(ProQuest: ... denotes non-US-ASCII text omitted.)
Recommended by Yansheng Liu
1, Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China
Received 17 May 2012; Accepted 12 July 2012
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.
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 [figure omitted; refer to PDF] 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 (Ω,...,...,...) on which an m -dimensional Brownian motion W(·) is defined with ...={...t}t...5;0 being its natural filtration augmented by the ... -null sets in ... . The mathematical expectation with respect to the given probability measure ... is denoted by E(·) . For convenience, we state Itô's formula in [1] as follows.
Definition 1.1.
A d -dimensional Itô process is an ...d -valued continuous adapted process x(t)=(x1 (t),...,xd (t))T on t...5;0 of the form [figure omitted; refer to PDF] where f=(f1 ,...,fd)T ∈[Lagrangian (script capital L)]1 (...+ ;...d ) and g=(gij)d×m ∈[Lagrangian (script capital L)]2 (...+ ;...d×m ) . We will say that x(t) has a stochastic differential dx(t) on t...5;0 given by [figure omitted; refer to PDF]
Two natural questions are whether the compound function V(x(t),t) is an Itô process, if x(t) is an Itô process and V∈C2,1 (...d ×...+ ;...) , And if it is, what is its stochastic differential? This leads to the following very famous Itô's formula.
Theorem 1.2.
Let x(t) be a d -dimensional Itô process on t...5;0 with the stochastic differential (1.3), and V∈C2,1 (...d ×...+ ;...) . Then V(x(t),t) is an Itô process with the stochastic differential given by [figure omitted; refer to PDF]
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): [figure omitted; refer to PDF] where [straight phi](t,ω) 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 Δ={(t,s):0...4;s...4;t<∞} . Let C2,1 (...d ×...+ ;...) stand for the family of all real-valued functions V(x,t) defined on ...d ×...+ such that they are continuously twice differentiable at x and once at t . For any p∈[1,∞) , we define [figure omitted; refer to PDF]
Definition 2.1.
A d -dimensional quasi-Itô process is an ...d -valued continuous adapted process x(t)=(x1 (t),... , xd (t))T on t...5;0 of the form [figure omitted; refer to PDF] where f=(f1 ,...,fd)T ∈[Lagrangian (script capital L)](...+ ,[Lagrangian (script capital L)]1 ((0,t);...d )) , g=(gij)d×m ∈[Lagrangian (script capital L)](...+ ,[Lagrangian (script capital L)]2 ((0,t);...d×m )) , for all s...5;0 , f(·,s) and g(·,s) are continuous, and [straight phi](t) is an ...t -adapted continuous stochastic process.
We will say that x(t) has quasistochastic differential dx(s) or Dx(s) for t...5;0 given by [figure omitted; refer to PDF] or [figure omitted; refer to PDF] in which Dx(s)=dx(s)-d[straight phi](s) .
Remark 2.2.
Definition 2.1 is well defined under the condition that ∫0t ...g(t,s)dW(s) is continuous for t . The proof of the continuity of ∫0t ...g(t,s)dW(s) is similar to Theorem 1.5.13 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 g:...d ×Ω[arrow right]... be ...(...d )×... -measurable, then g could be approximated pointwise by bounded functions of the form [figure omitted; refer to PDF]
Lemma 2.4.
Let [varphi](t,s):...2 [arrow right]... be ...(...2 ) -measurable, and for all s...5;0 , g(·,s) is continuous; then [varphi](t,s) is approximated by functions on the form [figure omitted; refer to PDF] where gk1 (t) is continuous and gk2 (s) is ...(...1 ) -measurable for every k∈{1,2,...,m} .
Similar to Theorem 1.2, it again raises the following question. If x(t) is a quasi-Itô process and V∈C2,1 (...d ×...+ ;...) , then whether the compound function V(x(t),t) 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 x(t) be a d -dimensional quasi-Itô process on t...5;0 with the quasistochastic differential [figure omitted; refer to PDF] or [figure omitted; refer to PDF] with Dx(s)=dx(s)-d[straight phi](s) . Here f , g are defined as Definition 2.1, [straight phi](t) is a continuous stochastic process, and for every t∈[0,∞) , [straight phi](t) is ...0 -measurable. Let V∈C2,1 (...d ×...+ ;...) and [figure omitted; refer to PDF] Then V(x(t),t) is a quasi-Itô process with the quasistochastic differential given by [figure omitted; refer to PDF] or [figure omitted; refer to PDF] with DV(x(s),s)=dV(x(s),s)-dV([straight phi](s),0) .
Proof.
Setting t* ∈[0,∞) is arbitrary and [figure omitted; refer to PDF] By Itô's formula, we can derive that for any t...5;0 , [figure omitted; refer to PDF] So V(y(t* ),t* )=V(x(t* ),t* ) . Setting t=t* , then we have [figure omitted; refer to PDF] Since t* is arbitrary, (2.10) must be required. The proof is complete.
Remark 2.6.
When [straight phi](t),f(t,s) and g(t,s) , are independent of t , that is, when [straight phi](t)=x(0),f(t,s)=f(s) and g(t,s)=g(s) , then it is easy to check that [figure omitted; refer to PDF] and the generalized Itô's formula becomes classical Itô's formula.
Example 2.7.
Suppose that [figure omitted; refer to PDF] where m(t) is a continuous function.
We find y(t)=x2 (t) . Here we have [figure omitted; refer to PDF] where W¯(s)[triangle, =]∫0s ...W(r)dr . Let V(x,t)=x2 . Then Vt =0,Vx =2x,Vxx =2 . So by (2.11) we obtain [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]
Sometimes function [straight phi](t,ω) is required to be ...t -adapted instead of ...0 -measurable. We suppose that ...9E; is the set of all absolutely continuous ...t -adapted processes. That is, if [varphi]∈...9E; , then [varphi](t,ω) is absolutely continuous for almost all ω∈Ω and [varphi](t,ω) is ...t -measurable for any t...5;0 .
Theorem 2.8.
Let x(t) be a d -dimensional quasi-Itô process on t...5;0 with the quasistochastic differential [figure omitted; refer to PDF] or [figure omitted; refer to PDF] with Dx(s)=dx(s)-d[straight phi](s) . Here f , g are defined as Definition 2.1 and [straight phi]∈...9E; . Let V∈C2,1 (...d ×...+ ;...) and [figure omitted; refer to PDF] Then V(x(t),t) is a quasi-Itô process with the quasistochastic differential given by [figure omitted; refer to PDF]
Proof.
It is easy to see that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] From Theorem 2.5 we have [figure omitted; refer to PDF] The proof is complete.
Example 2.9.
Let [figure omitted; refer to PDF] Find y(t)=x2 (t) . Here we have [figure omitted; refer to PDF] So by Theorem 2.8, we obtain [figure omitted; refer to PDF]
3. Stability in Probability of SVIEs
In this section, we use the generalized Itô's formula to investigate the stability for the d -dimensional SVIE: [figure omitted; refer to PDF] Assuming further that [straight phi](t,ω) is ...t0 -measurable, [straight phi](t0 ,ω)=1 and [figure omitted; refer to PDF] Hence, (3.1) has solution x(t)...1;0 corresponding to initial value x(t0 )=0 . This solution is called trivial solution. For any η∈C([t0 ,∞);...d ) , define [figure omitted; refer to PDF] Then by Theorem 2.5, [figure omitted; refer to PDF] Let ...A6; denote the family of all continuous nondecreasing functions μ:...+ [arrow right]...+ such that μ(0)=0 and μ(x)>0 if x>0 . For h>0 , let Sh ={x∈...d :|x|<h} . A continuous function V(x,t) defined on Sh ×[t0 ,∞) is said to be positive definite if V(0,t)=0 , and, for some μ∈...A6; , [figure omitted; refer to PDF] A function V(x,t) is said to be decrescent if V(x,t)...4;μ(|x|),(x,t)∈Sh ×[t0 ,∞) for some μ∈...A6; . A function V(x,t) defined on ...d ×[t0 ,∞) is said to be radially unbounded if lim inf |x|[arrow right]∞,t...5;t0 V(x,t)=∞ .
Definition 3.1.
(1) The trivial solution of (3.1) is said to be stochastically stable if for every pair [varepsilon]∈(0,1) and r>0 , there exists a δ=δ([varepsilon],r,t0 )>0 such that [figure omitted; refer to PDF] whenever |x0 |<δ .
(2) The trivial solution of (3.1) is said to be stochastically asymptotically stable if it is stochastically stable, and, moreover, for every [varepsilon]∈(0,1) , there exists a δ0 =δ0 ([varepsilon],t0 )>0 such that [figure omitted; refer to PDF] whenever |x0 |<δ0 .
(3) The trivial solution of (3.1) is said to be globally stochastically asymptotically stable if it is stochastically stable, and, moreover, for all x0 ∈...d [figure omitted; refer to PDF]
Lemma 3.2.
If there exists an [varepsilon]>0 , such that |[straight phi](t,ω)|>[varepsilon],t∈[t0 ,∞) a.s . Then for any x0 ∈...d and x0 ...0;0 , one has [figure omitted; refer to PDF]
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 K>0 , such that |[straight phi](t,ω)|...4;K a.s . If there exists a positive definite function V(x,t)∈C2,1 (Sh ×[t0 ,∞);...+ ) , such that for any (x,t,s)∈C([t0 ,∞);...d )×Δ , there is [figure omitted; refer to PDF] Then the trivial solution to (3.1) is stochastically stable.
Proof.
From the definition of the positive definite function, we know that V(0,t)=0 , and there exists nonnegative nondecreasing function μ(x) , such that V(x,t)...5;μ(|x|) for any (x,t)∈Sh ×[t0 ,∞) . Choose any [varepsilon]∈(0,1) , r>0 . Without loss of generality, we assume that r<h . Since V(x,t) is continuous and V(0,t0 )=0 , we could find δ=δ([varepsilon],r,t0 )>0 , such that for any t∈[t0 ,∞),ω∈Ω there is [figure omitted; refer to PDF] Define x(t)=x(t;t0 ,x0 ) . Choose any x0 ∈Sδ . Let τ be the first time of x(t) going out the ball Sr , that is, τ=inf {t...5;t0 :x(t) ∈¯ Sr } . By Theorem 2.5, for any t...5;t0 , there is [figure omitted; refer to PDF] Taking the expectation for both sides, and using [Lagrangian (script capital L)]V(x,t,s)...4;0 , we get [figure omitted; refer to PDF] On the other hand, we have [figure omitted; refer to PDF] So combining (3.11) and (3.14), it follows that [figure omitted; refer to PDF] Letting t[arrow right]∞ , we obtain [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] The proof is complete.
Theorem 3.4.
Suppose that the conditions in Lemma 3.2 hold. If there exists a positive definite function V(x,t)∈C2,1 (Sh ×[t0 ,∞);...+ ) , which has infinitesimal upper bounded and [figure omitted; refer to PDF] in which μ2 (x) 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 [varepsilon]∈(0,1) , there exists a δ=δ0 ([varepsilon],t0 )>0 , such that for any |x0 |<δ0 , [figure omitted; refer to PDF] holds. Fixing [varepsilon]∈(0,1) , in view of Theorem 3.3, there exists a δ0 =δ0 ([varepsilon],t0 )>0 , such that |x0 |<δ0 holds provided only that [figure omitted; refer to PDF] Fix x0 ∈Sδ0 , and denote x(t)=x(t;t0 ,x0 ) . Choose any 0<β<|x0 | and 0<α<β . Define stopping time [figure omitted; refer to PDF] From Theorem 2.5, for any t...5;t0 , there is [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Letting t[arrow right]∞ , it yields [figure omitted; refer to PDF] Clearly, from (3.20) it follows that ...(τh <∞)...4;[varepsilon]/4 . Therefore [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Choose sufficiently large θα , such that [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Again define two stopping times as [figure omitted; refer to PDF] By reason that [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] From Theorem 2.5, it follows that for any t...5;θα , there is [figure omitted; refer to PDF] Note that if ω∈{τα ...5;τh ⋀θα } , then [figure omitted; refer to PDF] Consequently, [figure omitted; refer to PDF] From the total probability formula, it yields that [figure omitted; refer to PDF] Since μ2 (·) is a concave function, and by (3.18), we have [figure omitted; refer to PDF] From (3.31) it follows that [figure omitted; refer to PDF] From Lemma 3.2, it is known that lim α[arrow right]0 τα =∞ . Hence lim α[arrow right]0 τβ =∞ . Letting α[arrow right]0 and taking the limit for (3.38), there is [figure omitted; refer to PDF] Thus we could choose sufficiently small [varepsilon]1 >0,α1 >0 , such that μ2 (α1 +[varepsilon]1 )/μ1 (β)<[varepsilon]/4 and [figure omitted; refer to PDF] hold. That is, [figure omitted; refer to PDF] Combining (3.35), (3.36), (3.37), and (3.41), it follows that for sufficiently large t there is [figure omitted; refer to PDF] Letting t[arrow right]∞ , it yields that ...(τβ ...4;∞)<[varepsilon]/4 . In view of (3.29), it deduces that [figure omitted; refer to PDF] which shows that [figure omitted; refer to PDF] By the arbitrariness of β , we have [figure omitted; refer to PDF] The proof is complete.
Theorem 3.5.
Suppose that the conditions in Theorem 3.4 are satisfied and V(x,t) 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 x0 ∈...d , there is [figure omitted; refer to PDF] Choose any x0 ∈...d and [varepsilon]∈(0,1) . Denote x(t)=x(t;t0 ,x0 ) . From that V(x,t) is radially unbounded and that [straight phi](t,ω) is bounded, we could find a sufficiently large h>|x0 | , such that [figure omitted; refer to PDF] Define stopping time τh ={t>t0 :|x(t)|...5;h} . Then from Theorem 2.5 and conditional property formula, we could prove that for any t>t0 , there is [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] Letting t[arrow right]∞ , it yields that ...(τh <∞)...4;[varepsilon]/4 , that is: [figure omitted; refer to PDF] In the following, applying the method in Theorem 3.4, we obatin that [figure omitted; refer to PDF] Hence, by the arbitrariness of [varepsilon] , (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: [figure omitted; refer to PDF] in which a(t)b(t)<0,t...5;0 , [straight phi](0,ω)=1 .
Letting y(t)=a-1 (t)x(t) , [straight phi]1 (t,ω)=a-1 (t)[straight phi](t,ω) , then (3.52) is changed into [figure omitted; refer to PDF] By Lemma 2.3, it follows that [figure omitted; refer to PDF] Setting V(x,t)=x2 , then [figure omitted; refer to PDF] If a-1 (t)[straight phi](t,ω) is increasing and bounded almost surely and 2|b(s)|...5;|a(s)||c(s)|2 , then [Lagrangian (script capital L)]V(y(s),t,s)<0 . From Theorem 2.5, the solution to (3.52) is stochastically stable.
Setting V(x,t)=x1/2 , then [figure omitted; refer to PDF] If a-1 (t)[straight phi](t,ω) is increasing and bounded almost surely, and a-1 (t)=-b(t) , then [Lagrangian (script capital L)]V(y(s),t,s)...4;-y(s)1/2 /2 . In view of Theorem 3.3, for any x0 >0 , there is [figure omitted; refer to PDF]
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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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.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer