Tính ổn định của phương trình động lực ngẫu nhiên trên thang thời gian (stability of stochastic dynamic equations on time scales)

VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Le Anh Tuan STABILITY OF STOCHASTIC DYNAMIC EQUATIONS ON TIME SCALES THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI – 2018 VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF SCIENCE LE ANH TUAN STABILITY OF STOCHASTIC DYNAMIC EQUATIONS ON TIME SCALES Speciality: Probability Theory and Mathematical Statistics Speciality Code: 62.46.01.06 THESIS FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY IN MATHEMATICS Supervisor: PROF DR NGUYEN HUU DU HANOI – 2018 This work has been completed at VNU-University of Science under the supervision of Prof Dr Nguyen Huu Du I declare hereby that the results presented in it are new and have never been used in any other thesis Author: Le Anh Tuan Acknowledgments First and foremost, I want to express my deep gratitude to Prof Dr Nguyen Huu Du for accepting me as a PhD student and for his help and advice while I was working on this thesis He has always encouraged me in my work and provided me with the freedom to elaborate my own ideas I would like to express my special appreciation to Professor Dang Hung Thang, Doctor Nguyen Thanh Dieu, other members of seminar at Department of Probability theory and mathematical statistics and all friends in Professor Nguyen Huu Du’s group seminar for their valuable comments and suggestions to my thesis I would like to thank the VNU of Science for providing me with such an excellent study environment Furthermore, I would like to thank the leaders of Faculty of Fundamental Science, Hanoi University of Industry, the Dean board as well as to the all my colleagues at Faculty of Fundamental Science for their encouragement and support throughout my PhD studies Finally, during my study, I always get the endless love and unconditional support from my family: my parents, my parents-in-law, my wife, my little children and my dearest aunt I would like to express my sincere gratitude to all of them Thank you all Abstract The theory of analysis on time scales was introduced by S Hilger in 1988 (see [26]) in order to unify the discrete and continuous analyses and simultaneously to construct mathematical models of systems that are unevenly evolving over time, reflecting real models Since was born, the theory of analysis on time scales has received much attentions from many research groups One of most important problems in analysis on time scales is to consider the quantity and quality of dynamic equations such as the existence and uniqueness of solutions, numerical methods for solving these solutions as well the stability theory However, so far, almost results related to the analysis on time scales are mainly in deterministic analysis, i.e., there are no random factors involved to dynamic equations Thus, these results only describe models developed in non-perturbed environmental conditions Obviously, such these models are not fitted to actual practice and we must take into account the random factors that affect the environment Therefore, the transfer of analytical results studying determinate models on time scales to stochastic models is an urgent need As far as we know, for the stochastic analysis on time scales, there are not many significant results, especially, results related to the stability of stochastic dynamic equations and stochastic dynamic delay equations Some results in this field can be referred to [13, 14, 40, 41, 44, 60, ] For the above reasons, we have chosen the doctoral thesis research topic as ”Stability of stochastic dynamic equations on time scales” Thesis is concerned with the following issues: • Studying the existence and uniqueness of solutions for ∇- stochastic dynamic delay equations: giving the definition of stochastic dynamic delay equations and the concept of solutions; proving theorems of existence and uniqueness of solutions; estimating the rate of the converi gence in Picard approximation for the solutions Proving theorem of existence and uniqueness of solutions under locally Lipschitz condition and estimating moments of solutions for stochastic dynamic equations on time scales • Studying the stability of ∇-stochastic dynamic equations and ∇-stochastic dynamic delay equations on time scale T by using methods of Lyapunov functions It is known that the theory of stochastic calculus is one of difficult topics in the probability theory since it relates to many basic knowledges like Brownian motions, Markov process and martingale theory Therefore, the theory of stochastic analysis on time scales is much more difficult because the structure of time scales is divert That causes very complicated calculations when we carry out familiar results from stochastic calculus to similar one on time scales Besides, some estimates of stochastic calculus for stochastic calculus on R are not automatically valid on an arbitrary time scale Therefore, it requires to reformulate these estimates and to find new suitable techniques to approach the problem ii List of Notations A Defined on the set C 1,2 (Ta × Rd ; R), is called generator; B Class of Borel sets in R; Crd Set of rd-continuous functions f : T −→ R ; Cld Set of ld-continuous functions f : T −→ R ; C 1,2 (Ta × Rd ; R) Family of all functions V (t, x) defined on Ta × Rd such that they are continuously ∇−differentiable in t and twice continuously differentiable in x; Ft+ = ∩s>t Fρ(s) ; (Ω, F, P, {Ft }t∈Ta )Stochastic basis; ft− = f (t−) = limσ(s)↑t f (s); I1 = {t : t is left-scattered}; I2 = {t : t is right-scattered}; I = I1 ∪ I2 ; Kt Density of M t ; Kt Density of M t ; L2 (M ) Space of all real - valued, predictable processes φ = {φt }t∈Ta satisfying φ t,M =E (a,t] |φτ | ∇ M τ < ∞ for all t ∈ Ta ; L2 ((a, b]; M ) Restriction of L2 (M ) on (a, b]; L1 ((a, T ]; Rd ) Set of all Ft −adapted process φt satisfying T a Lloc (Ta , R) φt ∇t < ∞; Family of real valued, Ft −adapted processes {f (t)}t∈Ta satisfying d Lloc (Tt0 ; R ) |f (τ )|∇τ < +∞ a.s for every T ∈ Ta ; Set of functions, valued in Rd , Ft -adapted such that T t0 d Lloc (Tt0 ; R , M ) T a f (τ )∇τ < +∞ for all T ∈ Ta ; Set of functions, valued in Rd , Ft -adapted such that E T t0 h (τ )∇ ∇ M τ < +∞) ∀ T ∈ Tt0 ; LV =V + AV ; Mloc Set of the locally square-integrable Ft − martingales; Mr2 Subspace of the space M2 consisting of martingales with continuous characteristics; M Characteristic of the martingale M ; iii Mt = Mt − Rn n− dimensional Euclidean space; R, Z, N, N0 Real numbers, the integers, the natural numbers, s∈(a,t] Ms − Mρ(s) ; and the nonnegative integers; R R Set of all regressive and rd-continuous functions f ; + T Ta kT k T Set of positive regressive element of R(T, R); Time scale; ={x ∈ T : x a}, a ∈ T; T \ {M } if T has a right-scattered minimum M min = T otherwise;  T \ {M } if T has a left-scattered maximum M max max = T otherwise; ρ(t) Backward operator; σ(t) Forward operator; µ(t) = σ(t) − t (Forward graininess); ν(t) = t − ρ(t) (Backward graininess); Ψt Density of jumps of Mt ; [a, b] = {t ∈ T : a t b}; iv Contents Page Abstract i List of Notations iii Introduction Chapter 1 Preliminaries 12 1.1 Survey on analysis on time scale 12 1.2 Differentiation 15 1.2.1 Continuous functions 15 1.2.2 Nabla derivative 16 1.2.3 Lesbesgue ∇− integral 18 1.2.4 Exponential function 21 1.3 1.4 Stochastic processes on time scales 23 1.3.1 Basic notations of probability theory 23 1.3.2 Stochastic processes on time scales 23 1.3.3 Martingales 25 ∇−stochastic integral 27 1.4.1 ∇−stochastic integral with respect to square integrable martingale 27 v 1.4.2 ∇−stochastic integral with respect to locally square integrable martingale 30 1.4.3 ∇−stochastic integral with respect to semimartingale 31 1.5 1.6 Itˆo’s formula 32 1.5.1 Quadratic co-variation 32 1.5.2 Itˆo’s formula 33 Martingale problem 35 1.6.1 Counting processes for discontinuous martingales 35 1.6.2 Martingale problem formulation 38 Chapter The stability of ∇-stochastic dynamic equations 40 2.1 Solutions of stochastic dynamic equations 41 2.2 Locally Lipschitz condition on existence and uniqueness of solutions 42 2.3 Finiteness of moments 47 2.4 Exponential p-stability of stochastic dynamic equations 49 2.4.1 Sufficient condition 50 2.4.2 Necessary condition 51 2.5 Stochastic stability of stochastic dynamic equations 64 2.5.1 Basic definitions 64 2.5.2 Sufficient conditions 65 2.6 Almost sure exponential stability of stochastic dynamic equations 71 2.7 Conclusion of Chapter 74 Chapter The stability of ∇−stochastic dynamic delay equations 3.1 76 ∇-stochastic dynamic delay equations 77 3.1.1 ∇-stochastic dynamic delay equations 77 3.1.2 Solutions of stochastic dynamic delay equations 78 vi Then, the equation (3.3) is uniformly exponentially p-stable Proof Let s t0 be fixed Denote by X(t) the solution of the equation (3.3) with the initial condition X(τ ) = ξ(τ ) for any τ ∈ Γs For each integer n > 0, define the stopping time θn = inf{t s: X(t) n} Obviously, θn → ∞ as n → ∞ almost surely By (3.13), (3.21) and calculating expectations we get E[eα1 (θn ∧ t, s)V (θn ∧ t, X(θn ∧ t))] θn ∧t = V (s, ξ(s)) + E eα1 (τ− , s) α1 V (τ− , X(τ− )) s + (1 + α1 ν(τ ))LV (τ, X(τ− ), X(r(τ ))) ∇τ V (s, ξ(s)) θn ∧t eα1 (τ− , s) α1 V (τ− , X(τ− )) + (1 + α1 ν(τ )) − +E s α1 V (τ− , X(τ− )) + α1 ν(τ ) α2 e α1 (τ− , r(τ )) V (r(τ ), X(r(τ ))) ∇τ V (s, ξ(s)) + α2 ν(τ ) θn ∧t α2 (1 + α1 ν(τ ))e α1 (τ− , r(τ ))eα1 (τ− , s) +E V (r(τ ), X(r(τ )))∇τ + α ν(τ ) s + Since the function α2 (1+α1 x) 1+α2 x α2 (1+α1 x) x→∞ 1+α2 x is increasing in x and lim α2 (1 + α1 ν(τ )) + α2 ν(τ ) = α1 , α2 (1 + α1 ν∗ ) =: α3 < α1 + α2 ν∗ Further, by [6, Theorem 2.36, pp 62], e Therefore, with t α1 (τ− , r(τ ))eα1 (τ− , s) = eα1 (r(τ ), s) s, sup Eeα1 (θn ∧ τ, s)V (θn ∧ τ, X(θn ∧ τ )) s τ t t V (s, ξ(s)) + α3 E eα1 (r(τ ), s)V (r(τ ), X(r(τ )))1[s,θn ] (τ )∇τ s 88 Let S = {t s} It is seen that S ⊂ [s, s + r∗ ] Therefore, s : r(t) sup Eeα1 (θn ∧ τ, s)V (θn ∧ τ, X(θn ∧ τ )) s τ t V (s, ξ(s)) + r∗ α3 sup Eeα1 (r(τ ), s)V (θn ∧ r(τ ), X(θn ∧ r(τ )) τ ∈S E[eα1 (θn ∧ τ, s)V (θn ∧ τ, X(θn ∧ τ ))]∇τ + α3 (s,t]\S c2 ξ(s) p + r∗ α3 c2 eα1 (h, s) ξ p s t sup E[eα1 (θn ∧ u, s)V (θn ∧ u, X(θn ∧ u))]∇τ + α3 s s u τ− t p s c4 ξ + α3 sup E[eα1 (θn ∧ u, s)V (θn ∧ u, X(θn ∧ u))]∇τ s s u τ− where c4 = c2 + r∗ α3 c2 eα1 (h, s) and h = min{t ∈ T : t s + r∗ } Using Gronwall-Bellman inequality yields sup Eeα1 (θn ∧ τ, s)V (θn ∧ τ, X(θn ∧ τ )) c4 ξ s τ t p s eα3 (t, s) Hence, c1 E X(t) where α = p EV (t, X(t)) eα (t, s) c4 ξ ps = c4 ξ ps eα3 eα1 (t, s) α1 −α3 1+α3 ν∗ α1 (t, s) c4 ξ ps e α (t, s), Thus, E X(t) p c4 ξ ps e c1 α (t, s) The proof is complete 3.2.2 Examples We now consider a special case Let P be a positive definite matrix and V (t, x) = x P x, where x is the transpose of a vector x By using 89 (3.12) and by directly calculating we obtain LV (t, x, y) = x P f (t, x, y) + f (t, x, y) P x + f (t, x, y) P f (t, x, y)ν(t) + g(t, x, y) P g(t, x, y)Kt (3.22) Example 3.2.3 Let T be a time scale containing and r(t) be a delay function Consider the stochastic dynamic delay equation on time scale T  d∇ X(t) = AX(t )d∇ t + BX(r(t))d∇ W (t) − (3.23) X(s) = ξ(s) ∀ s ∈ Γ , t ∈ T , 0 where A and B are d×d matrices and W (t) is an one dimensional Brownian motion on time scale defined as in [23] It is known from [24, Theorem 2.1, pp 1678] Kt = By using V (t, x) = x and (3.22) we have LV (t, x, y) = x (A + A + A Aν(t))x + y B BKt y (3.24) Suppose that the spectral abscissa of the matrix A + A + A Aν(t) is uniformly bounded by a negative constant −α1 From the equation (3.24) we see that LV (t, x, y) −α1 x + B y (3.25) It is easy to see that e−α1 (t−s) e α1 (t, s) ∀ t ∈ Ts , (see [41] for details) Suppose that there exists a positive constant α2 such α2 that α2 < α1 and B er∗ α1 1+ν ∗ α2 For this assumption and (3.25) we obtain LV (t, x, y) − α1 x + α1 ν(t) + α2 e α1 (t− , r(t)) y + α2 ν(t) Therefore, assumptions of Theorem 3.2.2 are satisfied with p = 2, it means the trivial solution of the equation (3.23) is exponentially stable in mean square Example 3.2.4 Let T be a time scale defined by ∞ T = P ,1 = k=1 90 5k 5k + , 4 Let r(t) be a delay function satisfying r∗ = supt∈T (t − r(t)) = 14 Consider the stochastic dynamic delay equation on time scale T  d∇ X(t) = AX(t ) + X(r(t)) d∇ t + BX(t )d∇ W (t), t − − X(s) = ξ(s) ∀ s ∈ Γ , t0 (3.26) t0 where W (t) is a Brownian motion as in Example 3.2.3 and A, B are the × matrices defined by     11 − −3 18 18             2 ; B =   A= −2 − − −   3 18          25 − 23 − 73 18 − 18 36 With the Lyapunov function V (t, x) = x , we obtain by (3.22) 2A + A Aν(t) + B BKt x + x (I + A ν(t))y + y y In this case Kt = Therefore,  73 17  19 − 36 36 72       17 91 53  H := 2A + A Aν∗ + B BKt =  − −  36 36 72      19 379 − 53 72 72 − 144 LV (t, x, y) = x Further, I + Aν∗ = η(H) = − 27 16 Thus, LV (t, x, y) and the spectral abscissa of the matrix H is x Hx + I + A ν∗ y + y 27 143 x 2+ x y + y − x + y (3.27) 16 144 143 Setting α1 := 144 , α2 := then α1 , α2 satisfy the inequalities 12 eα1 r∗ < α2 1+ν∗ α2 Combining these estimations and (3.27), we obtain α2 LV (t, x, y) −α1 x + e−α1 r∗ y + ν∗ α α1 α2 e α1 (t− , r(t)) − x 2+ y + α1 ν(t) + α2 ν(t) − 91 By virtue of Theorem 3.2.2 the trivial solution of the equation (3.26) is exponentially stable in mean square 3.3 Almost sure exponential stability of dynamic delay equations Definition 3.3.1 The trivial solution X(t) ≡ of the equation (3.3) is said to be almost surely exponentially stable if for any s ∈ Tt0 the relation lim sup t→∞ ln X(t, s, ξ) < a.s t (3.28) holds for any ξ ∈ C(Γs ; Rd ) Theorem 3.3.2 Let α1 , α2 , p, c1 be positive numbers with α1 > α2 Let α < α1 and let η be a non-negative α be a positive number satisfying 1+αν(t) ld-continuous function defined on Tt0 such that ∞ eα (τ− , t0 )ηt ∇t < ∞ a.s t0 Suppose that there exists a positive definite function V ∈ C 1,2 (Tt0 ×Rd ; R+ ) satisfying c1 x p V (t, x) ∀(t, x) ∈ Tt0 × Rd , (3.29) and for all t t0 , x ∈ Rd V ∇t (t, x) + AV (t, x, y) −α1 V (t− , x) + ηt a.s (3.30) Then, the trivial solution of the equation (3.3) is almost surely exponentially stable Proof By (3.13), (3.30) and calculating expectations we get t eα (t, t0 )V (t, X(t)) = V (t0 , ξ(t0 )) + eα (τ− , t0 ) αV (τ− , X(τ− )) t0 t + (1 + αν(τ )) V ∇τ (τ, X(τ− )) + AV (τ, X(τ− ), X(δ(τ ))) ∇τ + eα (τ, t0 )∇Hτ t0 t V (t0 , ξ(t0 )) + eα (τ− , t0 ) αV (τ− , X(τ− )) t0 t + (1 + αν(τ )) − α1 V (τ− , X(τ− )) + ητ ∇τ + eα (τ, t0 )∇Hτ t0 92 Using the inequality α 1+αν(t) < α1 gets eα (t, t0 )V (t, X(t)) V (t0 , ξ(t0 )) + Ft + Gt , where t t (1 + αν(τ ))eα (τ− , t0 )ητ ∇τ ; Gt = Ft = eα (τ, t0 )∇Hτ t0 t0 In view of the hypotheses we see that F∞ = limt→∞ Ft < ∞ Define Yt = V (t0 , ξ(t0 )) + Ft + Gt for all t ∈ Tt0 Then Y is a nonnegative special semimartingale By Theorem on page 139 in [39], one sees that {F∞ < ∞} ⊂ { lim Yt exists and finite} a.s t→∞ By P {F∞ < ∞} = So we must have P { lim Yt exists and finite} = t→∞ Note that that eα (t, t0 )V (t, X(t)) Yt for all t t0 a.s It then follows P {lim sup eα (t, t0 )V (t, X(t)) < ∞} = t→∞ So lim sup [eα (t, t0 )V (t, X(t))] < ∞ a.s (3.31) t→∞ Consequently, there exists a pair of random variables υ > t0 and ξ > such that eα (t, t0 )V (t, X(t)) ξ for all t υ a.s Using (3.29), we have c1 eα (t, t0 ) X(t) p eα (t, t0 )V (t, X(t)) ξ for all t υ a.s Since the time scale T has bounded graininess, there is a constant β > such that eα (t, t0 ) > eβ(t−t0 ) for any t ∈ T Therefore, ln X(t) β + p lim for all t υ a.s t→∞ t Thus, ln X(t) β lim − for all t υ a.s t→∞ t p The proof is completed 93 3.4 Conclusion of Chapter In this chapter, the thesis has resolved to the problems: - Defined delay functions, given some basic notions of stochastic dynamic delay equations on time scales - Stated and proved the theorems of existence and uniqueness of solutions for stochastic dynamic delay equations under Lipschitz condition and locally Lipschitz condition - Estimated the rate of the convergence in Picard approximation for the solutions - Gave the concept of the exponential p; given theorems of sufficient condition for the exponential p-stability via Lyapunov functions - Provided some illustrative examples for the exponential p-stability - Gave the concept of the almost sure exponential stability; constructed the Lyapunov function and given theorems of sufficient condition for the almost sure exponential stability The result of this chapter is written on the basis of the paper • N H Du, L A Tuan and N T Dieu (2017), Stability of stochastic dynamic equations with time-varying delay on time scales, it has been accepted to Asian-European Journal of Mathematics 94 Conclusion In the dissertation, we have obtained the following main results: • Given the theorem for existence and uniqueness of solutions for stochastic dynamic equations on time scales under locally Lipschitz condition • Constructed the Lyapunov function to evaluate the exponential pmoment stability, stochastic stability and exponential almost sure stability of stochastic dynamic equations on time scales • Introduced concepts and theorems, examples of exponential p-moment stability, stochastic stability, exponential almost sure stability of stochastic dynamic equations on time scales • Defined delay function and stochastic dynamic delay equations on time scales • Given the theorems of existence and uniqueness of solutions for stochastic dynamic delay equations on time scales • Introduced concepts and theorems, examples of the exponential pmoment stability, exponential almost sure stability of stochastic dynamic delay equations on time scales Here are some of our future research directions: • Give necessary conditions for the exponential p-moment stability, stochastic stability, exponential almost sure stability of stochastic dynamic equations and stochastic dynamic delay equations on time scales • Release some conditions in the theorems of the thesis to obtain the most general theorems • Provides formulas to calculate the stable radius for stochastic dynamic equations on time scales • Consider theorems of convergence of the solution of dynamic equations on different time scales 95 Parts of the thesis have been published in: [1] N H Du, N T Dieu and L A Tuan (2015), Exponential p-stability of stochastic ∇-dynamic equations on disconnected sets, Electron J Diff Equ., 285, 1-23 [2] L A Tuan, N H Du and N T Dieu (2017), On the stability of stochastic dynamic equations on time scales, Journal Acta Mathematica Vietnamica, (online), 1-14 DOI: 10.1007/s40306-017-0220-5 [3] N H Du, L A Tuan and N T Dieu (2017), Stability of stochastic dynamic equations with time-varying delay on time scales, it has been accepted to Asian-European Journal of Mathematics And have been presented at: Seminar about ”Stochastic dynamic equations on time scales” at the 7th floor, VIASM, 2013, 2014, 2015, 2016, 2017 Seminar about ”Stochastic dynamic equations on time scales” at group Math - Biology, Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, 2012, 2013, 2014 5th National Conference ”Probability - Statistics: Research, Application and Teaching”, DaNang, Vietnam, 23–25, May, 2015 Scientific Conference, Faculty of Mathematics, Mechanics and Informatics, VNU University of Science, 2016 National Conference ”Vietnam - Korea workshop on selected topics in Mathematics”, DaNang, Vietnam, 20-24, February, 2017 96 Bibliography [1] E Akin-Bohner and Y N Raffoul (2006), Boundedness in Functional Dynamic Systems on Time scales, Advances in Difference the equations, 1-18 [2] K B Athreya and S N Lahiri (2006), Measure Theory and Probability Theory, Springer Science Business Media, LLC [3] L Arnold (1974), Stochastic Difference the equations: Theory and Applications, John Wiley and Sons [4] V B Baji´c, D LJ Debeljkovi´c, B B Bogi´cevic and M B Jovanovic (1998), Non-Lyapunov stability robustness consideration for discrete linear descriptor systems, IMA Journal of Mathematical Control & Information, 15, 105-115 [5] S Bhamidi, S N Evans, R Peled and P Ralph (2008), Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan, IMS Collections in Probability and Statistics: Essays in Honor of David A Freedman, 42-75 [6] M Bohner and A Peterson (2001), Dynamic the equations on Time scales, Birkhăauser Boston, Massachusetts [7] M Bohner and A Peterson (2003), Advances in Dynamic the equations on Time scales, Birkhăauser Boston, Basel, Berlin [8] M Bohner, O M Stanzhytskyi and A O Bratochkina (2013), Stochastic dynamic equations on general time scales, Electronic Journal of Differential the equations, 57, 1-15 97 [9] A Cabada and D R Vivero (2006), Expression of the Lebesgue ∆integral on time scales as a usual Lebesgue integral: Application to the calculus of ∆-antiderivatives, Mathematical and Computer Modeling, 43, 194 - 207 [10] J J Dacunha (2005), Stability for time varying linear dynamic systems on time scales, J of Computational and Applied Mathematics, 176, 381 - 410 [11] J M Davis, I A Gravagne, R J Marks II and A A Ramos (2010), Algebraic and Dynamic Lyapunov the equations on Time Scales, 42nd South Eastern Symposium on System Theory University of Texas at Tyler, TX, USA, March, 7-9 ă Ufuktepe (2009), Lebesgue - Stieltjes measure on [12] A Denizand and U time scales, Turk J Math, 33, 27 - 40 [13] N H Du and N T Dieu (2011), The first attempt on the stochastic calculus on time scale, Stochastic Analysis and Applications, 29, 1057 - 1080 [14] N.H Du and N.T Dieu (2013), Stochastic dynamic equations on time scales, Acta Mathematica Vietnamica, 38, 317-338 [15] N H Du, N T Dieu and L A Tuan (2015), Exponential p-stability of stochastic ∇-dynamic equations on disconnected sets, Electron J Diff Equ., 285, 1-23 [16] N H Du and L H Tien (2007), On the exponential stability of dynamic equations on time scales, J Math Anal Appl., 331, 1159 1174 [17] Q Feng and B Zheng (2012), Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications, Appl Math Comput., 15, 7880-7892 [18] S Foss and T Konstantopoulos (2004), An overview of some stochastic stability methods, J of the Operations Research, 47, 275 - 303 98 [19] I I Gihman and A V Skorokhod (1979), The theory of stochastic processes III, Springer - Verlag, New York Inc [20] I I Gihman and A V Skorokhod (1972), Stochastic differential equations, Springer - Verlag, Berlin [21] T E Govindan (2012), Existence and stability of solutions of stochastic semilinear functional differential equations, Stochastic Analysis and Applications, 6, 1257 - 1280 [22] V M Gundlach and O Steinkamp (2000), Product of random rectangular matrices, Math Nachr., 212, 54 - 76 [23] D Grow and S Sanyal (2011), Brownian motion indexed by a time scale, Stochastic Analysis and Applications, 29, 457 - 472 [24] D Grow and S Sanyal (2012), The quadratic variation of Brownian motion on a time scale, Statist Probab Lett., 9, 1677-1680 [25] R Z Has’minskii (1980), Stochastic stability of differential equation, Sijthoff & Noordhoff [26] S Hilger (1988), Ein Maòkettenkalkă aul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D thesis, Universităaat Wăaurzburg [27] J Hoffacker and C.C Tisdell (2005), Stability and instability for dynamic equations on time scales, J Computers and Mathematics with Applications, 49, 1327 - 1334 [28] K Itˆo (1944), Stochastic Integral, Proc Imp Acad Tokyo, 20, 519 524 [29] K Itˆo (1951), On a formula concerning stochastic differentials, Nagoya Math J., 3, 55 - 65 [30] K Itˆo (1951), On stochastic differential equations, Mem Amer Math Soc., 4, - 51 [31] L Kallenberg (2001), Foundations of modern probability, Springer Verlag, New York Berlin Heidelberg 99 [32] D Kannan and B Zhan (2002), A discrete - time Itˆo’s formula, Stochastic Analysis and Applications, 20, 1133 - 1140 [33] N Kazamaki (1972), On the existence of the solutions of martingale integral equations, Tohoku Math Journ., 24, 463 - 468 [34] R Z Khas’minskii (1986), Stochastic Stability of Difference the equations, Alphen: Sijtjoff and Noordhoff (translation of the Russian edition, Nauka, Moscow) [35] V B Kolmanovskii and V R Nosov (1986), Stability of Functional Differential equations, Academic Press [36] H Kunita and S Wantanabe (1967), On square integrable martingales, Nagoya Math J., 30, 209 - 245 [37] H J Kushner (1967), Stochastic Stability and Control, Academic Press [38] V Lakshmikantham, S Sivasundaram and B Kaymakcalan (1996), Dynamic systems on measure chains, Kluwer Academic Publishers, Dordrecht, The Netherlands [39] R Sh Lipster and A N Shiryayev (1986), Theory of Martingales, Kluwer Academic Publishers (translation of the Russian edition, Nauka, Moscow) [40] X L Liu, W X Wang and J Wu (2010), Dynamic delay equations on time scales, Appl Anal., 8, 1241-1249 [41] X L Liu ( 2007), Boundedness and exponential stability of solutions to dynamic equations on time scales, Electron J Diff Equ., 12, 1-14 [42] C Lungan and V Lupulescu (2012), Random dynamical systems on time scales, Electronic Journal of Differential the equations, 86, 14 [43] Li Yan, Zhang Weihai and Liu Xikui (2013), Stability of nonlinear stochastic discrete-time systems, J Appl Math Art ID 356746 100 [44] Y Ma and J Sun (2007), Stability criteria of delay impulsive systems on time scales, Nonlinear Anal., 4, 1181-1189 [45] X Mao (1990), Exponential stability for stochastic differential equations with respect to semimartingale, Stochastic Processes and their Applications, 35, 267 - 277 [46] X Mao (1989), Lyapunov functions and almost sure exponential stability of stochastic differential equations based on semimartingale with spatial parameters, SIAM Journal on Control and Optimization, 28, 343-355 [47] X Mao (1997), Stochastic differential equations and their applications, Horwood publishing chichester [48] X Mao (2001), Almost sure exponential stability of delay equations with damped stochastic perturbation, Stochastic Analysis and Application, 19, 67-84 [49] X Mao (2003), Asymptotic Stability and Boundedness of stochastic differential equations with respect to semimartingales, Stochastic Analysis and Applications, 21, 737 - 751 [50] X Mao, D J Higham and A M Stuart (2002), Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40, 1041 - 1063 [51] H P McKean(1969), Stochastic Integrals, Academic Press, New York [52] P Medvegyev (2007), Stochastic integration theory, Oxford University Press Inc, New York [53] P A Meyer (1967), Intdgrales stochastiques I, II, Lecture notes in mathematics, Springer - Verleg, Berlin, 39, 72 - 117 [54] P A Meyer (1962), A decomposition theorem for supermartingales, Ill J Math., 6, 193 - 205 101 [55] P A Meyer and C Dolans-Dade (1970), Intgrales stochastiques par rapport aux martingales locales, S´eminaire de Probabilit´es IV, Lecture Notes in Mathematics, 124, 77 - 107 [56] M M Mili´c and V B Baji´c (1987), Quanlitative Analysis of Motion Properties of Semistate Models of Large-Scale Systems, Circuits Systems Signal Process, 6, 315-334 [57] A C Peterson and C C Tisdell (2004), Boundedness and uniqueness of solutions to dynamic equations on time scales, J Difference Equ Appl., 10, 1295-1306 [58] P E Protter (1977), On the existence, uniqueness, convergence and explosions of system of stochastic integral equations, The Annals of Probability, 5, 243 - 261 [59] P E Protter (2004), Stochastic integration and differential equations, Springer-Verlag Berlin Heidelberg [60] S Sanyal (2008), Stochastic dynamic equations, Ph.D Dissertation, Applied Mathematics, Missouri University of Science and Technology [61] L Socha (2000), Exponential stability of singularly perturbed stochastic systems, IEEE Transactions on Automatic Control, 45, 576 - 580 [62] A Tartakovsky (1998), Asymptotically optimal sequential tests for nonhomogeneous processes [63] T N Thiele (1880), Surla compensation de quelques erreurs quasisystematiques par la methode des moindres carres, Reitzel, Copenhagen [64] W Vervaat (1979), On the stochastic difference equation and a representation on nonnegative infinitely random variables, Adv Appl Probab., 11, 750 - 783 [65] Z Yang and D Xu (2007), Mean square exponential stability of impulsive stochastic difference equations, Applied Mathematics Letters, 20, 938 - 945 102 ... Stochastic dynamic equations on time scales With the concept of stochastic integral on time scales, we can consider the notion of stochastic dynamic equations on time scales Here, we mention some of. .. proposed stochastic dynamic equations on time scales: definition of solutions of stochastic dynamic equations; stating and proving the theorems of existence and uniqueness of solutions for stochastic. .. stability of ∇ -stochastic dynamic equations and ∇ -stochastic dynamic delay equations on time scale T by using methods of Lyapunov functions It is known that the theory of stochastic calculus is one of