Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 292740, 16 pages
doi:10.1155/2012/292740
Research Article
A Generalization of Itô’s Formula and the Stability
of Stochastic Volterra Integral Equations
Wenxue Li, Meng Liu, and Ke Wang
Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China
Correspondence should be addressed to Meng Liu, liumeng0557@sina.com
Received 17 May 2012; Accepted 12 July 2012
Academic Editor: Yansheng Liu
Copyright q 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.
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
dXt fXt, tdt gXt, tdWt,
1.1
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
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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, F, F, P on which an m-dimensional Brownian motion W· is defined
with F {Ft }t≥0 being its natural filtration augmented by the P-null sets in F. The
mathematical expectation with respect to the given probability measure P 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 Rd -valued continuous adapted process xt x1 t, . . . , xd tT on t ≥ 0 of the form
xt x0 t
fsds t
0
gsdWs,
1.2
0
where f f1 , . . . , fd T ∈ L1 R ; Rd and g gij d×m ∈ L2 R ; Rd×m . We will say that xt
has a stochastic differential dxt on t ≥ 0 given by
dxt ftdt gtdWt.
1.3
Two natural questions are whether the compound function V xt, t is an Itô process,
if xt is an Itô process and V ∈ C2,1 Rd × R ; R, And if it is, what is its stochastic differential?
This leads to the following very famous Itô’s formula.
Theorem 1.2. Let xt be a d-dimensional Itô process on t ≥ 0 with the stochastic differential 1.3,
and V ∈ C2,1 Rd × R ; R. Then V xt, t is an Itô process with the stochastic differential given by
1
T
dV xt, t Vt xt, t Vx xt, tft trace g tVxx xt, tgt dt
2
1.4
Vx xt, tgtdWt a.s.
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:
xt ϕt, ω t
t0
fxs, t, s, ωds t
gxs, t, s, ωdWs,
1.5
t0
where ϕt, 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.
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3
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 ≤ s ≤ t < ∞}. Let C2,1 Rd × R ; R stand for the family of all real-valued functions V x, t
defined on Rd × R such that they are continuously twice differentiable at x and once at t. For
any p ∈ 1, ∞, we define
L
p
R ;R
d×m
ϕ : R × Ω −→ Rd×m | ϕ· is measurable Ft -adapted and
T
p
ϕt dt < ∞ a.s. for every T > 0 ,
2.1
0
L R , Lp 0, t; Rd×m
ψ : Δ × Ω −→ Rm×d | ψt, · ∈ Lp 0, t; Rd×m .
Definition 2.1. A d-dimensional quasi-Itô process is an Rd -valued continuous adapted process
xt x1 t, . . .,xd tT on t ≥ 0 of the form
xt ϕt t
0
ft, sds t
gt, sdWs,
2.2
0
where f f1 , . . . , fd T ∈ LR , L1 0, t; Rd , g gij d×m ∈ LR , L2 0, t; Rd×m , for
all s ≥ 0, f·, s and g·, s are continuous, and ϕt is an Ft -adapted continuous stochastic
process.
We will say that xt has quasistochastic differential dxs or Dxs for t ≥ 0 given by
dxs dϕs ft, sds gt, sdWs,
2.3
Dxs ft, sds gt, sdWs,
2.4
or
in which Dxs dxs − dϕs.
t
Remark 2.2. Definition 2.1 is well defined under the condition that 0 gt, sdWs is
t
continuous for t. The proof of the continuity of 0 gt, sdWs 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.
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Lemma 2.3 see 2, page 116. Letting g : Rd × Ω → R be BRd × F-measurable, then g could
be approximated pointwise by bounded functions of the form
m
φk xψk ω.
2.5
k1
Lemma 2.4. Let φt, s : R2 → R be BR2 -measurable, and for all s ≥ 0, g·, s is continuous;
then φt, s is approximated by functions on the form
m
gk1 tgk2 s,
2.6
k1
where gk1 t is continuous and gk2 s is BR1 -measurable for every k ∈ {1, 2, . . . , m}.
Similar to Theorem 1.2, it again raises the following question. If xt is a quasi-Itô
process and V ∈ C2,1 Rd × R ; R, then whether the compound function V xt, t is a quasiItô 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 xt be a d-dimensional quasi-Itô process on t ≥ 0 with the quasistochastic
differential
dxs dϕs ft, sds gt, sdWs,
2.7
Dxs ft, sds gt, sdWs,
2.8
or
with Dxs dxs − dϕs. Here f, g are defined as Definition 2.1, ϕt is a continuous stochastic
process, and for every t ∈ 0, ∞, ϕt is F0 -measurable. Let V ∈ C2,1 Rd × R ; R and
ht, s ϕt s
ft, rdr 0
s
gt, rdWr,
t ∈ 0, ∞.
2.9
0
Then V xt, t is a quasi-Itô process with the quasistochastic differential given by
dV xs, s
dV ϕs, 0
2.10
1
Vt ht, s, s Vx ht, s, sft, s trace g T t, sVxx ht, s, sgt, s ds
2
Vx ht, s, sgt, sdWs a.s.,
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5
or
1
DV xs, s Vt ht, s, s Vx ht, s, sft, s trace g T t, sVxx ht, s, sgt, s ds
2
Vx ht, s, sgt, sdWs a.s.
2.11
with DV xs, s dV xs, s − dV ϕs, 0.
Proof. Setting t∗ ∈ 0, ∞ is arbitrary and
∗
yt ϕt t
∗
ft , sds 0
t
gt∗ , sdWsa.s.
2.12
0
By Itô’s formula, we can derive that for any t ≥ 0,
V yt, t
V ϕt∗ , 0
t
1
Vt ys, s Vx ys, s ft∗ , s trace g T t∗ , sVxx ys, s gt∗ , s ds 2.13
2
0
t
Vx ys, s gt∗ , sdWs a.s.
0
So V yt∗ , t∗ V xt∗ , t∗ . Setting t t∗ , then we have
V xt∗ , t∗ V ϕt∗ , 0
t∗ 1
Vt ys, s Vx ys, s ft∗ , s trace g T t∗ , sVxx ys, s gt∗ , s ds
2
0
t∗
Vx ys, s gt∗ , sdWs
0
V ϕt∗ , 0
t∗ Vt ht∗ , s, s Vx ht∗ , s, sft∗ , s
0
t∗
1
trace g T t∗ , sVxx ht∗ , s, sgt∗ , s ds
2
Vx ht∗ , s, sgt∗ , sdWs.
0
Since t∗ is arbitrary, 2.10 must be required. The proof is complete.
2.14
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Remark 2.6. When ϕt, ft, s and gt, s, are independent of t, that is, when ϕt x0, ft, s fs and gt, s gs, then it is easy to check that
ht, s x0 s
frdr s
0
grdWr xs,
2.15
0
and the generalized Itô’s formula becomes classical Itô’s formula.
Example 2.7. Suppose that
xt mt t
tWsds t
0
e− cos t−s/2Ws dWs,
2.16
0
where mt is a continuous function.
We find yt x2 t. Here we have
ht, s mt s
0
tWrdr s
e− cos t−r/2Wr dWr
0
2.17
mt tWs e− cos t−s/2Ws − e− cos t ,
where Ws obtain
Dyt s
0
Wrdr. Let V x, t x2 . Then Vt 0, Vx 2x, Vxx 2. So by 2.11 we
2mt 2tWs 2e− cos t−s/2Ws − 2e− cos t tWs e−2 cos t−s2Ws ds
2 mt tWs e− cos t−s/2Ws − e− cos t e− cos t−s/2Ws dWs.
2.18
Therefore,
x t 2
t 2mt 2tWs 2e− cos t−s/2Ws − 2e− cos t tWs e−2 cos t−s2Ws ds
0
t 2 mt tWs e− cos t−s/2Ws − e− cos t e− cos t−s/2Ws dWs e2t .
2.19
0
Sometimes function ϕt, ω is required to be Ft -adapted instead of F0 -measurable. We
suppose that C is the set of all absolutely continuous Ft -adapted processes. That is, if φ ∈ C,
then φt, ω is absolutely continuous for almost all ω ∈ Ω and φt, ω is Ft -measurable for
any t ≥ 0.
Theorem 2.8. Let xt be a d-dimensional quasi-Itô process on t ≥ 0 with the quasistochastic
differential
dxs dϕs ft, sds gt, sdWs,
2.20
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7
or
Dxs ft, sds gt, sdWs,
2.21
with Dxs dxs − dϕs. Here f, g are defined as Definition 2.1 and ϕ ∈ C. Let V ∈ C2,1 Rd ×
R ; R and
ht, s s
ϕ r ft, r dr 0
s
gt, rdWr
, t ∈ 0, ∞.
2.22
0
Then V xt, t is a quasi-Itô process with the quasistochastic differential given by
dV xs, s Vt ht, s, s Vx ht, s, s ϕ s ft, s
1
T
trace g t, sVxx ht, s, s gt, s ds
2
Vx ht, s, s gt, sdWs a.s.
2.23
Proof. It is easy to see that
dxs ϕ s ft, s ds gt, sdWs ft, sds gt, sdWs.
2.24
Let
ht, s s
ft, rdr 0
s
gt, rdWr,
t ∈ 0, ∞.
2.25
0
From Theorem 2.5 we have
dV xs, s Vt ht, s, s Vx ht, s, s ϕ s ft, s
1
T
trace g t, sVxx ht, s, s gt, s ds
2
Vx ht, s, s gt, sdWs a.s.
2.26
The proof is complete.
Example 2.9. Let
xt e
sin Wt
t
0
tWsds t
0
e− cos t−s/2Ws dWs.
2.27
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Find yt x2 t. Here we have
ht, s s
tWr Wre
sin Wr
cos Wr dr s
0
e− cos t−r/2Wr dWr
0
tWs e
sin Ws
e
− cos t−s/2Ws
−e
− cos t
2.28
.
So by Theorem 2.8, we obtain
x2 t t 2 tWs esin Ws e− cos t−s/2Ws − e− cos t
0
× tWs Wresin Wr cos Wr e−2 cos t−s2Ws ds
t 2.29
2tWs 2esin Ws 2e− cos t−s/2Ws − 2e− cos t e− cos t−s/2Ws dWs.
0
3. Stability in Probability of SVIEs
In this section, we use the generalized Itô’s formula to investigate the stability for the ddimensional SVIE:
xt ϕt, ωx0 t
fxs, t, s, ωds t0
t
gxs, t, s, ωdWs.
3.1
t0
Assuming further that ϕt, ω is Ft0 -measurable, ϕt0 , ω 1 and
f0, t, s 0,
g0, t, s 0,
t, s ∈ Δ a.s.
3.2
Hence, 3.1 has solution xt ≡ 0 corresponding to initial value xt0 0. This solution is
called trivial solution. For any η ∈ Ct0 , ∞; Rd , define
h η, t, s ηs ϕt, ωx0 − ϕs, ωx0 ,
LV η, t, s Vt h η, t, s , s Vx h η, t, s , s f ηs, s, ω
1
trace g T ηs, s, ω Vxx h η, t, s g ηs, s, ω .
2
3.3
Then by Theorem 2.5,
V xt, t V ϕt, ωx0 , t0 t
t0
LV x, t, sds t
t0
Vx hx, t, s, sgxs, s, ωdWs a.s.
3.4
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9
Let K denote the family of all continuous nondecreasing functions μ : R → R such that
μ0 0 and μx > 0 if x > 0. For h > 0, let Sh {x ∈ Rd : |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 ∈ K,
V x, t ≥ μ|x|
∀x, t ∈ Sh × t0 , ∞.
3.5
A function V x, t is said to be decrescent if V x, t ≤ μ|x|, x, t ∈ Sh × t0 , ∞ for some
μ ∈ K. A function V x, t defined on Rd × t0 , ∞ is said to be radially unbounded if
lim inf|x| → ∞,t≥t0 V x, t ∞.
Definition 3.1. 1 The trivial solution of 3.1 is said to be stochastically stable if for every pair
ε ∈ 0, 1 and r > 0, there exists a δ δε, r, t0 > 0 such that
P|xt; t0 , x0 | < r, t ≥ t0 ≥ 1 − ε,
3.6
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 ε ∈ 0, 1, there exists a δ0 δ0 ε, t0 > 0 such
that
P lim xt; t0 , x0 0 ≥ 1 − ε,
t→∞
3.7
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 ∈ Rd
P lim xt; t0 , x0 0 1.
t→∞
3.8
Lemma 3.2. If there exists an ε > 0, such that |ϕt, ω| > ε, t ∈ t0 , ∞ a.s. Then for any x0 ∈ Rd
and x0 / 0, one has
P|xt; t0 , x0 | /
0, t ≥ t0 1.
3.9
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 |ϕt, ω| ≤ K a.s. If there exists a positive
definite function V x, t ∈ C2,1 Sh × t0 , ∞; R , such that for any x, t, s ∈ Ct0 , ∞; Rd × Δ,
there is
LV x, t, s ≤ 0.
Then the trivial solution to 3.1 is stochastically stable.
3.10
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Proof. From the definition of the positive definite function, we know that V 0, t 0, and there
exists nonnegative nondecreasing function μx, such that V x, t ≥ μ|x| for any x, t ∈
Sh × t0 , ∞. Choose any ε ∈ 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 δε, r, t0 > 0, such that for any
t ∈ t0 , ∞, ω ∈ Ω there is
1
supV ϕt, ωx, t0 < μr.
ε x∈Sδ
3.11
Define xt xt; t0 , x0 . Choose any x0 ∈ Sδ . Let τ be the first time of xt going out the ball
Sr , that is, τ inf{t ≥ t0 : xt ∈ Sr }. By Theorem 2.5, for any t ≥ t0 , there is
V xτ ∧ t, τ ∧ t V ϕτ ∧ t, ωx0 , t0 τ∧t
LV x, τ ∧ t, sds
t0
τ∧t
3.12
Vx hx, τ ∧ t, s, sgxs, s, ωdWr.
t0
Taking the expectation for both sides, and using LV x, t, s ≤ 0, we get
EV xτ ∧ t, τ ∧ t ≤ EV ϕτ ∧ t, ωx0 , t0 .
3.13
On the other hand, we have
EV xτ ∧ t, τ ∧ t E I{τ≤t} I{τ>t} V xτ ∧ t, τ ∧ t
≥ E I{τ≤t} V xτ ∧ t, τ ∧ t
≥ μrE I{τ≤t}
3.14
μrPτ ≤ t,
So combining 3.11 and 3.14, it follows that
EV ϕτ ∧ t, ωx0 , t0
Pτ ≤ t ≤
< ε.
μr
3.15
Pτ ≤ ∞ < ε,
3.16
P|xt; t0 , x0 | < r, t ≥ t0 > 1 − ε.
3.17
Letting t → ∞, we obtain
that is,
The proof is complete.
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11
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 , ∞; R , which has infinitesimal upper bounded and
x, t ∈ Sh × t0 , ∞,
x, t, s ∈ C t0 , ∞; Rd × Δ,
μ1 |x| ≤ V x, t ≤ μ2 |x|,
LV x, t, s ≤ −μ3 |xt|,
3.18
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 ε ∈ 0, 1, there exists a
δ δ0 ε, t0 > 0, such that for any |x0 | < δ0 ,
P lim xt; t0 , x0 0 ≥ 1 − ε
t→∞
3.19
holds. Fixing ε ∈ 0, 1, in view of Theorem 3.3, there exists a δ0 δ0 ε, t0 > 0, such that
|x0 | < δ0 holds provided only that
ε
h
P |xt; t0 , x0 | < , t ≥ t0 ≥ 1 − .
2
4
3.20
Fix x0 ∈ Sδ0 , and denote xt xt; t0 , x0 . Choose any 0 < β < |x0 | and 0 < α < β. Define
stopping time
h
.
τh inf t ≥ t0 : |xt| ≥
2
τα inf{t ≥ t0 : |xt| ≤ α},
3.21
From Theorem 2.5, for any t ≥ t0 , there is
0 ≤ EV xτα ∧ τh ∧ t, τα ∧ τh ∧ t
EV ϕτα ∧ τh ∧ t, ωx0 , t0 E
τα ∧τh ∧t
LV x, τα ∧ τh ∧ t, sds
3.22
t0
≤ EV ϕτα ∧ τh ∧ t, ωx0 , t0 − μ3 αEτα ∧ τh ∧ t − t0 .
Therefore
Eτα ∧ τh ∧ t − t0 ≤
EV ϕτα ∧ τh ∧ t, ωx0 , t0
.
μ3 α
3.23
So
EV ϕτα ∧ τh ∧ t, ωx0 , t0
Pt ≤ τα ∧ τh ≤
.
μ3 αt − t0 3.24
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Letting t → ∞, it yields
Pτα ∧ τh < ∞ 1.
3.25
Clearly, from 3.20 it follows that Pτh < ∞ ≤ ε/4. Therefore
ε
1 Pτα ∧ τh < ∞ ≤ Pτα < ∞ Pτh < ∞ ≤ Pτα < ∞ .
4
3.26
ε
Pτα < ∞ ≥ 1 − .
4
3.27
So
Choose sufficiently large θα , such that
ε
Pτα < θα ≥ 1 − .
2
3.28
Then
Pτα < τh ∧ θα ≥ P{τα < θα } ∩ {τh ∞} ≥ Pτα < θα − Pτh ∞ ≥ 1 −
3ε
.
4
3.29
Again define two stopping times as
σ
τα ,
τα < τh ∧ θα ,
∞, otherwise,
and τβ inf t > σ : |xt| ≥ β .
3.30
By reason that
xt ϕt, ωx0 t
fxs, t, s, ωds t0
ϕ3 t, r, x0 , ω t
gxs, t, s, ωdWs
t0
t
fxs, t, s, ωds r
3.31
t
gxs, t, s, ωdWs,
t ≥ r,
r
in which
ϕ3 t, r, x0 , ω ϕt, ωx0 r
fxs, t, s, ωds r
t0
gxs, t, s, ωdWs.
3.32
t0
From Theorem 2.5, it follows that for any t ≥ θα , there is
EV x τβ ∧ t , τβ ∧ t ≤ EV ϕ3 τβ ∧ t, σ ∧ t, x0 , ω , σ ∧ t .
3.33
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13
Note that if ω ∈ {τα ≥ τh ∧ θα }, then
V x τβ ∧ t , τβ ∧ t V ϕ3 τβ ∧ t, σ ∧ t, x0 , ω , σ ∧ t .
3.34
Consequently,
E I{τα <τh ∧θα } V x τβ ∧ t , τβ ∧ t ≤ E I{τα <τh ∧θα } V ϕ3 τβ ∧ t, σ ∧ t, x0 , ω , σ ∧ t .
3.35
From the total probability formula, it yields that
E I{τα <τh ∧θα } V x τβ ∧ t , τβ ∧ t ≥ E I{τα <τh ∧θα } V x τβ ∧ t , τβ ∧ t | τβ ≤ t P τβ ≤ t
3.36
≥ μ1 β P τβ ≤ t .
Since μ2 · is a concave function, and by 3.18, we have
E I{τα <τh ∧θα } V ϕ3 τβ ∧ t, σ ∧ t, x0 , ω , σ ∧ t ≤ E I{τα <τh ∧θα } μ2 ϕ3 τβ ∧ t, σ ∧ t, x0 , ω ≤ Eμ2 ϕ3 τβ ∧ t, τα , x0 , ω ≤ μ2 Eϕ3 τβ ∧ t, τα , x0 , ω .
3.37
From 3.31 it follows that
Eϕ3 τβ ∧ t, τα , x0 , ω − E|xτα | ≤ Eϕ τβ ∧ t, ω x0 − ϕτα , ωx0 .
3.38
From Lemma 3.2, it is known that limα → 0 τα ∞. Hence limα → 0 τβ ∞. Letting α → 0 and
taking the limit for 3.38, there is
lim Eϕ3 τβ ∧ t, τα , x0 , ω − E|xτα | 0.
α→0
3.39
Thus we could choose sufficiently small ε1 > 0, α1 > 0, such that μ2 α1 ε1 /μ1 β < ε/4 and
Eϕ3 τβα1 ∧ t, τα1 , x0 , ω − E|xτα1 | ≤ ε1
3.40
Eϕ3 τβα1 ∧ t, τα1 , x0 , ω ≤ E|xτα1 | ε1 α1 ε1 .
3.41
hold. That is,
Combining 3.35, 3.36, 3.37, and 3.41, it follows that for sufficiently large t there is
μ2 α1 ε1 ε
P τβ ≤ t ≤
< .
4
μ1 β
3.42
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Letting t → ∞, it yields that Pτβ ≤ ∞ < ε/4. In view of 3.29, it deduces that
P τβ ∞ ∩ {σ < ∞} ≥ Pτα < τα ∧ θ − P τβ < ∞ ≥ 1 − ε,
3.43
which shows that
P ω : lim sup|xt| ≤ β ≥ 1 − ε.
t→∞
3.44
By the arbitrariness of β, we have
P ω : lim sup|xt| 0 ≥ 1 − ε.
t→∞
3.45
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 ∈ Rd , there is
P lim xt; t0 , x0 0 1.
t→∞
3.46
Choose any x0 ∈ Rd and ε ∈ 0, 1. Denote xt xt; t0 , x0 . From that V x, t is radially
unbounded and that ϕt, ω is bounded, we could find a sufficiently large h > |x0 |, such that
inf
t≥t0 ,|x|≥h
EV ϕt, ωx0 , t0
ε
≤ .
EV x, t
4
3.47
Define stopping time τh {t > t0 : |xt| ≥ h}. Then from Theorem 2.5 and conditional
property formula, we could prove that for any t > t0 , there is
EV xτh , τh Pτh ≤ t ≤ EV xτh ∧ t, τh ∧ t ≤ EV ϕτh ∧ t, ωx0 , t0 .
3.48
Therefore
EV ϕτh ∧ t, ωx0 , t0
ε
≤ .
Pτh < t ≤
EV xτh , τh 4
3.49
Letting t → ∞, it yields that Pτh < ∞ ≤ ε/4, that is:
ε
P|xt; t0 , x0 | ≤ h, t ≥ t0 ≥ 1 − .
4
3.50
Journal of Applied Mathematics
15
In the following, applying the method in Theorem 3.4, we obatin that
P lim xt; t0 , x0 0 ≥ 1 − ε.
3.51
t→∞
Hence, by the arbitrariness of ε, 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:
xt ϕt, ωx0 t
atbsxsds 0
t
atcsxsdWs,
3.52
t ∈ 0, ∞,
0
in which atbt < 0, t ≥ 0, ϕ0, ω 1.
Letting yt a−1 txt, ϕ1 t, ω a−1 tϕt, ω, then 3.52 is changed into
yt ϕ1 t, ωx0 t
asbsysds 0
t
ascsysdWs
, t ∈ 0, ∞.
3.53
0
By Lemma 2.3, it follows that
ht, s ys ϕ1 t, ωx0 − ϕ1 s, ωx0 .
3.54
Setting V x, t x2 , then
2
LV ys, t, s 2asbsy2 s ϕ1 t, ω − ϕ1 s, ω asbsx0 ys |ascs|2 ys .
3.55
If a−1 tϕt, ω is increasing and bounded almost surely and 2|bs| ≥ |ascs|2 , then
LV ys, t, s < 0. From Theorem 2.5, the solution to 3.52 is stochastically stable.
Setting V x, t x1/2 , then
1
−1/2
LV ys, t, s ys ϕ1 t, ωx0 − ϕ1 s, ωx0
asbsys
2
−3/2 2
1
a sb2 sy2 s.
− ys ϕ1 t, ωx0 − ϕ1 s, ωx0
8
If a−1 tϕt, ω is increasing and bounded almost surely, and a−1 t
LV ys, t, s ≤ −ys1/2 /2. In view of Theorem 3.3, for any x0 > 0, there is
P lim xt; t0 , x0 0 1.
t→∞
3.56
−bt, then
3.57
16
Journal of Applied Mathematics
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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