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[.]
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 z 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 z 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 z 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 z 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 z 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 z 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 bt K Density of hM it ; cit ; Density of hM L2 (M ) Space of all real - valued, predictable processes φ = {φt }t∈Ta satisfying R kφk2t,M = E (a,t] |φτ |2 ∇hM iτ < ∞ 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 RT a kφt k∇t < ∞; Lloc (Ta , R) Family of real valued, Ft −adapted processes {f (t)}t∈Ta RT satisfying a |f (τ )|∇τ < +∞ a.s for every T ∈ Ta ; d Lloc (Tt0 ; R ) Set of functions, valued in Rd , Ft -adapted such that RT t0 f (τ )∇τ < +∞ for all T ∈ Ta ; d Lloc (Tt0 ; R , M ) Set of functions, valued in Rd , Ft -adapted such that RT E t0 h2 (τ )∇hM iτ < +∞) ∀ 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; hM i Characteristic of the martingale M ; iii z P Ms − Mρ(s) ; ct M = Mt − Rn n− dimensional Euclidean space; R, Z, N, N0 Real numbers, the integers, the natural numbers, s∈(a,t] 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) bt Ψ = t − ρ(t) (Backward graininess); ct ; Density of jumps of M [a, b] = {t ∈ T : a t b}; iv z 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 z 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 z ... 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. .. 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. .. 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