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

Existence, Uniqueness And Asymptotic Properties of 
Neutrastochastic Functional Differential Equations with 

Markovian Switching 
Hu Rong 1,2, Hu Shigeng 1 

1Department of Mathematics, Huazhong University of Science and Technology, 

 Wuhan 430074. China 
2Department of Mathematics in Science Institute,  Wuhan University of Technology,  

Wuhan 430070. China 

Abstract    This paper considers the existence and uniqueness of solution to neutral stochastic 
functional  differential equation with Markovian switching with local Lipschitz condition but 
neither the linear growth condition. And we discuss the asymptotic properties of this solution 
including moment boundedness and moment average boundedness in time. A One-dimension 
nonlinear example is discussed to illustrate the theory. 

Keywords   Moment boundedness  Lyapunov function  Stochastic functional di fferential 
equations Markovian Switching  Generalized Ito formula 

 1. Introduction and Preliminaries 

Many practical systems may experience abrupt changes in their structure and parameters 

caused by phenomena such as component failures or repairs, changing subsystem 

interconnections, and abrupt environmental disturbances. The hybrid systems driven by 

continuous-time Markov chains have recently been developed to cope with such situation, which 

have therefore received a great deal of attention, and have played a more and more important 

role in recent years. Stochastic functional differential equations with Markovian switching have 

been studied by many authors, and we here mention [8-12], in which they mainly discuss the 

asymptotic property of the solution, including the stability and moment boundedness and so on 

with the linear growth condition. Kolmanovskii [1] studied the neutral stochastic differential 

delay equations with Markovian switching, and discussed the existence and uniqueness of the 

solution of the equation and the moment asymptotic boundedness and moment exponential 

stability. Mao [2] discussed the almost surely asymptotic stability of NSDDE. In this paper, we 

will mainly consider neutral stochastic functional differential equations with Markovian 

switching and discuss the existence and uniqueness of a global solution without the linear 

growth condition, and asymptotic properties including moment boundedness and moment 

average boundedness in time of the this global solution.  

42-54 Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2



 

 

Consider the neutral stochastic functional differential equations with Markovian switching 

of the form: 

[ ( ) ( , ( ))] ( ( ), , ( )) ( ( ), , ( )) ( ),t t td x t u x r t f x t x r t dt g x t x r t dW t               (1) 

where ( ) ( ), [ ,0]tx x t       which is regarded as in ([ ,0]; ), ( )( 0)nC R r t t   is a 

right-continuous Markovian chain on the probability space taking values in a finite state space 

 1, 2, ,S N  . Moreover 

   : ([ ,0]; ) ,n n nf R C R S R          : ([ ,0]; ) ,n n n mg R C R S R      

: ([ ,0]; ) .n n mu C R S R     

Let  },}{,, 0 PFF tt  be a complete probability space with a filtration 0}{ ttF satisfying 

the usual conditions (i.e. it is right continuous and 0F contains all P-null sets). Let 

( )( 0)W t t   be an m-dimensional Brownian motion defined on this space. Let 0  and 

)];0,([ nRC  denote the family of continuous functions  from ]0,[   to nR with the 

norm |)(|sup|||| 0    . If A is a vector or matrix, its transpose is denoted by TA . Let 

( ) 0r t t  , be a right-continuous Markovian chain on the probability space taking values in a 

finite state space },,2,1{ NS   with generator NNij  )(  given by 

jio

jio
ij

ii
itrjtrP 

 ),(

),(1{})(|)({ 
  

where 0 . Here 0ij is the transition rate from i to j if ji  while 



ij

ijii  . We 

assume that the Markovian chain )(r  is independent of the Brownian motion )(W .  

For any 2( , ) ( ; )nV x i C R S R  , define an operator LV  from 

([ ,0]; )n nR C R S    to R by 

Advances in Systems Science and Applications (2011), Vol.11, No.1-2                   43 



 

1
( , , ) ( ( , ), ) ( , , ) [ ( , , ) ( ( , ), ) ( , , )]

2
T

x xxLV x i V x u i i f x i trace g x i V x u i i g x i                      

1

( ( , ), ),
N

ij
j

V x u i j 


                                           (2) 

where 

2

1

( , ) ( , ) ( , )
( , ) , , , ( , ) .x xx

n i j n n

V x i V x i V x i
V x i V x i

x x x x


    
           

  

If x(t) is a solution to Eq.(1) and let ( ) ( ) ( , ( ))tz t x t u x r t  (as is the following), then by the 

generalized Ito


 formula, we have  

         
0

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

EV z t r t EV z r E LV z s r s ds   , 

where ( ( ), ( )) ( ( ), , ( ))tLV z t r t LV x t x r t .    

In this paper the following assumptions are imposed as standing hypothesis.        

Assumption 1.1 Both f and g are locally Lipschitz continuous. 

Assumption 1.2 For each ,i S there is constant (0,1)i   such that 

0
( , ) ( , ) ( ) ( ) ( ),iu i u i d


        


                          (3) 

where   is a probability measure and those , ([ ,0]; ).nC R     

Assume moreover that u(0,i)=0, f(0,0,i)=0, g(0,0,i)=0. 

In general, these assumptions will only guarantee a unique maximal local solution to Eq.(1) 

for any given initial data ([ ,0]; )nC R    and 0(0)r i S  . However, the additional 

conditions imposed in it, we will guarantee that this maximal local solution is in fact a unique 

global solution, which is denoted by 0( , , ),x t i  and this solution has properties 

0limsup ( , , ) ,
p

p
x

E x t i K


   *
00

1
limsup ( , , ) ,

t p

p
x

E x t i ds K
t


 


          (4) 

where 0   and 0p   are proper parameters, pK  and *
pK   are positive constants 

independent of    and 0i . 

44        Rong: Existence, Uniqueness and Asymptotic Properties of Neutral……



 

 

For the convenience of reference, several elementary inequalities are given in the following 

which will be used frequently. For any , nx y R , 

, , 0.
x y

x y
   

     
 

 
 


                      (5) 

1 1( ) (1 ) , 1,0 1.p p p p px y x y p                        (6) 

( ) ,0 1.p p px y x y p                          (7) 

2
2 2

( ) ,0 1.
1

x y
x y 

 
    


                     (8) 

Before we state our main results, let us cite several useful lemmas. 

Lemma 1.3 For any ( ) ( ; ), , 0,nh x C R R b  when ,x  ( ) ( ),h x o x
 then 

sup[ ( ) ] .
nx R

h x b x



    

In this paper, when we use the notation ( )o x


, it is always under the condition x  .  

In addition, throughout this paper, const represents a positive constant, whose precise value or 

expression is not important. ( )I x const always implies that ( )( )nI x x R is bounded above. 

Note that the notation ( )o x


 includes the continuity. Hence Lemma 1.3 can be rewritten as 

( ) .b x o x const
     

In this paper, let 2( , ) ( ) ( )
p

T n
iV x i x Q x x R  . ( )n n

iQ R i S   are positive definite 

matrices and 0p  . Clearly, we have 

2 2( , ) ,
p pp p

i iq x V x i Q x                      (9) 

here min ( )i iq Q . By (2), we have  

12( , , ) ( ) [2 ( , , ) ( , , ) ( , , )]
2

p
T T T

i i i

p
LV x i z Q z z Q f x i g x i Q g x i      

Advances in Systems Science and Applications (2011), Vol.11, No.1-2                   45 



 

2 22 2
( 2)

( ) [ ( , , )] ( ) ,
2

p p
T T T

i i ij i
j

p p
z Q z z Q g x i z Q z 

    (10) 

where ( , )z x u i  . 

Lemma 1.4 Let I be the last term of (10), then we have 

,
p

pI M z                             (11) 

Where 2 2max ( ) 0.
p p

p i ii i j i ij jM q Q     

Proof  Clearly, (11) is obtained directly. We only need to prove that 0pM  . We may 

suppose 1 2 .NQ Q Q   Noting that 1 1Q q , then  

2 2 22 2
11 1 1 11 1 1 1 1 1

1 1 1

0.
p p pp p

p j j j j
j j j

M q Q q Q q    
  

         

Lemma 1.5 Assume 1p  , let x(t) be a solution of Eq.(1) with 0x  , we have 

limsup ( ) (1 ) limsup ( ) ,
p pp

t t
E x t E z t 

 
                  (12) 

Where  1( ) ( ) ( , ( )), max .t i n iz t x t u x r t       

Proof  By (3) and (6), we have 

01( ) (1 ) ( ) ( )
p

p pE x t E x t d


    


   

 
 

            1

0
(1 ) sup ( ) sup ( ) .

p pp

s t s t
E z s E x s


 

    
   

 

This implies 1

0
0

sup ( ) (1 ) sup ( ) sup ( ) .
p p p pp

s t s t s t
E x s E z s E x s

 
  

       
    

So, we can 

get sup ( )
p

s t E x s     . Then limsup ( ) (1 ) limsup ( )
p pp

t t
E x t E z t 

 
   . 

2. A Basic Lemma 

The following lemma plays a key role in this paper. 

46        Rong: Existence, Uniqueness and Asymptotic Properties of Neutral……



 

 

Lemma 2.1 Under Assumptions 1.1 and 1.2, let 1,p   if there exist constants 

00, , , , , 0( ,1 ),i j i ija K K i S j m        positive definite matrices iQ  and probability 

measures j , such that 

( , , ) ( ( , ), )LV x i V x u i i      
0

0 ( ( ) ( ) ),j jp

i i ij j
j

a x K K d e x
  


   


                (13) 

then for any initial data ([ ,0], )nC R    and 0(0)r i S   there exists a unique global 

solution 0( , , )x t i  to Eq.(1) and this solution satisfies (4). 

Proof  For any given initial data ([ ,0]; )nC R    and 0i S , write 0( , , ) ( )x t i x t  , 

we will divide the whole proof into three steps. 

Step 1 Let us first show the existence of the global solution x(t). Under Assumption 1.1 and 1.2, 

Eq.(1) admits a unique maximal local solution ( )( )x t t    , where   is the explosion 

time. Let ( ) ( ) ( , ( ))tz t x t u x r t  , define the stopping time 

 inf 0 : ( ( ), ( )) , ( )k t V z t r t k k N      . 

Since   is bounded, when k is large enough, ( ( ), ( ))V z r k    for      , thus, 

0k  .  If    , when t  , z(t) may explode. Hence, 

 : ( ( ), ( )) , ( )t V z t r t k k N         

shows that k  . Thus, we may assume 0 ( )k k N    . Obviously, k is increasing 

and ( ) . .k k a s     . If we could show , . .a s   , then . .a s   . Thus it 

need only, for any 0t  , ( ) 0kP t    as k  . 

 

Fix 0t  . Now we prove that ( ) 0kP t    as k  . First note that if k   , 

then by the continuity of x(t) and the right continuity of r(t), ( ( ), ( ))k kV z r k   . Hence, by 

(13), we have 

Advances in Systems Science and Applications (2011), Vol.11, No.1-2                   47 



 

( ) ( ( ), ( )) ( ) ( ( ), ( ))k k k k k kkP t V z r P t EV z t r t            

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

kt
EV z i LV z s r s ds


    

       0( (0), )EV z i +
0

00
[ ( ) ( ) ( ) ]

k j j
t

r rj j
j

E K K x s d x s ds
  


  





 
   

 
   

 0

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

k kj j
t t

j j
j

EV z i K t K E d x s ds x s ds
  

 
 

 

 
      

 
 

0

0 0 0( (0) ( (0), ), ) ( ) : ,j

j t
j

V u i i K t K d K



    


     

 
 

where the index r represents r(t), max (0 )j i ijK K j m  


, and tK  is a positive constant 

independent of k. So we can get 1( ) 0( ).k tP t k K k      That shows that x(t) is a 

global solution to Eq.(1). 

Step 2 Let us now show inequality (4). By (13), we obtain that 

0
( ( ), ( )) ( (0), (0)) [ ( ( ), ( ))]

t se EV z t r t EV z r E L e V z s r s ds     

0

00
( (0), (0)) [ ( ) ( ) ( ) ]j j

t s
r rj j

j

EV z r E e K K x s d e x s ds
  


  



 
     

 
   

01 ( )
0 0 0 1( (0) ( (0), ), ) ( 1) ( ) : ,jt t

j
j

V u i i K e K e d c Ke
    


      


       

 
 

where 1c  is a positive constant independent of t and 1
0K K 


 is a positive constant 

independent of   and 0i . Hence, we have limsup ( ( ), ( )) .
t

EV z t r t K


  Then the required 

assertion (4) follows from (9) and (12).  

Step 3 Finally, using (13), we obtain that 

0
( )

t P
a E x s ds






 

0

00
( ( ), ( )) [ ( ) ( ) ( ) ]j j

r rj j
j

t
LV z s r s K K x s d x s dE s

 


  



 
     





   

0

0 0 0 2 0( (0) ( (0), ), ) ( ) : ,j

j
j

V u i i K t K d c K t



    


      

  
 

48        Rong: Existence, Uniqueness and Asymptotic Properties of Neutral……



 

 

where min i ia a  and 2c  is a positive constant independent of t. The assertion (4) follows 

directly. The proof is therefore complete. 

Denote the left side of (13) by   and establish the inequality 
0

( ( ) ( ) ) ,j j

ij j
j

K d e x I
 


    


                (14) 

where  

( ).
p p

iI a x o x
                          (15) 

By Lemma 2.1, we have ( ) .
2

p pia
x o x const
      This together with (15) yields 

.
2

pia
I x const

    Substituting this into (14) shows that the condition (13) are required. 

To get (14) and (15), some conditions imposing on the coefficients f and g. These conditions are 

considered in the next section. 

3. Main Results

Recall   to denote the left hand of (13). If p>2, by (9) (10) and (11) 

1 12 2( ) ( , , ) ( ) ( , ) ( , , )
p p

T T T
i i i ip z Q z x Q f x i p z Q z u i Q f x i       

2 22 2
1 2 3 4

( 1)
( , , ) [ ] : .

2

p pp p

i p i

p p
Q z g x i M Q z I I I I 

          (16) 

We firstly list the following conditions that we will need: 

(H1)  There exist , , 0,i ia    positive-definite matrices iQ  and a probability measure  , 

such that 

02 2 2
( , , ) ( ) ( ) ( ).T

i i ix Q f x i a x d o x
  


       


     

(H2)  There exist 0, , 0i ir r    and a probability measure  , such that 

01 1 1
( , , ) ( ) ( ) ( ).i if x i r x r d o x

  


      


    

(H3)  There exist 0, , 0,i i     positive-definite matrices iQ  and a probability measure 

 , such that 

01 1 1
( , , ) ( ) ( ) ( ).i ig x i x d o x

  


        


    

Advances in Systems Science and Applications (2011), Vol.11, No.1-2                   49 



 

We can now state our main result in this paper.  

Theorem 4.1 Under Assumptions 1.1 and 1.2, if the conditions (H1)-(H3) hold, 

2 , 2 3p     and 

2 2

2 2

1 ( )( 2)
[

1 ( 1)(1 )

p p
i i i i

i i ip p
i i

r r p p
a Q

p

 
 

 

 

   
 

  
 

2( 1)( )
(1 sgn( 2 ))],

2
i i

i

p
Q

    
                        (17) 

then for any initial data ([ ,0], )nC R    and 0(0)r i S   there exists a unique global 

solution 0( , , )x t i  to Eq.(1) and this solution satisfies (4). 

Proof Let 0( ) ( , , )x t x t i  and   be sufficiently small. Now we estimate 1 4I I  

respectively. First, by the condition (H1), the inequalities (5) and (7), we can have  

01 2 2 222
1 [ ]

p p pp
i i iI a p Q x x d




     


     

0 01 2 2 2 222 [ ][ ( )]
p p pp

i i iQ x d d op x
 

 
         

 
     

01 22
( 2) ( 2)

[
p p

p p p
i i i i

p x
Q a x ap d

p

 




 
 



 
  



  


     

0 0( 2) ( 2)
( ) ( )

p p
p p

i

p x
d o d o x

p

 
 

 

 
  




 
 

 

  
 

   

0 02 ( 2) ( ) ( 2) ( )
( ) ( )].i

p p

p
i

p s
d s d

p

 

 

   
   




 


 

  


          (18) 

Next, by the condition (H2), the inequalities (5) and (7), we obtain  

01 112
2 [( 2) ]

1

p p pp
i i

p
I Q p x p d

p 
   


  

   

01 1 1
[ ( ) ( ) ( )]i ir x r d o x

  


     


   

0
2

( 1) ( 1)
[( 2) ( 2)

1

p p
p p

i i i

p xp
Q p r x p r d

p p

 




 




 




  
   

   

50        Rong: Existence, Uniqueness and Asymptotic Properties of Neutral……



 

 

  
0 01 ( 1) ( 1)

( ) ( )
p p

p pp
i i

p x
p r d o d o x

p

 
 

 

 
   



 
 

 

  
  

   

  
0 01 ( 1) ( ) ( 1) ( )

( ) ( )].
p p

p
i i

p s
p r d s d

p

 

 

   
   



 


 

  


           (19) 

Then by the condition (H3) and the inequalities (5), (7) and (8), we can get 

02 222
3

( 1)
[ ]

2

p p pp
i i

p p
I Q x d


  


 




    

0 2 222 22
2 2

[ ( )]
1

ii

i i

dx
o x




  
 


 




 

2 22 2 2
022

( 2) (2 2)( 1)
[

2 2

p ppp pi i i
i

i i

p xp p
Q x d

v p

 




    
  

 




  
 

  

  

2 22
0 0 2 2(2 2) ( 2)

( ) ( )
1 2

p p
p pi

i

p x
d o d o x

p

 
 

 

    
 

 
 

 

  
  

    

  

2 22 2
0 0 ( 2) ( ) (2 2) ( )

( ) ( )],
1 2

p pp
i i

i

p s
d s d

p

 

 

       
 

 

 

  


        (20) 

where , (0,1)i v   are constants.  

It is easy to see that 

012
4 [ ][(1 ) ].

p p pp
p i i iI M Q x d


    


                  (21) 

Then substituting (18)-(21) into (16), we can get   whose form is similar to (14), where 

1 22
( 2) ( 2) ( 2) ( 2)

{
p

p
i i i i i

e p p e
Q a aI

p p
p

 


 


       


 


   

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

[( 2)
1

p

ip p
i ii i i

Q p e e p
e p r p r

p p p

 
   

 
      

   
  

 

2 2 2 2 2
1 2

( 1)
( 2) ]} [

2 1

p pppp i i i i i
i i i i

i i i

p p
p r p re x Q e

v
     

  

 
 

     


 

Advances in Systems Science and Applications (2011), Vol.11, No.1-2                   51 



 

2
2 2( 2) (2 2) (2 2) ( 2)

] ( ) ( ).
2 1 2

p p pi

i

e p e p
x o x o x

p p

 
   

  
       

  
  

 

If 2  , then we have 
12 ( ),

p p p

i iI p Q a x o x
       where 

2 2( 2) ( 2) ( 2) ( 2
] {

)
[1 p p

i i i ii i

e p p e
a a e

p p

 
  

 
       

  
 

  

1( 1) ( 1) ( 1) ( 1)
[( 2)

1
i p

i i i

Q p e e p
p r p r

p p p

  
 

     
  

  
 

1( 2) ]} : ( ).p
i i i ip r p re a      

By (17), we have (0) 0ia  . Since   is sufficiently small, we get 0ia  . Therefore, the 

form of I  is similar to (15). 

If 2  , then we can get 
12 ( ),

p p p

i iI p Q a x o x
       where 

2 2 2 2 21 ( 2) ( 2)
[

2 1

p p
i i i i i

i i i
i i i

p e p
a a Q e

v p


     

   

    
   

 
  

2 ( 2) ( 2)
] : ( , ).

1
i

i
i

e p
a v

p

  
 

  
 

 
  

Choosing that ( )i i i i      and by (17), we get ( , ) 0ia v  . Then we also have 

0ia  , and the form of I  is similar to (15). 

Thus, by Lemma 2.1, we can get that for any initial data   and 0i , there exists a unique global 

solution 0( , , )x t i to Eq.(1) and this solution satisfies (4). 

Theorem 4.2 Under Assumptions 1.1 and 1.2, if the conditions (H1)-(H3) hold, 

2 , 2 3p     and 

2 1 2 21 ( )( 2)
[

1 ( 1)(1 )

p p
i i i i i i

i i i
i i

r r p p
a Q

p

   
 

       
     

 

52        Rong: Existence, Uniqueness and Asymptotic Properties of Neutral……



 

 

2( 1)( )
(1 sgn( 2 ))],

2
i i

i

p
Q

    
                          (22) 

then for any initial data ([ ,0], )nC R    and 0(0)r i S   there exists a unique global 

solution 0( , , )x t i  to Eq.(1) and this solution satisfies (4). 

The proof is mostly the same as the one we provided previously, only when we estimate the 

1 3I I , we use the inequality (6) not (7). 

4. One-dimension Nonlinear Example

Let us discuss a one-dimension nonlinear neutral stochastic functional differential with 

Markovian switching to illustrate our theory. 

04 2 4 2

1
[ ( ) 0.1 ( 1)] [ ( ) ( ) ( ) ] ( ) ( ).rd x t x t b x t cx t d x t d dt x t dW t  


          

Let 3, , 3, 1, 0.1, 0, 0i i ip Q E b          , then 

0 05 5 55 3 4

1 1

1 4
( , , ) ( ) ( ),

5 5
T

i i ix Q f x i b x cx dx x d b x d o x    
 

            

04 4 4

1
( , , ) ( ),if x i b x d o x  


        

2
( , , ) .g x i x   

Now,
1 4

, , , 1, , 0.
5 5i i i i i i i i ia b r b r           By Theorem 4.1, when 

166

25ib  , we 

can conclude that there exists a unique global solution to Eq.(1), and the solution has properties 

(4) . 

References 

[1] Kolmanovskii, V.B., Neutral stochastic differential delay equations with Markovian 

switching. Stoch. Anal. Appl., 2003, 21 : 819-847. 

[2] X. Mao, Y. Shen and C. Yuan, Almost surely asymptotic stability of neutral stochastic 

differential delay equations with Markovian switching. Stochastic processes and their 

applications, 2008, 118: 1385-1406. 

[3] X. Ouyang, S. Hu, Stochastic optimization of firms investment decision under indeterminate 

conditions. Advances in systems science and applications.2008, 1: 46-52. 

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[4] S. Qu, M. Gong, A sliding mode control strategy for uncertain systems with time delays. 

Advances in systems science and applications.2008, 3: 506-512. 

[5] J. Luo, Comparison principle and stability criteria for stochastic delay differential equations 

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[6] X. Mao, Stability of stochastic differential equations with Markovian switching. Stochastic 

Processes Appl., 1999, 79: 45-67. 

[7] L. Wang, S. Hu, Comparison principle and stability of stochastic functional differential 

equations with Markovian switching. Advances in systems science and applications.2008, 2: 

220-227. 

[8] X. Mao, Exponential stability of stochastic delay interval systems with Markovian switching. 

IEEE Trans. Automat. Control., 2002, 47(10):1604-1612. 

[9] X. Mao, Robustness of stability of stochastic differential delay equations with Markovian 

switching. Stability Control:Theory Appl., 2000, 3(1): 48-61. 

[10] A. V. Svishchuk, Yu. I. Kazmerchuk,  Stability of stochastic  delay equations of It ô form 

with jumps and Markovian switchings,  and their applications in finance.  Theor. Probab. 

Math. Stat., 2002, 64: 167-178. 

[11] X. Mao, Asymptotic stability for stochastic differential delay equations with Markovian 

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54        Rong: Existence, Uniqueness and Asymptotic Properties of Neutral……