Microsoft Word - 10 Li Guangyu  Wei Fengying  Wang Ke--Some Results for Self-Stabilization of Stochastic DifferentialEquation.d


280-285                             Advances in Systems Science and Applications (2011), Vol.11, No.3-4            

                           ISSN   1078-6236   International Institute for General Systems Studies, Inc. 

Some Results for Self-Stabilization of Stochastic Differential 
Equation * 

Li Guangyu1, Wei Fengying 2 and Wang Ke 3 
2,1

 
 1Dept of  Math, Wenzhou Univercity Oujiang College, Wenzhou 325027, Zhejiang,PR China； 

2College of  Mathematics and Computer Science, Fuzhou University, Fuzhou 350002, Fujian, PR China； 
3Dept of  Math, Harbin Institute of Technology, Weihai 264209, Shandong, PR  China 

Abstract  A class of stochastic differential equation is studied about the self-stabilization in this  paper. By 
constructing suitable hypothesis, sufficient criteria for the stochastic differential equation stabilized itself are 
established. 

Keywords  Self-stabilization; Stochastic neural networks; It’s formula; Borel-Cantelli lemma  
MR(2000) Subject Classification  92B20; 93E15 

1. Introduction 

The stability of solutions is very important in the theory of differential equations. Many 
authors have done very excellent work [1-3] on the fields of stability. However, the stability for 
stochastic differential equations are somewhat difficult than that of deterministic differential 
equations, related literature can be found in recent work. For stochastic differential equations, [4] 
discussed a criterion that determines the region of mean square stability for second-order weak 
numerical schemes, [5] had given some sufficient conditions corcerning stability of solutions of 
stochastic differential evolution equations with general decay rate and [6] considered the one-step 
approximations of solutions, respectively. Here we just mentioned the work for stability about 
Mao Xuerong, for example, he established stochastic versions of the well-known Lasalle 
stability theorem in [7] and investigated exponential stability of paths for a class of Hilbert 
space-valued non-linear stochastic evolutions in [8]. Our work is motivated by Mao [1], he had 
showed that the trivial solution of equation (1.1) in [1], namely,  

( ) ( ( ), ) ( ( ), ) ( )dx t f x t t dt ug x t t dB t= +                    (1)  
was almost surely exponentially stable for all sufficiently large u ,let 0u >  be the noise 
intensity parameter and )(tB be an m -dimensional Brownian motion. Mao [1] had discussed if 

the intensity parameter u  was replaced by 0
( ) ( )

pt
r s x s ds∫ , then the equation (1.1) becomes  

0
( ) ( ( ), ) ( ( ) ( ) ) ( ( ), ) ( )

pt
dx t f x t t dt r s x s ds g x t t dB t= + ∫       (2) 

Where 0p > and ( )r s  was a continuous $R^{n\times d}$-valued function defined  

on R+ satisfying ( ) tr t Meγ≤  for all 0t ≥ ,which be called a convergence rate function.  

The standing hypothesis (H1) and (H2) are imposed in [1] as follows: 

                                                        
*Supported by NNSF of PR China(No.10701020) ; Supported by NNSF of P.R.China (No.10726062). 
 



Advances in Systems Science and Applications (2011), Vol.11, No.3-4                                281            

                        

(H1.1) There exists a symmetric positive-definite d d× -matrixQ  and three positive constants 
, ,K α β with 2β α> , such that 

2

2 2

( , ) ,

( ( , ) ( , )) ,

( , ) ,

T

T T

T T

x Qf x t K x

trace g x t Qg x t x Qx

x Qg x t x Qx

α

β

≤

≤

≥
  for all 0t ≥  and dx R∈ . 

Now, we will prove that equation (4) stabilizes itself, that is a alternative theorem given by 
expressions (7)and (8). As far as the author's knowledge that there is no corresponding results. 
To make the statement more clear, Mao [1] had stated the condition on the convergence rate 
function ( )r t as another hypothesis: 

(H1.2) There exists a pair of constants 0M > and 0γ ≥ such that  

( ) tr t Meγ≤
     for all 0t ≥ 。 

Mao [1] had proved that if (H1.1) and (H1.2) hold, then for every 0
dx R∈ , the solution of 

equation (1.2), either  

                 
0

min

2( ) ( )
(2 ) ( )

p Kr t x t dt
Qβ α λ

∞
≤

−∫
                       (3)  

or            ( )0
1limsup log ( ; ) 0

t
x t x

t→∞
<                                    (4) 

holds for almost allω∈Ω . 

If u is replaced by
0
sup ( ) ( )

s t
r s x s

≤ ≤
, then equation (1.1) becomes   

     0
( ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( )

s t
dx t f x t t dt r s x s g x t t dB t

≤ ≤
= +

                (5) 
for general 0 0t t≥ = and denote 0(0) dx x R= ∈ . 

Mao [1] had also shown that if (H1) and (H2) hold, then for every 个 0
dx R∈ , the solution of 

equation (5) has the property  

0
sup ( ) ( )

t
r t x t

≤ <∞
< ∞

   as.                           (6) 
Furthermore,     
(i) if t →∞  as min ( ( ) ( ))r t x tλ →∞ , then  

                           lim ( ) 0
t

x t
→∞

=   as.                          (7) 

(ii) if minliminf log[ ( ( ) ( ))] / 0T

t
r t r t tλ λ

→∞
≥ > ，then 

           ( )1limsup log ( )
2t

x t
t

λ
→∞

≤ −   as.                              (8) 

It is very interesting for us to analyze whether the system  

                  0
( ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( )

s t
dx t f x t t dt r s x s g x t t dB t

≤ ≤
= +

     (5) 
has the altermnative theorem like the results given by expressions (3) and (4).  



                Li: Some Results for Self-Stabilization of Stochastic Differential Equation                            

 

282 

 

To the author's knowledge, there is no corresponding results. Now we will put our efforts on 
system (5), by use of exponential martingale formula, Lyapunov function and some special 
inequalities in our paper, the trivial solution of equation (5) is self-stabilization will been shown 
in the next section. Let us begin with our paper now.  

2. Main Results 
Lemma  2.1 Let hypothesis (H1) hold. Then the solution of equation (5) has the property 

that 

0{ ( ; ) 0p x t x ≠ for all 0} 1t ≥ =  

provided 0 0x ≠ .        
   We consider the following problem of stochastic self-stabilization in this section. Suppose 
we are given a stochastic differential equation 

0
( ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( )

s t
dx t f x t t dt r s x s g x t t dB t

≤ ≤
= +

           (9) 
on 0 0t t≥ =  with initial value 0(0) dx x R= ∈  (it is just for convenience to set 0 0t =  and 

the theory clearly works for general 0 0t ≥ .  
We have the following result. 

Theorem 2.1 Let (H1) and (H2) hold. Then for every 0
dx R∈ , the solution of equation (9), 

either 

0 min

2sup ( ) ( )
(2 ) ( )s

Kr s x s
Qβ α λ≤ ≤∞

≤
−                         (10) 

or              

( )0
1limsup log ( ; ) 0

t
x t x

t→∞
<

                            (11) 
Proof. Since hypothesis (1) guarantees ( ;0) 0x t ≡ , one only need to show the conclusions 

for all 0 0x ≠ . Fix 0 0x ≠ arbitrarily, and write 0( ; ) ( )x t x x t= , by lemma 2.1 0t ≥  for all 

0( ; ) 0x t x ≠ almost surely. Suppose (10) is false, then there exists some 0 0x ≠  for 

which
_

( ) 0P Ω > ,  where   

           
0 min

2: sup ( ) ( ) .
(2 ) ( )s

Kr s x s
Q

ω
β α λ

−

≤ ≤∞

⎧ ⎫⎪ ⎪Ω = ∈Ω >⎨ ⎬−⎪ ⎪⎩ ⎭  
Clearly, one only needs to show that (11) holds for almost allω∈Ω , For each 1, 2, ,i = …  

define     

1
1 2

0 min

2: sup ( ) ( ) (1 ) .
(2 ) ( )i

s

Kr s x s i i
Q

ω
β α λ

− −
−

≤ ≤∞

⎧ ⎫⎪ ⎪Ω = ∈Ω > +⎨ ⎬−⎪ ⎪⎩ ⎭         

Now, 
1

i
i

− −∞

=
Ω∈ Ω∪ , and hence one only needs to show that for each 1i ≥ ,（10）holds for 

almost iω
−

∈Ω . Fix any 1i ≥  let ( ( ), ) log( ( ) ( ))TV x t t x t Qx t= , one then derives that  



Advances in Systems Science and Applications (2011), Vol.11, No.3-4                                283            

                        

2

2 ( ) 2 ( ) ( ) 2 ( ) 2 ( )( ( ), ) 0, ( ( ), ) , ( ( ), ) .
( ) ( ) ( ( ) ( ))

T T T

t T T

x t Q Qx t Qx t Qx t x t QV x t t Vx x t t Vxx x t t
x t Qx t x t Qx t

−
= = =

i

 
By It ô ’s formula, we obtain, we obtain 

0 0

0

0

( ( ), ) ( ( ), ) ( ( ), )
1 [ ( ( ), )(sup ( ) ( ) ) ( ( ), )(sup ( ) ( ) ) ( ( ), )]
2

( ( ), ) ( ( ), ) (sup ( ) ( ) ) ( ( ), ) ( ( ), ) ( )

1 (sup ( ) ( ) )
2

t x

T
xx

s t s t

x x
s t

s t

dV x t t V x t t dt V x t t dx

trace g x t t r s x s V x t t r s x s g x t t dt

V x t t f x t t dt r s x s V x t t g x t t dB t

r s x s

≤ ≤ ≤ ≤

≤ ≤

≤ ≤

= +

+

= +

+ 2

0

2
2

0

0

( ( ( ), ) ( ( ), ) ( ( ), ))

2 ( ) ( ( ), ) ( ) ( ( ), )2(sup ( ) ( ) ) ( )
( ) ( ) ( ) ( )

1(sup ( ) ( ) ) ( ( ( ), ) ( ) ( ) ( ( ), ))
( ( ) ( ))

2(sup

T
xx

T T

T T
s t

T T
T

s t

trace g x t t V x t t g x t t dt

x t Qf x t t x t Qg x t tdt r s x s dB t
x t Qx t x t Qx t

r s x s trace g x t t Qx t Qx t g x t t dt
x t Qx t

≤ ≤

≤ ≤

≤

= +

+

− 2 T
2

1( ) ( ) ) ( ( ( ), ) ( ) ( ) ( ( ), )) ,
( ( ) ( ))

T
T

s t
r s x s trace g x t t Qx t x t Qg x t t dt

x t Qx t≤  
this yields that 

0 0
00 0

2

00

2

0

log( ( ) ( ))

2 ( ) ( ( ), ) ( ) ( ( ), )log( ) 2 (sup ( ) ( ) ) ( )
( ) ( ) ( ) ( )

( ( ( ), ) ( ( ), ))(sup ( ) ( ) )
( ) ( )

( )
2 (sup ( ) ( ) )

T

t tT T
T

T T
s t

t T

T
s t

T

s t

x t Qx t

x s Qf x s s x s Qg x s sx Qx ds r s x s dB s
x s Qx s x s Qx s

trace g x s s Qg x s sr s x s ds
x s Qx s

x s
r s x s

≤ ≤

≤ ≤

≤ ≤

= + +

+

−

∫ ∫

∫
2

2
0

( ( ), )
.

( ( ) ( ))

t

T

Qg x s s
ds

x s Qx s∫
By hypothesis (H1.1) and the fact that

2 2
min max( ) ( ) ( ) ( ) ( ) ( )TQ x t x t Qx t Q x tλ λ≤ ≤  

for Q  is a symmetric d d×  matrix, one can show that for any 0t ≥  

2
0 0

0min 0

2

2
2

00

log( ( ) ( ))

2log( ) ( ) (sup ( ) ( ) )
( )

( ) ( ( ), )
2 (sup ( ) ( ) )

( ( ) ( ))

T

t
T

v s

Tt

T
v s

x t Qx t

Ktx Qx M t r v x v ds
Q

x s Qg x s s
r v x v ds

x s Qx s

α
λ ≤ ≤

≤ ≤

≤ + + +

−

∫

∫
              (12)      

Where        
00

( ) ( ( ), )( ) 2 (sup ( ) ( ) ) ( )
( ) ( )

t T

T
v s

x s Qg x s sM t r v x v dB s
x s Qx s≤ ≤

= ∫  

is a continuous martingale vanishing at 0t = , Let 1, 2, ,k = … Then by the exponential 
martingale inequality  



                Li: Some Results for Self-Stabilization of Stochastic Differential Equation                            

 

284 

 

1

1 2
0

2 4 (1 ) log 1( : sup ( ) ( ), ( ) ) ,
4 (1 ) 2t k

i kp M t M t M t
i k

β α βω
β β α

−

−
≤ ≤

⎡ ⎤− +
− > ≤⎢ ⎥+ −⎣ ⎦  

where 

2

2
2

00

( ) ( ( ), )
( ), ( ) 4 (sup ( ) ( ) ) .

( ( ) ( ))

Tt

T
v s

x s Qg x s s
M t M t r v x v ds

x s Qx s≤ ≤
= ∫  

whereHence the well-known Borel-Cantelli lemma yields that for almost all ω∈Ω  there 
exists a  random integer 1( )k ω  uch that for almost all 1k k≥ , 

             

1

1
0

2 4 (1 ) logsup ( ) ( ), ( )
4 (1 ) 2t k

i kM t M t M t
i

β α β
β β α

−

−
≤ ≤

⎡ ⎤− +
− ≤⎢ ⎥+ −⎣ ⎦  

that is，for 0 ,t k≤ ≤  

   

1

1

2
1

2
1 2

00

4 (1 ) log 2( ) ( ), ( )
2 4 (1 )

( ) ( ( ), )4 (1 ) log 2 (sup ( ) ( ) ) .
2 (1 ) ( ( ) ( ))

Tt

T
v s

i kM t M t M t
i

x s Qg x s si k r v x v ds
i x s Qx s

β β α
β α β

β β α
β α β

−

−

−

−
≤ ≤

+ −
≤ +

− +

+ −
= +

− + ∫
 (13) 

Substituting (13）into (12) and then applying (H1) one obtains that for each 
^

,ω∈Ω−Ω  

with 
^
Ω  a P-null set, there exists a random integer 2 ( ),k ω such that, for every 个 iω

−

∈Ω there 

exists a random number 3( )k ω such that 

      

1
1 2

0 min

2sup ( ) ( ) (1 )
(2 ) ( )s t

Kr s x s i i
Qβ α λ

−

≤ ≤
≥ +

−        for almost all 3.t k≥     

It then follows from that for almost all
^

iω
−

∈Ω −Ω， if 2 31 , ( 1)k t k k k k− ≤ ≤ ≥ ∨ +  

3

1
2

0 0 1
0min

1 1

0 0 3
min min

1
3

0 0
min

log( ( ) ( ))

2 4 (1 ) log 2log( ) (sup ( ) ( ) )
( ) 2 (1 )

2 4 (1 ) log 2 (1 )log( ) ( 1 )
( ) 2 ( )

2 ( 1) 4 (1 ) llog( )
( )

T

t
T

v sk

T

T

x t Qx t

Kt i kx Qx r v x v ds
Q i i

Kt i k K ix Qx k k
Q i Q

K k ix Qx
Q

β β α
λ β α

β
λ β α λ

β
λ

−

−
≤ ≤

− −

−

+ −
≤ + + −

− +

+ +
≤ + + − − −

−

+ +
≤ + +

∫

3
min

og 2 ( 1 ).
2 ( )

k K k k
i Qβ α λ

− − −
−  

This implies that 
1

3
0 0 3

min min

2 ( 1)1 1 4 (1 ) log 2log( ( ) ( )) log( ) ( 1 ) .
1 ( ) 2 ( )

T T K k i k Kx t Qx t x Qx k k
t k Q i Q

β
λ β α λ

−⎡ ⎤+ +
≤ + + − − −⎢ ⎥− −⎣ ⎦

  
It then follows that 

min

1 2limsup log( ( ) ( ))
( )

T

t

Kx t Qx t
t i Qλ→∞

≤ −      for almost all
^

.iω
−

∈Ω −Ω    (14) 



Advances in Systems Science and Applications (2011), Vol.11, No.3-4                                285            

                        

Thus, for almost all 
^

,iω
−

∈Ω −Ω  there exists a random number 4 ( )k ω and 
0δ > arbitrarily such that 

    
min

1 2log( ( ) ( )) 2
( )

T Kx t Qx t
t i Q

δ
λ

≤ − +            for almost all 4 ,t k≥  

from the expression 
2

min ( ) ( ) ( ) ( )TQ x t x t Qx tλ ≤  one derives that  

min

min

exp{( ) }
( )( )
( )

K t
i Qx t

Q

δ
λ
λ

− +
≤       for almost all 4.t k≥  

Consequently, 

            
0

min

1 2limsup log( ( ; ) )
( )t

Kx t x
t i Q

δ
λ→∞

≤ − +    for almost all  
^

.iω
−

∈Ω −Ω  

Since 0δ > is arbitrary, we must have that 

             0
min

1 2limsup log( ( ; ) ) 0
( )t

Kx t x
t i Qλ→∞

< − <     for almost all 

^
.iω

−

∈Ω −Ω  
The proof is now complete.  

References 

[1] X. Mao, Stochastic Differential Equations and There Applications, Horwood Publication, 
Chichester, 1997:100-141 

[2] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York Inc, 
1977:50-70 

[3] Geng Liu, Study on Capacity Expansion Problems on Directed Networks, Advances in 
Systems Science and Applications 2007, 7(4): 679-684  

[4] Xianguo Wu, Lieyun Ding, Xintian Cai, Deformation Predication of Deep Excavation Pit 
Support Structure Based on Fuzzy Neural Network, Advances in Systems Science and 
Applications 2006, 6(2), 119-126 

[5] Xiaoya Hu, Desen Zhu, Bingwen Wang, Delay Analysis of Switched Ethernet for 
Networked Control Systems, Advances in Systems Science and Applications 2006, 6(2) : 
201-206  

[6] Chen Wanyi, Globally Exponential Asymptotic stability of Hopfield Neural Network with 
Time-varying Delays, Act Scientiarwn Naturaliwn Universitatis Nankaiensis, 2005, 5 
(3):70-75   

[7] Jiang Minghui, Shen Yi, Liao Xiaoxin, Stability of Stochastic Neural Networks with 
Multi-Delay, Mathemics Applicata, 2006, 3 (19): 61-65 

[8] X. Mao, Some Contribution to Stochastic Asymptotic Stability and Boundedness Via 
Multiple Lyapunov Functions, J. Math. Anal. Appl. 2001 260: 325-340.