ABSTRACT

The objective of this paper is to attempt to apply the theoretical techniques of probabilistic functional analysis to answer the question of existence and Uniqueness of a Random Solution to Itô Stochastic Integral Equation. Another type of stochastic integral equation which has been of considerable importance to applied mathematicians and engineers is that involving the Itô or Itô-Doob form of stochastic integrals.

1. Introduction

We shall give some historical remarks concerning the development of this type of equation and point out the essential difference between them and other random integral equations.

In 1930 N. Wiener introduced an integral of the form where a deterministic real-valued function and is a scalar Brownian motion process.

Author of [1] in 1944 generalized Wiener’s integral to include those cases where the integrand is random. That is he obtained an integral of the form

Which is referred to as the Itô stochastic integral or simply the stochastic integral. Since that time many scientists have contributed to the general development of this type of stochastic integral. For example see [2- 10].

In 1946 Author of [5] formulated a stochastic integral equation of the form

(1.0)

where, is a scalar Brownian motion process, and C is a constant Restrictions are usually placed on the functions f and g so that the first integral is interpreted as the usual Lebesgue integral of the sample functions which can then be related to the sample integral of the process and the second integral is an Itô stochastic integral.

The principal feature which distinguishes the type of equation studied from an equation of the Itô type is the fact that in the former case each of the integrals involved is interpreted as a Lebesgue integral for almost all. That is, almost all sample functions are Lebesgue integrable. Since in the Itô stochastic integral the limit is taken in the mean-square or in the probability sense, the theory of such integrals has been developed as self-contained and self-consistent.

One of the main purposes of subsequent work in connection with the Itô stochastic integral equation has been to construct Markov processes such that their transition probabilities satisfy given Kolmogorov equations and to investigate the continuity of the processes, among other properties of the sample function.

The method of successive approximation was used by Itô and Doob to show the existence and uniqueness of a random solution to Equation (1.0).

2. Preliminaries

Let be a scalar Brownian motion process. In this section we shall be concerned with the integral

(1.1)

for a fairly general class of functions. This integral will be called the Itô stochastic integral as we mentioned previously. As is well known, almost all the sample functions of the Brownian motion process are of unbounded variation and hence the integral (1.1) cannot be defined as an ordinary Stieltjes integral.

First we shall define the integral (1.1) for the class of step functions. That is, functions of the form

(1.2)

where are measurable with respect to the -algebra, and

for such functions we define the Itô integral by

(1.3)

Now suppose that is any function satisfying the following conditions.

1) is a product-measurable function from, assuming the usual Lebesgue measure on.

2) For each, , is measurable with respect to -algebra, where is the smallest -algebra on, such that, is measurable.

3)

In view of Equation (1.2) it is evident that the class of step functions satisfy conditions 1)-3).

For the function satisfying conditions 1)-3) we shall define their norm as follows:

(1.4)

For this case author of [2] has shown the following 1) can be approximated in the mean-square sense by a sequence of step functions. That is

as

2) The sequence of integrals

Possesses a mean-square limit. That is there exists a such that

(1.5)

as

Now we shall define the integral (1.1) for a class of functions satisfying conditions 1)-3) by

(1.6)

As with the ordinary integrals, we shall define

(1.7)

Definition 1.1 Let, where L denote the collection of Lebesgue measurable subsets of. Define a function from by

Lemma 1.1 The function defined by

where satisfies conditions 1)-3), and is as defined earlier, also satisfies conditions 1)-3).

Proof. The proof is a straightforward result of the definition of and the fact that satisfies conditions 1)-3).

We are now in a position to define exactly what is meant by the expression

Definition 1.2 We define for G a Lebesgue-measurable subset of by

Note that lemma 1.4 guarantees the expression on the right exists and is well defined Definition 1.3 We shall denote by

the space of all continuous functions from into. We shall define the norm of by

Lemma 1.2

Lemma 1.3

Lemma 1.4 If we define a distance between two functions and each satisfying conditions 1)-3) by

and the distance between and by

Then.

For the proof of the Lemmas see [2].

Lemma 1.5 Let,

Then

For the proof see [4].

3. On an Itô Stochastic Integral Equation

In this section we shall investigate a stochastic integral equation of the type

(2.1)

where is the unknown random process defined for and.

We shall place the following restrictions on the random functions which constitute the stochastic integral Equation (2.1).

1') is an element of and is continuous where .

2') is an operator on the set S with values in the Banach space B satisfying

for.

3') Conditions 1)-3) of section 1 hold.

Thus with the given assumptions the first integral of (2.1) can be interpreted as a Lebesgue integral and the second as an Itô stochastic integral.

We shall now proceed to state and prove a theorem concerning the behavior of the Itô integral. More precisely, if we show that the Itô integral is an element of the space, we can apply the theory of admissibility to Equation (2.1) to show the existence of a random solution. By a random solution to Equation (2.1) we mean a random function from into such that for each, satisfies the integral equation P-a.e. showing that the Itô integral is an element of will make feasible the assumption that we wish to make that the integral is an element of D, a Banach space contained in the topological space mentioned For convenient we shall denote the Itô integral by

Theorem 2.1 For

Proof Fix Then

Thus

by lemma 1.3.

Hence.

Therefore for fixed t,. Now let in. To show that in, it is sufficient to show that

can be made arbitrarily small. That is, we must show that

Can be made arbitrarily small. Choose. Consider the nonnegative function. By condition 3) is integrable over. Hence there exists a such that for every set of Lebesgue measure less than,. Thus

Since for and and since the Lebesgue measure of the interval is its length, we conclude that the Lebesgue measure of is less than.

Hence

Implying that is continuous from into and the proof is complete.

Since we have shown that , we can conclude that the stochastic integral Equation (2.1) possesses a unique random solution

4. On Itô-Doob-Type Stochastic Integral Equations

In this section we shall study the existence and uniqueness of a random solution to a stochastic integral equation of the form

(3.1)

where. As before, the first integral is a Lebesgue integral, while the second is an Itô-type stochastic integral defined with respect to a scalar Brownian motion process.

Recall that

, We shall define the operators and from into by

(3.2)

and

(3.3)

Note that in view of lemma 1.5 . Its clear that and are linear operators.

Theorem 3.1 The operators and defined by (3.2) and (3.3) respectively, are continuous operators from into.

Lemma 3.1 Let T be a continuous operator from into itself. If B and D are Banach spaces stronger than and the pair (B, D) is admissible with respect to T. Then T is a continuous operator from B to D.

Proof of theorem 3.1 The fact that is a continuous operator from into follows from lemma 3.1. From (3.3) we have

Furthermore

Therefore

Thus and are continuous operators from into.

An Existence Theorem

We shall assume that lemma 3.1 holds with respect to the operators and. Therefore there exist positive constants and less than one such that

and

The following theorem gives sufficient conditions for the existence of a unique random solution, a second order stochastic process, to the Itô-Doob stochastic integral Equation (3.1).

Theorem 3.2 Consider the stochastic integral equation (3.1) under the following condition:

1) B and D are Banach spaces in

which are stronger than

such that is admissible with respect to the operators and

2) a) is an operator on

With values in B satisfying

b) is an operator on S into B satisfying

where and are constants. Then there exists a unique random solution to Equation (3.1) provided that. And

Proof. Define an operator U from the set S into D as follows

We need to show that U is a contraction operator on S and that.

Let.

Then because D is a Banach space. Further, we have

Thus U is a contraction operator.

For any element in S we have

Since it follows that

from the assumptions in the theorem.

Thus the existence and uniqueness of a random solution to Equation (3.1) follow from the Banach fixed-point theorem.

Theorem 3.4 (S. Banach’s fixed-point principle) ([11]).

If T is a contraction operator on a complete metric space H. then there exists a unique point for which.

5. Conclusion

We investigated the existence and uniqueness of Itô stochastic integral equation by applying the theoretical techniques of probabilistic functional analysis. In fact author of [12] refers to probabilistic functional analysis as being concerned with the applications and extensions of the methods of functional analysis to the study of the various concepts, processes, and structures which arise in the theory of probability and its applications. Finally to develop and unify the theory of stochastic or random equations see [13-15].

REFERENCES