (7.6)

The optimal stochastic control problem is to minimize the cost functional (7.6), for each

given x0 ∈ R, over the set of all Markovian linear feedback controls k(t), 0 < t ≤ τ .

Theorem 7.0.1.

X(t) = x0 exp

[∫ t

0

(a(u) + b(u)k(u)) du+

∫ t

0

c(u)dBm(u)− 1

2‖Tm (1tc(·)) ‖2

](7.7)

is the unique solution of (7.5).

Proof: Since in this work we have defined the stochastic integral by means of the Sm

transform, we use the Sm differential version of (7.5) which is

d

dtXs(t) = (a(t) + b(t)k(t) + c(t)B′s(t))Xs(t), (7.8)

where Xs(t) , Sm (X(t)) (s). By the existence and uniqueness theory for ordinary differ-

ential equations, it follows that for every s ∈ S , there is a unique solution Xs(t), t ≥ 0,

which satisfies (7.8), and is given by

Xs(t) = Xs(0) exp

∫ t

0

(Au +BuKu) du+

∫ t

0

C(u)B′s(u)du

=

X(0) exp

∫ t

0

(Au +BuKu) du

exp (Tms, Tm (1tC(·))) .

It follows from Lemma 5.3.7 and Theorem 5.3.2 that the (unique) inverse S-transform of

the last expression is

X(t) = X(0) exp

∫ t

0

(a(u) + b(u)k(u)) du

exp 〈ω,1tc(·)〉 ,

which equals (7.7).

Note that the proof of the last theorem is significantly shorter than its counterpart for the

special case of the fractional Brownian motion appears in [29], thanks to the S-transform

approach.

7. Application in optimal control 53

7.1 Solution

The solution for the control problem is given in the following theorem.

Theorem 7.1.1. Assume g > 0 and that almost everywhere a(t) 6= 0,b(t) 6= 0, q(t) > 0,

R(t)− εI > 0. The optimal Markovian linear feedback control u(t) is given by

u(t) = k(t)X(t) with k(t) = −R−1(t)b∗(t)p(t), (7.9)

where p(t), t ∈ [0, τ ] is the unique non-negative solution of the backward Riccati equa-

tion p(t) + 2p(t)

[a(t) + d

dt‖T1tc(·)‖2]+ q(t)− b(t)R−1(t)b∗(t)p(t)2 = 0

p(τ) = g(7.10)

Proof: Under the assumptions on the coefficients, the unique solvability of the Ricatti

equation (7.10) is a classical result which was proved in, e.g., [17]. Since X(t) from (7.7)

is an exponent of a Gaussian random variable, conditions (6.8)-(6.10) are trivially met

for

p(t)X(t)2 = p(t)x02 exp

2

∫ t

0

(a(u) + b(u)k(u)) du

exp

2

∫ t

0

c(u)dBm(u)− 2‖Tm (1tc(·)) ‖2

,

hence by applying Ito’s formula (6.12) we obtain

p(τ)X(τ)2 = p(0)x20 +

∫ τ

0

[p(t)X(t)2 + 2p(t)X(t)2

(a(t) + b(t)k(t) +

d

dt‖Tm (1tc(·)) ‖2

)]dt

+

∫ τ

0

2c(t)p(t)X(t)2dBm(t)dt.

Rearranging and taking expectation at both sides we get

E[p(τ)X(τ)2

]= p(0)x2

0+E∫ τ

0

X(t)2

[p(t) + 2p(t)

(a(t) + b(t)k(t) +

d

dt‖Tm (1tc(·)) ‖2

)]dt.

7. Application in optimal control 54

Since p(τ) = g, we have

J (x0,k(·)) = p(0)x20+

E∫ τ

0

X(t)2

[p(t) + (q(t) + k∗(t)R(t)k(t)) + 2p(t)

(a(t) + b(t)k(t) +

d

dt‖Tm (1tc(·)) ‖2

)]dt

= p(0)x20 + E

∫ τ

0

X(t)2

[p(t) + 2p(t)a(t) + 2p(t)

d

dt‖Tm (1tc(·)) ‖2 + q(t)

+(k(t) + R−1(t)b∗(t)p(t)

)∗R(t)

(k(t) + R−1(t)b∗(t)p(t)

)− b(t)R−1(t)b∗(t)p(t)2

]dt.

Substituting the Ricatti equation (7.10) into the last expression and rearranging yields

J (x0,k(·)) = p(0)x20+E

[∫ τ

0

(k(t) + R−1(t)b∗(t)p(t)

)∗R(t)

(k(t) + R−1(t)b∗(t)p(t)

)dt

].

It follows that the cost function achieves its minimum when k(t) = −R−1(t)b∗(t)p(t),

and that minimum is p(0)x20.

7.2 Simulation

In this section we check the validity of our state-space model and the solution to the

optimal control problem by means of numeric approximation and computer simulation.

We use Monte Carlo simulation in order to compute the cost function (7.6) which is

associated here with two different linear Markovian controllers. The first controller

kop(t), 0 ≤ t ≤ τ is the optimal controller obtained from Theorem 7.1.1. The second

controller kna(t), 0 ≤ t ≤ τ would have been the optimal controller if the noise was white

and with the same drift as the real noise, i.e. kna corresponds to a naive design of the

system, which does not takes into account the true correlated nature of the noise.

In order to simulate a sample path of the fundamental stationary increment process Bm,

we use 10,000 independent normally distributed pseudo-random numbers, multiplied by

the square root of a covariance matrix of size 10, 000 × 10, 000, obtained by using the

spectral measure

E [Bm(t)Bm(s)] = mint, s+(t2H + s2H − |t− s|2H

), 0 < H < 1,

So in this case Bm can be viewed as the sum of standard Brownian motion and a fractional

Brownian motion with Hurst parameter H. The corresponding spectral function is

m(ξ) = 1 + c(H)|ξ|1−2H ,

7. Application in optimal control 55

Fig. 7.1: The ratio between the resulting cost using the naive controller and the resulting costusing the optimal Markovian controller, averaged over 5,000 runs, for the system (7.3)with τ = 5, x0 = 3, b = 0.3, q = 1, g = 2, r = 2, H = 0.8 (In order to simulate onesample path of the process Bm in n times samples, we draw the vector v consistedof n independent standard normal samples, and then compute the square root

√A of

the n× n covariance matrix A of Bm in these n times. The vector√Av can be seen

as a vector of samples taken from one sample path of Bm).

where C(H) is a constant depends on H. We pick non-time varying coefficients for the

state-space model (7.3), and define the signal to noise ratio in our model as SNR , a2

c2.

This is motivated by view of (7.7), since this is the ratio between the free term in the

exponent after removing the drift, and the noise amplitude. The measured ratio between

the two cost ratio is depicted in Figure 7.1, in which we can observe that for example for

a noise level of −6dB the expected cost of the system (7.5) in which k = kna is about

1.6 times than the expected cost in the optimal case.

8. CONCLUSIONS

In this work we have developed the basics of stochastic distribution theory which is

based on noises with arbitrary spectrum subject to the condition (4.1), and stochastic

integration theory using the analogue of the S-transform in our new settings. This was

done using a variation on Hida’s white noise space theory. Unlike related extension of

the white noise theory such as [5] for the fractional noise and [2] for the same family

of noises considered here, our introduction of the characteristic functional (4.3) and the

space Wm allows a natural and simple representation of the noise as the underlying

stochastic process of this space; it plays the same role in Wm that the white noise has

in the white noise space W1. The fact that the Ito formula for stochastic integrals in

Wm takes a similar structure as the original, suggests that many results in stochastic

analysis which are based on the Brownian motion can be extended to our more general

setting. On Chapter 7 we gave an example for such an extension in the field of stochastic

optimal control. In view of Chapter 7, this work can be continued by further extending

other stochastic differential equations driven by white noise to our non-white settings.

Additional directions in which this work can be extended are given below.

• The Wiener-Ito chaos expansion plays a key role in the white noise space theory

in the sense that the spaces of stochastic distributions are defined by the chaos

coefficients, and seems to be depended in the particular choice of the basis element

in L2(R) as suggested for example by the definition of the singular white noise

(3.8). Moreover, as mentioned in the end of Section 5.3, an alternative approach to

define Hitsuda-Skorokhod integral in Wm can be carried out by means of the chaos

expansion and the Wick product (which does not depends on the basis). In our

more general case, we have replaced L2(R) with the domain of Tm, and considered

the Hilbert space generated by the Gaussian random variables 〈ω, f〉, f ∈ D(Tm).Chaos expansion for Wm involves a particular basis for that Hilbert space, and this

is given by inverting the operator Tm.

• Spaces of stochastic distributions and corresponding Gelfand triples for the space

Wm has not yet been defined, as there are few issues that needed to be considered

in this regard. If we wish to follow the same line as in the Kondratiev spaces of the

8. Conclusions 57

white noise space, there is still the question of whether the definition depends on

the basis elements, and in that case which basis to pick. An alternative definition

can be done in terms of the S-transform, similar to what was given in [35, Chapter

8] for Hida’s spaces of stochastic distributions. We ask how different are these two

approaches? It is also interesting to note that for a particular family of spectral

functions m, the time derivative of Bm is a stationary (ordinary) process already

belongs to Wm (This is the case of Example 5.2.2 since eiξt belongs to L1 (m(ξ)dξ)

in that case), so it may happen that some properties of Wm are fundamentally

different than the standard white noise space.

• It turns out that the chaos expansion is useful in the solution of some linear pro-

jection problems in Wm. This approach is now being investigated by the author.

A special version of the chaos expansion for the space Wm and application of it in

prediction problems for Gaussian process will be presented and discuss in a future

work [4].

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