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VNU Journal of Science: Mathematics – Physics, Vol 32, No (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 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 we survey some basic notation and properties of the analysis on time scales Section 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 (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 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, bT , 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 (2016) 64-75 66 b a f ( ) [a;b) f ( )d f (s)(s) asb f ( )d f (s ) (s) ab f ( ) asb Therefore, if pR then the exponential function e p (t , t ) defined by [2, Definition 2.30, pp (a,b] 59] is solution of the initial value problem y (t ) p(t ) y(t ), y(t0 ) 1, t t0 Also if pR , e! p (t, t ) is the solution of the equation 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 2V V ( X (t )) V ( X (a)) at ( X ( ))Xi ( ) at ( X ( ))[ Xi , X j ] i, j xi x j i1 xi d V s(a,t](V ( X (s)) V ( X (s ))) s(a,t] ( X (s ))*Xi (s) x i1 i t 2V ( X (s ))(*X (s))(*X (s)), a i j s(a,t ] i, j xi x j where *Xi (s) Xi (s) Xi (s ) Almost sure exponential stability of stochastic dynamic delay equations Let T be a time scale and with fixed aT 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 sT , 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 }tT , P) be a probability space with filtration t0 ss {Ft }tT satisfying the usual conditions (i.e., {Ft } tTt is increasing and right continuous while Ft0 t0 L.A Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (2016) 64-75 67 contains all P -null sets) Denote by M the set of the square integrable Ft -martingales and by M 2r the subspace of the space M consisting of martingales with continuous characteristics Let M M with the characteristic M t (see [5]) We write L ([t ,T ], ¡ d , M ) for the set of the processes h(t ) , valued in ¡ , Ft -adapted such that E tT h2 (t )M t For any f L ([t ,T ], ¡ d , M ) we can define the stochastic integral b t f (s)M s d (see [5] in detail) Denote also by L ([t ,T ]; ¡ d ) the set of functions f :[t ,T ] ¡ d such that T t0 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 with E‖ ‖ t g : T ¡ d ¡ d ¡ d are two Borel C(t ; ¡ d ) -valued, Ft -measurable 0 Definition 2.1 An stochastic process {X (t)} t[bt ,T ] 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 ); (iii) X (t ) (t ) t t and for any t [t ,T ] and there holds the equation 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 L.A Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (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 two processes satisfying (2.2) then P{X (t) X (t) t [bt ,T ]} 1 t It is seen that t g (s, X (s ), X ( (s)))M s is Ft -martingale so it has a cadlag modification 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 M M Mˆ t Mt s(t ,t ] s (s) It is clearly that M M (2.3) Mˆ t M t s s(t ,t] (s) Denote by B the class of Borel sets in ¡ whose closure does not contain the point Let (t, A) be the number of jumps of M on the (t0 , t ] whose values fall into the set AB Since the sample functions of the martingale M are cadlag, the process (t, A) is defined with probability for all t Tt0 , AB We extend its definition over the whole by setting (t, A) 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 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 (2016) 64-75 69 Mˆ td tt ¡ uˆ( , du), we get Mˆ t Mˆ c t Mˆ d t , Mˆ d t tt ¡ u2ˆ ( , du) (2.5) 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) Further, for any T Tt , P{ sup | Kt | N} 1, (2.7) t0tT 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 Kˆc , Mˆ d t tt Kˆd , 0 and the following relation holds P{ sup Kˆtc Kˆtd N} t tT 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) (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) 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 (2016) 64-75 70 : (t, A) s(t ,t ] E[1A(M s M (s) ) | F (s) ] : Putting (t, A) E[1A(Mt M (t ) ) | F (t ) ] (t ) if ~ (t ) and (t, A) if (t) yields ~ ~ (t, A) tt ( , A) (2.9) Further, by the definition if (t) we have E Mt M ~ (t ) | F (t ) 0,(2.10) ¡ u (t, du) (t ) and E Mt M 2~ ¡ u (t, du) |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) Let C1,2 (Tt ¡ d ; ¡ ) be the set of continuous -derivative in t and all functions V (t, x) defined on Tt ¡ d , having 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 defined by d V (t, x) AV (t, x, y) (11I (t)) fi (t, x, y) (V (t, x f (t, x, y) (t)) V (t, x))(t) i1 xi d V (t, x) 2V (t, x) gi (t, x, y) g j (t, x, y)Kˆ tc gi (t, x, y)¡ u (t, du) i, j xi x j i1 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 ) (t ) if t left-scattered L.A Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (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) 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 i1 d V ( , X ( )) tt ¡ (( ) u gi ( , X ( ), X ( ( ))))ˆ( , du).(2.14) xi i1 Using the Ito formula in [13], we see that for any V C1,2 (Tt ¡ d ; ¡ ) V (t, X (t )) V (t0, X (t0)) tt (V ( , X ( )) AV ( , X ( ), X ( ( )))) (2.15) 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)‖ ‖ g(t, x1, y1) g(t, x2, y2)‖ K‖( x2 x1‖ ‖ y2 y1‖ ).(2.16) (ii) (Linear growth condition) for all (t, x, y) [t ,T ] ¡ d ¡ d ‖ f (t, x, y)‖ ‖ g(t, x, y)‖ 2 K(1‖ x‖ ‖ 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 (2016) 64-75 72 d 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 ) of the equation (2.1) is said to be almost surely exponentially stable if for any s Tt the relation log‖ X (t, s, )‖ (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 (t ) t e (t, t0 )t t a.s Suppose that there exists a positive definite function V C1,2 (Tt ¡ d ; ¡ ) satisfying c1‖ x‖ p V (t, x) (t, x) Tt ¡ d ,(2.20) and for all t t , V t (t, x) AV (t, x, y) 1V (t, x) 2V ( (t ), y) t for all x¡ d and t t 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 e (t, t0 )V (t, X (t )) V (t0, (t0 )) tt e ( , t0 )[V ( , X ( )) (1 ( ))(V ( , X ( )) AV ( , X ( ), X ( ( ))))] t e ( , t0)H t0 V (t0 , (t0 )) tt e ( , t0 )[V ( , X ( )) (1 ( ))(1V ( , X ( )) 2V ( ( ), X ( ( ))) )] t e ( , t0 )H t0 L.A Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (2016) 64-75 73 [1 (1 (t0 bt ))(t0 bt )]max V (s, (s)) bt st 0 0 t t e ( , t0 )[V ( , X ( )) (1 ( ))(3V ( , X ( )) )] t e ( , t0)H t0 Using the inequality (t ) 3 gets e (t, t0 )V (t, X (t)) [1 (1 (t0 bt ))(t0 bt )]max V (s, (s)) Ft Gt , bt st 0 0 where Ft tt (1 ( ))e ( , t0 ) ; Gt tt e ( , t0 )H 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 bt st 0 Then Y is a nonnegative special semimartingale By Theorem 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 e (t, t )V (t, X (t )) Yt for all t t a.s It then follows that 0 P{limsup e (t, t0 )V (t, X (t)) } 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 e (t, t0 )V (t, X (t)) 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 such that e (t, t0 ) e (t t0) for any t T Therefore, L.A Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (2016) 64-75 74 p lim t ln‖ X (t)‖ 0 t for all t a.s Thus, lim t ln‖ X (t )‖ t p for all t a.s The proof is completed We now consider a special case where V (t, x) ‖ x‖ Using (2.13) we have LV(t, x, y) 2(11I (t)) xT f (t, x, y) (‖ x f (t, x, y) (t)‖ ‖ x‖ 2)(t) ‖ g(t, x, y)‖ Kˆtc 2xT g(t, x, y)¡ u (t, du) ¡ (‖ x f (t, x, y) (t ) g(t, x, y)u‖ ‖ x f (t, x, y) (t )‖ 2)(t, du) We have 2(11I (t))xT f (t, x, y) (‖ x f (t, x, y) (t)‖ ‖ x‖ 2)(t) 2xT f (t, x, y)‖ f (t, x, y)‖ (t).(2.23) ~ Paying attention that (t )¡ u (t, du) and (t, du) (t, du) (t, du) yields 2 ¡ (‖ x f (t, x, y) (t ) g(t, x, y)u‖ ‖ x f (t, x, y) (t )‖ )(t, du) ¡ ‖ g(t, x, y)‖ u2(t, du) 2xT g(t, x, y)¡ u (t, du)(2.24) ~ ~ Since Kt Kˆ tc Kˆ td Kt and Kˆ td K t ¡ u2(t, du) , we can combine (2.23) and (2.24) to obtain LV(t, x, y) 2xT f (t, x, y)‖ f (t, x, y)‖ (t)‖ g(t, x, y)‖ Kt (2.25) We can impose conditions on the functions f and g such that there are 1,2 with 1 2 and a non-negative ld-continuous function satisfying 2xT f (t, x, y)‖ f (t, x, y)‖ (t)‖ g(t, x, y)‖ Kt 1‖ x‖ 2‖ y‖ t Example 2.6 Let T be a time scale t t t L tn L where tn Consider the 1 positive numbers stochastic dynamic equation on time scale T d X (t ) X (t )d t X ( ( (t )))d W (t ), t T (2.26) X (t ) a, X (0) d , 1 where W (t) is an one dimensional Brownian motion on time scale defined as in [9] We can choose 1 1,2 , 0, By directly calculating, we obtain L.A Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (2016) 64-75 75 LV(t, x, y) ( (t) 2) x2 y2,(2.27) where f (t, x, y) x, g(t, x, y) y If (t ) 1; 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Delay dynamic equations on time scales, Appl Anal., 89 (2010), no 8, 12411249 [12] M Bohner, O M Stanzhytskyi and A O Bratochkina, Stochastic dynamic equations on general time scales, Electron... Tuan / VNU Journal of Science: Mathematics – Physics, Vol 32, No (2016) 64-75 65 corollaries for the almost sure exponential stability for -stochastic dynamic delay equations on time scale and