Research Article | Open Access

# A Generalization of Itô's Formula and the Stability of Stochastic Volterra Integral Equations

**Academic Editor:**Yansheng Liu

#### Abstract

It is well known that Itô’s formula is an essential tool in stochastic analysis. But it cannot be used for general stochastic Volterra integral equations (SVIEs). In this paper, we first introduce the concept of quasi-Itô process which is a generalization of well-known Itô process. And then we extend Itô’s formula to a more general form applicable to some kinds of SVIEs. Furthermore, the stability in probability for some SVIEs is analyzed by the generalized Itô’s formula. Our work shows that the generalized Itô’s formula is powerful and flexible to use in many relevant fields.

#### 1. Introduction

Nowadays, more and more people have realized that stochastic differential equation (SDE) is an important subject which provides more realistic models in many areas of science and applications, such as in biomathematics, filtering problems, physics, stochastic control, and mathematical finance. It is known that Itô SDEs of the form have been used and applied broadly, and their fundamental theories have been well developed [1–3].

In [1–4] and many other references, we see that Itô’s formula plays a key role in the study of stochastic analysis. It is applied in the studying of stochastic control, stochastic neural network, backward SDEs, and numerical solutions of SDEs. Itô’s formula can be seen as a stochastic version of chain rule in calculus. It is very useful in evaluating Itô integral, in investigating the existence and uniqueness, the stability and the oscillation of solutions to SDEs, and so does in many other aspects of stochastic calculus [5–15]. Hence, we can imagine that if there was no Itô’s formula, many known results might be very difficult to get.

Here, and throughout this paper, what we mentioned is all in a complete filtered probability space on which an -dimensional Brownian motion is defined with being its natural filtration augmented by the -null sets in . The mathematical expectation with respect to the given probability measure is denoted by . For convenience, we state Itô’s formula in [1] as follows.

*Definition 1.1. *A -dimensional Itô process is an -valued continuous adapted process on of the form
where and . We will say that has a stochastic differential on given by

Two natural questions are whether the compound function is an Itô process, if is an Itô process and , And if it is, what is its stochastic differential? This leads to the following very famous Itô’s formula.

Theorem 1.2. *Let be a -dimensional Itô process on with the stochastic differential (1.3), and . Then is an Itô process with the stochastic differential given by
*

From Theorem 1.2, it is easy to see that the differential form of the Itô process is more convenient to apply than its integral form. To describe the realistic world better, it is natural to extend SDE (1.1) to a more general case as the following stochastic Volterra integral equation (SVIE): where is a continuous stochastic process. It is easy to see that SDE (1.1) is a special case of SVIE (1.5). Many scholars have given some results for SVIE (1.5) (see [16, 17]). However, it is noted that the solutions decided by (1.5) are not Itô processes; hence they do not satisfy the conditions in Theorem 1.2. So the Itô’s formula cannot be used for these SVIEs. It is one of the reasons that many basic theories of SVIEs have not been accomplished.

Motivated by the previous discussions, in this paper, we extend Itô’s formula to a more general form applicable to SVIEs, by employing the technique in stochastic analysis. Based on the generalized Itô’s formula and Lyapunov method, the stochastic stability to some kinds of SVIEs is investigated. Consequently, some sufficient conditions, which ensure the global stochastic asymptotic stability of the trivial solution, are established. By constructing an appropriate Lyapunov function, a condition ensuring global stochastic asymptotic stability of a linear SVIE is given. Our work shows that the generalized Itô’s formula is powerful and flexible to use. Obviously, it can also be used in many other relevant fields.

#### 2. Quasi-ItÔ Process and Generalized Itô’s Formula

In this section, we begin with introducing the concept of quasi-Itô process. Set . Let stand for the family of all real-valued functions defined on such that they are continuously twice differentiable at and once at . For any , we define

*Definition 2.1. *A -dimensional quasi-Itô process is an -valued continuous adapted process , on of the form
where , , for all , and are continuous, and is an -adapted continuous stochastic process.

We will say that has quasistochastic differential or for given by
or
in which .

*Remark 2.2. *Definition 2.1 is well defined under the condition that is continuous for . The proof of the continuity of is similar to Theorem in [1]. Here we do not verify it. But in the proof we will need two approximation theorems as follows.

Lemma 2.3 (see [2, page 116]). *Letting be -measurable, then could be approximated pointwise by bounded functions of the form
*

Lemma 2.4. *Let be -measurable, and for all , is continuous; then is approximated by functions on the form
**
where is continuous and is -measurable for every .*

Similar to Theorem 1.2, it again raises the following question. If is a quasi-Itô process and , then whether the compound function is a quasi-Itô process. And if it is, then what is its quasistochastic differential? We now have the result which is a well generalization of Itô’s formula.

Theorem 2.5. *Let be a -dimensional quasi-Itô process on with the quasistochastic differential
**
or
**
with . Here , are defined as Definition 2.1, is a continuous stochastic process, and for every , is -measurable. Let and
**
Then is a quasi-Itô process with the quasistochastic differential given by
**
or
**
with .*

*Proof. *Setting is arbitrary and
By Itô’s formula, we can derive that for any ,
So . Setting , then we have
Since is arbitrary, (2.10) must be required. The proof is complete.

*Remark 2.6. *When and , are independent of , that is, when and , then it is easy to check that
and the generalized Itô’s formula becomes classical Itô’s formula.

*Example 2.7. *Suppose that
where is a continuous function.

We find . Here we have where . Let . Then . So by (2.11) we obtain Therefore,

Sometimes function is required to be -adapted instead of -measurable. We suppose that is the set of all absolutely continuous -adapted processes. That is, if , then is absolutely continuous for almost all and is -measurable for any .

Theorem 2.8. *Let be a -dimensional quasi-Itô process on with the quasistochastic differential
**
or
**
with . Here , are defined as Definition 2.1 and . Let and
**
Then is a quasi-Itô process with the quasistochastic differential given by
*

* Proof. *It is easy to see that
Let
From Theorem 2.5 we have
The proof is complete.

*Example 2.9. *Let
Find . Here we have
So by Theorem 2.8, we obtain

#### 3. Stability in Probability of SVIEs

In this section, we use the generalized Itô’s formula to investigate the stability for the -dimensional SVIE: Assuming further that is -measurable, and Hence, (3.1) has solution corresponding to initial value . This solution is called trivial solution. For any , define Then by Theorem 2.5, Let denote the family of all continuous nondecreasing functions such that and if . For , let . A continuous function defined on is said to be positive definite if , and, for some , A function is said to be decrescent if for some . A function defined on is said to be radially unbounded if .

*Definition 3.1. *(1) The trivial solution of (3.1) is said to be stochastically stable if for every pair and , there exists a such that
whenever .

(2) The trivial solution of (3.1) is said to be stochastically asymptotically stable if it is stochastically stable, and, moreover, for every , there exists a such that
whenever .

(3) The trivial solution of (3.1) is said to be globally stochastically asymptotically stable if it is stochastically stable, and, moreover, for all

Lemma 3.2. *If there exists an , such that . Then for any and , one has
*

Similar to the proof of Lemma 3.2 in [2, pp. 120], it is easy to get the lemma. Here we do not recount it.

Theorem 3.3. *Suppose that there exists a , such that . If there exists a positive definite function , such that for any , there is
**
Then the trivial solution to (3.1) is stochastically stable.*

*Proof. *From the definition of the positive definite function, we know that , and there exists nonnegative nondecreasing function , such that for any . Choose any , . Without loss of generality, we assume that . Since is continuous and , we could find , such that for any there is
Define . Choose any . Let be the first time of going out the ball , that is, . By Theorem 2.5, for any , there is
Taking the expectation for both sides, and using , we get
On the other hand, we have
So combining (3.11) and (3.14), it follows that
Letting , we obtain
that is,
The proof is complete.

Theorem 3.4. *Suppose that the conditions in Lemma 3.2 hold. If there exists a positive definite function , which has infinitesimal upper bounded and
**
in which is concave function. Then the trivial solution to (3.1) is stochastically asymptotically stable.*

*Proof. *It is clear that the conditions in Theorem 3.3 are satisfied. Hence, the solution to (3.1) is stochastically stable. So it is only necessary to show that for any , there exists a , such that for any ,
holds. Fixing , in view of Theorem 3.3, there exists a , such that holds provided only that
Fix , and denote . Choose any and . Define stopping time
From Theorem 2.5, for any , there is
Therefore
So
Letting , it yields
Clearly, from (3.20) it follows that . Therefore
So
Choose sufficiently large , such that
Then
Again define two stopping times as
By reason that
in which
From Theorem 2.5, it follows that for any , there is
Note that if , then
Consequently,
From the total probability formula, it yields that
Since is a concave function, and by (3.18), we have
From (3.31) it follows that
From Lemma 3.2, it is known that . Hence . Letting and taking the limit for (3.38), there is
Thus we could choose sufficiently small , such that and
hold. That is,
Combining (3.35), (3.36), (3.37), and (3.41), it follows that for sufficiently large there is
Letting , it yields that . In view of (3.29), it deduces that
which shows that
By the arbitrariness of , we have
The proof is complete.

Theorem 3.5. *Suppose that the conditions in Theorem 3.4 are satisfied and is radially unbounded. Then the trivial solution to (3.1) is globally stochastically asymptotically stable.*

*Proof. *In view of Theorem 3.4, it is known that the trivial solution to (3.1) is stochastically asymptotically stable. Therefore it is only necessary to explain that for any , there is
Choose any and . Denote . From that is radially unbounded and that is bounded, we could find a sufficiently large , such that
Define stopping time . Then from Theorem 2.5 and conditional property formula, we could prove that for any , there is
Therefore
Letting , it yields that , that is:
In the following, applying the method in Theorem 3.4, we obatin that
Hence, by the arbitrariness of , (3.46) holds. It completes the proof.

To illustrate the theorem developed in this section, an example now is discussed.

*Example 3.6. *Consider a scale linear SVIE:
in which , .

Letting , , then (3.52) is changed into
By Lemma 2.3, it follows that
Setting , then
If is increasing and bounded almost surely and , then . From Theorem 2.5, the solution to (3.52) is stochastically stable.

Setting , then
If is increasing and bounded almost surely, and , then . In view of Theorem 3.3, for any , there is

*Remark 3.7. *The generalized Itô’s formula provides a powerful tool to deal with SVIEs. But we also remind of its complexity, which will bring some difficulties when the almost sure exponential stability and the moment exponential stability for SVIEs are discussed. In this point, we shall go on to discuss in another papers.

#### Acknowledgments

This work was supported by the NNSF of China (nos.11126219, 11171081, and 11171056), the NNSF of Shandong Province (no. ZR2010AQ021), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (no. HIT.NSRIF. 2011104), and NCET-08-0755.

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#### Copyright

Copyright © 2012 Wenxue Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.