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VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 Almost Sure Exponential Stability of Stochastic Differential Delay Equations on Time Scales Le Anh Tuan* Faculty of Fundamental Science, Hanoi University of Industry, Tu Liem, Hanoi, Vietnam Received 16 August 2016 Revised 15 September 2016; Accepted 09 September 2016 Abstract: The aim of this paper is to study the almost sure exponential stability of stochastic differential delay equations on time scales. This work can be considered as a unification and generalization of stochastic difference and stochastic differential delay equations. Keywords: Delay equation, almost sure exponential stability, Ito formula, Lyapunov function. 1. Introduction The stochastic differential/difference delay equations have come to play an important role in describing the evolution of eco-systems in random environment, in which the future state depends not only on the present state but also on its history. Therefore, their qualitative and quantitative properties have received much attention from many research groups (see [1, 2] for the stochastic differential delay equations and [3-6] for the stochastic difference one). In order to unify the theory of differential and difference equations into a single set-up, the theory of analysis on time scales has received much attention from many research groups. While the deterministic dynamic equations on time scales have been investigated for a long history (see [7-11]), as far as we know, we can only refer to very few papers [12-15] which contributed to the stochastic dynamics on time scales. Moreover, there is no work dealing with the stochastic dynamic delay equations. Recently, in [14], we have studied the exponential p -stability of stochastic  -dynamic equations on time scale, via Lyapunov function. Continuing the idea of this article [14], we study the almost sure exponential stability of stochastic dynamic delay equations on time scales. Motivated by the aforementioned reasons, the purpose of this paper is to use Lyapunov function to consider the almost sure exponential stability of  -stochastic dynamic delay equations on time scale T . The organization of this paper is as follows. In Section 1 we survey some basic notation and properties of the analysis on time scales. Section 2 is devoted to giving definition and some theorems, _______  Tel.: 84-915412183 Email: tuansl83@yahoo.com 64 L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 65 corollaries for the almost sure exponential stability for  -stochastic dynamic delay equations on time scale and some examples are provided to illustrate our results. 2. Preliminaries on time scales Let T be a closed subset of ¡ , enclosed with the topology inherited from the standard topology on ¡ . Let  (t )  inf{s T : s  t}, (t)   (t)  t and (t)  sup{s T : s  t}, (t)  t  (t) (supplemented by sup  inf T,inf   sup T ). A point t T is said to be right-dense if  (t )  t , right-scattered if  (t)  t , left-dense if (t)  t , leftscattered if (t )  t and isolated if t is simultaneously right-scattered and left-scattered. The set k T is defined to be T if T does not have a right-scattered minimum; otherwise it is T without this rightscattered minimum. A function f defined on T is regulated if there exist the left-sided limit at every left-dense point and right-sided limit at every right-dense point. A regulated function is called ldcontinuous if it is continuous at every left-dense point. Similarly, one has the notion of rd-continuous. For every a, bT , by [a,b], we mean the set {t T : a  t  b} . Denote Ta  {t  T : t  a} and by R (resp. R  ) the set of all rd-continuous and regressive (resp. positive regressive) functions. For   any function f defined on T , we write f for the function f  ; i.e., ft  f ((t)) for all t  T f (s) by f (t ) or ft if this limit exists. It is easy to see that if t is left-scattered k and  lim (s)t  then ft  ft . Let  I ={ t: t is left-scattered}. Clearly, the set I of all left-scattered points of T is at most countable. Throughout of this paper, we suppose that the time scale T has bounded graininess, that is *  sup{ (t ): t k T}   . Let A be an increasing right continuous function defined on T . We denote by A the Lebesgue t  -measure associated with A . For any A -measurable function f : T ¡ we write a f A for the integral of f with respect to the measures A on (a, t ] . It is seen that the function t  at f A is cadlag. It is continuous if A is continuous. In case A(t)  t we write simply t t a f  for a f A . For details, we can refer to [7]. In general, there is no relation between the  -integral and  -integral. However, in case the integrand f is regulated one has b b k a f (  )  a f ( ) ,  a, b T . Indeed, by [7, Theorem 6.5], L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 66 b a f ( )  [a;b) f ( )d   f (s)(s) asb f (  )d   f (s ) (s)  ab f (  ) . asb Therefore, if pR then the exponential function e p (t , t ) defined by [2, Definition 2.30, pp. 0  (a,b] 59] is solution of the initial value problem y (t )  p(t ) y(t ), y(t0 )  1, t  t0. Also if pR , e! p (t, t ) is the solution of the equation 0 y (t )   p(t ) y(t ), y(t0 ) 1, t  t0, where ! p(t )   p(t ) 1 (t ) p(t ) . Theorem 1.1 (Ito formula, [16]). Let X  ( X1 ,L , X d ) be a d  tuple of semimartingales, and let V : ¡ d  ¡ d be a twice continuously differentiable function. Then V ( X ) is a semimartingale and the following formula holds d V 1 2V V ( X (t ))  V ( X (a))   at ( X (  ))Xi ( )   at ( X (  ))[ Xi , X j ] 2 i, j xi x j i1 xi d V s(a,t](V ( X (s)) V ( X (s )))  s(a,t]  ( X (s ))*Xi (s)  x i1 i 1 t 2V ( X (s ))(*X (s))(*X (s)),      a i j 2 s(a,t ] i, j xi x j where *Xi (s)  Xi (s)  Xi (s ). 3. Almost sure exponential stability of stochastic dynamic delay equations Let T be a time scale and with fixed aT . We say that the rd-map  (): T  T is a delay function if  (t)  t  for all t T and sup{t   (t ): t T}   . For any sT , we see that bs : min{ (t): t  s}   . Denote s  { (t): t  s}[bs , s] . We write simply  for  s if there is no confusion. Let C(s ; ¡ d ) be the family of continuous functions from  s to ¡ d with the norm ‖ ‖ s  sup | (s)|. Fix t0 T and let (, F ,{Ft }tT , P) be a probability space with filtration t0 ss {Ft }tT satisfying the usual conditions (i.e., {Ft } tTt is increasing and right continuous while Ft0 0 t0 L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 67 contains all P -null sets). Denote by M 2 the set of the square integrable Ft -martingales and by M 2r the subspace of the space M 2 consisting of martingales with continuous characteristics. Let M M 2 with the characteristic  M t (see [5]). We write L ([t ,T ], ¡ d , M ) for the set of the processes 2 0 h(t ) , valued in ¡ , Ft -adapted such that E tT h2 (t )M t  . 0 For any f L ([t ,T ], ¡ d , M ) we can define the stochastic integral 2 0 b t f (s)M s 0 d (see [5] in detail). Denote also by L ([t ,T ]; ¡ d ) the set of functions f :[t ,T ] ¡ d such that 1 0  T t0 0 f (t )t  . We now consider the  -stochastic dynamic delay equations on time scale d  X (t )  f (t, X (t ), X ( (t )))d t  g (t, X (t ), X ( (t )))d M (t ), t  T   t0  (2.1)  X ( s   ( s )  s  ,  t0  where f : T  ¡ d  ¡ d  ¡ d ;   { (s): bt  s  t0} is a 0 2 with E‖ ‖ t   . 0 g : T  ¡ d  ¡ d  ¡ d are two Borel C(t ; ¡ d ) -valued, Ft -measurable 0 0 Definition 2.1. An stochastic process {X (t)} t[bt ,T ] 0 functions and and random variable , valued in ¡ d , is called a solution of the equation (2.1) if (i) {X (t )} is Ft -adapted; (ii) f (, X ( ), X ( ())) L ([t ,T ]; ¡ d ) and g(, X ( ), X ( ())) L ([t ,T ], ¡ d , M ); 1 0 2 0 (iii) X (t )   (t )  t t and for any t [t ,T ] and there holds the equation 0 0 X (t )   (t0 )  tt f (s, X (s ), X ( (s)))s  tt g(s, X (s ), X ( (s)))M s ,  t [t0,T ], (2.2) 0 0 L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 68 The equation (2.1) is said to have the uniqueness of solutions on [bt ,T ] if X (t ) and X (t ) are 0 two processes satisfying (2.2) then P{X (t)  X (t) t [bt ,T ]} 1. 0 t It is seen that t g (s, X (s ), X ( (s)))M s is Ft -martingale so it has a cadlag modification. 0 Hence, if X (t ) satisfies (2.2) then X (t ) is cadlag. In addition, if M t is rd-continuous, so is X (t ) . For any M M , set 2 M  M . Mˆ t  Mt   s(t ,t ]  s  (s)  0 It is clearly that  M   M  . (2.3) Mˆ t  M t   s s(t ,t]   (s)  0 Denote by B the class of Borel sets in ¡ whose closure does not contain the point 0 . Let  (t, A) be the number of jumps of M on the (t0 , t ] whose values fall into the set AB . Since the sample functions of the martingale M are cadlag, the process  (t, A) is defined with probability 1 for all t  Tt0 , AB . We extend its definition over the whole  by setting  (t, A)  0 if the sample t  Mt () is not cadlag. Clearly the process  (t, A) is Ft -adapted and its sample functions are nonnegative, monotonically nondecreasing, continuous from the right and take on integer values. We also define ~ ˆ(t, A) for Mˆ t by a similar way. Let  (t, A)  é{s (t0, t]: Ms  M (s)  A} . It is evident that ~  (t, A)  ˆ(t, A)   (t, A). (2.4)  (t,),ˆ(t,) and  (t,.) are measures. ~ Further, for fixed t , The functions ~  (t, A),ˆ(t, A) and  (t, A), t Tt are Ft -regular submartingales for fixed A . 0 By Doob-Meyer decomposition, each process has a unique representation of the form ~ ~ ~  (t, A)   (t, A)   (t, A), ˆ(t, A)  ˆ(t, A)  ˆ(t, A),  (t, A)   (t, A)   (t, A), ~ where  (t, A),ˆ (t, A) and  (t, A) are natural increasing integrable processes ~ and  (t, A),ˆ(t, A) ,  (t , A) are martingales. We find a version of these processes such that they are measures when t is fixed. By denoting Mˆ tc  Mˆ t  Mˆ td , Where L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 69 Mˆ td  tt ¡ uˆ( , du), 0 we get Mˆ t  Mˆ c t  Mˆ d t , Mˆ d t  tt ¡ u2ˆ ( , du). (2.5) 0 Throughout this paper, we suppose that  M t is absolutely continuous with respect to Lebesgue measure  , i.e., there exists Ft -adapted progressively measurable process Kt such that M t  tt K  . (2.6) 0 Further, for any T Tt , 0 P{ sup | Kt | N}  1, (2.7) t0tT where N is a constant (possibly depending on T ). ˆ c  and Mˆ d  are absolutely continuous with respect to The relations (2.3), (2.5) imply that M t t  on T . Thus, there exists Ft -adapted, progressively measurable bounded process Kˆ tc and Kˆ td satisfying Mˆ c t  tt Kˆc , Mˆ d t  tt Kˆd  , 0 0 and the following relation holds P{ sup Kˆtc  Kˆtd  N}  1. t tT 0 Moreover, it is easy to show that ˆ (t, A) is absolutely continuous with respect to  on T , that is, it can be expressed as  ˆ (t, A)  tt ( , A) ,(2.8) 0  (t , A) . Since B is generated by a ˆ (t , A) such that the map t   ˆ (t, A) is countable family of Borel sets, we can find a version of  ˆ (t , ) is a measure.Hence, from [2.5] we see that measurable and for t fixed,  with an Ft -adapted, progressively measurable process  Mˆ d t  tt ¡ u2 ( , du) . 0 This means that  Kˆtd  ¡ u2 (t, du). ~ The process  (t , A) is written in the specific form as following L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 70 :  (t, A)  s(t ,t ] E[1A(M s  M  (s) ) | F  (s) ]. 0 : Putting (t, A)  E[1A(Mt  M  (t ) ) | F  (t ) ]  (t ) if ~  (t )  0 and (t, A)  0 if  (t)  0 yields ~ ~  (t, A)  tt ( , A) .(2.9) 0 Further, by the definition if  (t)  0 we have   E Mt  M ~  (t ) | F  (t )    0,(2.10) ¡ u (t, du)   (t ) and  E  Mt  M   2~ ¡ u (t, du)   2   |F    (t )   (t )  M t  M   (t )  .  (t )  (t ) ~ Let (t, A)  (t, A)  (t, A) . We see from (2.4) that  (t, A)  tt ( , A) . 0 Let C1,2 (Tt ¡ d ; ¡ ) be the set of 0 continuous  -derivative in t and all functions V (t, x) defined on Tt ¡ d , having 0 continuous second derivative in x. For any V C1,2 (Tt ¡ d ; ¡ ) , define the operators AV : Tt  ¡ d  ¡ d  ¡ with respect to (2.1) is 0 0 defined by d V (t, x) AV (t, x, y)   (11I (t)) fi (t, x, y)  (V (t, x  f (t, x, y) (t)) V (t, x))(t) i1 xi  d V (t, x) 1 2V (t, x)   gi (t, x, y) g j (t, x, y)Kˆ tc   gi (t, x, y)¡ u (t, du) 2 i, j xi x j i1 xi ¡ (V (t, x  f (t, x, y) (t)  g(t, x, y)u) V (t, x  f (t, x, y) (t)))(t, du),(2.11) where 0 if t left-dense (t )   1  (t ) if t left-scattered  L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 71 Theorem 2.2 (Ito formula, [13]). Let X  ( X1 ,L , X d ) be a d  tuple of semimartingales, and let V : ¡ d  ¡ d be a twice continuously differentiable function. Then V ( X ) is a semimartingale and the following formula holds V (t, X (t ))  V (t0, X (t0 ))  tt LV( , X (  ), X ( ( )))  Ht .(2.12) 0 Where  LV(t, x, y)  V t (t, x)  AV (t, x, y),(2.13) and ( )  V ( , X (  )  f ( , X (  ), X ( ( ))) ( )  g( , X (  ), X ( ( )))u) V ( , X (  )  f ( , X (  ), X ( ( ))) ( )). ~ d V ( , X (  )) ·  t Ht   tt gi ( , X (  ), X ( ( )))M t ¡ ( ) ( , du)  xi 0 i1 0 d V ( , X (  )) tt ¡ (( )   u gi ( , X (  ), X ( ( ))))ˆ( , du).(2.14) xi 0 i1 Using the Ito formula in [13], we see that for any V C1,2 (Tt  ¡ d ; ¡  ) 0  V (t, X (t )) V (t0, X (t0))  tt (V  ( , X (  ))  AV ( , X ( ), X ( ( )))) (2.15) 0  is a locally integrable martingale, where V t is partial  -derivative of V (t, x) in t . We now give conditions guaranteeing the existence and uniqueness of the solution to the equation (2.1). Theorem 2.3. (Existence and uniqueness of solution). Assume that there exist two positive constants K and K such that (i) (Lipschitz condition) for all xi , yi  ¡ d i  1,2 and t [t0 ,T ] ‖ f (t, x1, y1)  f (t, x2, y2)‖ 2 ‖ g(t, x1, y1)  g(t, x2, y2)‖ 2  K‖( x2  x1‖ 2 ‖ y2  y1‖ 2 ).(2.16) (ii) (Linear growth condition) for all (t, x, y) [t ,T ] ¡ d  ¡ d 0 ‖ f (t, x, y)‖ 2 ‖ g(t, x, y)‖ 2 K(1‖ x‖ 2 ‖ y‖ 2).(2.17) Then, there exists a unique solution X (t ) to equation (2.1) and this solution is a square integrable semimartingale. L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 72 d 0 and  C(s ; ¡ ) , there exists a unique solution X (t, s, ), t  bs of the equation 2.1 satisfying X (t, s, )   (t) for any t s . Further, f (t,0,0)  0; g(t,0,0)  0, t Ta.(2.18) that for any s  t We suppose Definition 2.4. The trivial solution X (t )  0 of the equation (2.1) is said to be almost surely exponentially stable if for any s  Tt the relation 0 log‖ X (t, s, )‖  0 (2.19) t holds for any  C(s ; ¡ d ). limsup t Theorem 2.5. Let satisfying that 1,2 , p, c1 be positive numbers with 1  2 . Let  be a positive number    and let  be a non-negative ld-continuous function defined on Tt such 1   (t ) 3 0  t e (t, t0 )t t   a.s.. 0 Suppose that there exists a positive definite function V C1,2 (Tt  ¡ d ; ¡  ) satisfying 0 c1‖ x‖ p  V (t, x) (t, x) Tt ¡ d ,(2.20) 0 and for all t  t , 0  V t (t, x)  AV (t, x, y)  1V (t, x)  2V ( (t ), y) t for all x¡ d and t  t . 0 a.s.,(2.21) Then, the trivial solution of equation (2.1) is almost surely exponentially stable. Proof. Let      . By (2.12), (2.21) and calculating expectations we get 3 1 2 e (t, t0 )V (t, X (t ))  V (t0, (t0 ))  tt e ( , t0 )[V ( , X (  )) 0  (1  ( ))(V  ( , X (  ))  AV ( , X (  ), X ( ( ))))]  t e ( , t0)H t0  V (t0 , (t0 ))  tt e (  , t0 )[V ( , X (  )) 0 (1   ( ))(1V (  , X (  ))  2V ( ( ), X ( ( )))  )]  t e ( , t0 )H t0 L.A. Tuan. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 73  [1 (1  (t0  bt ))(t0  bt )]max V (s, (s)) bt st 0 0 0 0 t t e (  , t0 )[V (  , X (  ))  (1   ( ))(3V (  , X (  ))  )]  t e ( , t0)H . t0 0 Using the inequality  1   (t )  3 gets e (t, t0 )V (t, X (t))  [1 (1  (t0  bt ))(t0  bt )]max V (s, (s))  Ft  Gt , bt st 0 0 0 0 where Ft  tt (1   ( ))e (  , t0 )  ; Gt  tt e ( , t0 )H . 0 0 In view of the hypotheses we see that F  lim Ft  . t Define Yt  [1 (1  (t  bt ))(t  bt )]max V (s, (s))  Ft  Gt for all t Tt . 0 0 0 0 bt st 0 0 0 Then Y is a nonnegative special semimartingale. By Theorem 7 on page 139 in [17], one sees that {F  } { lim Yt exists and finite} a.s.. t By P{F  } 1 . So we must have P{ lim Yt exists and finite} 1. t Note that 0  e (t, t )V (t, X (t ))  Yt for all t  t a.s.. It then follows that 0 0 P{limsup e (t, t0 )V (t, X (t))  }  1. t So limsup e (t, t0 )V (t, X (t ))    t  a.s..(2.22) Consequently, there exists a pair of random variables   t and 0 e (t, t0 )V (t, X (t))     0 such that for all t   a.s.. Using (2.20), we have c1e (t,t0)‖ X (t)‖ p  e (t,t0)V (t, X (t))   for all t  a.s.. Since the time scale T has bounded graininess, there is a constant   0 such that e (t, t0 )  e  (t t0) for any t  T . Therefore,