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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 TIEU LUAN MOI download : skknchat@gmail.com 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: 62.06 THESIS FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY IN MATHEMATICS Supervisor: PROF. NGUYEN HUU DU HANOI – 2018 TIEU LUAN MOI download : skknchat@gmail.com This work has been completed at VNU-University of Science under the supervision of Prof. 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 TIEU LUAN MOI download : skknchat@gmail.com Acknowledgments First and foremost, I want to express my deep gratitude to Prof. 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 Depart- ment 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 Fundamen- tal Science, Hanoi University of Industry, the Dean board as well as to the all my colleagues at Faculty of Fundamental Science for their encourage- ment and support throughout my PhD studies. Finally, during my study, I always get the endless love and uncondi- tional 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.

TIEU LUAN MOI download : skknchat@gmail.com 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 un- evenly 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., 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 ex- istence and uniqueness of solutions; estimating the rate of the conver- i TIEU LUAN MOI download : skknchat@gmail.com 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 Lya- punov 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 TIEU LUAN MOI download : skknchat@gmail.com 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 hM it ; K bt Density of hM cit ; L2 (M ) Space of all real - valued, predictable processes φ = {φt }t∈Ta satisfying kφk2t,M = E (a,t] |φτ |2 ∇hM iτ < ∞ for all t ∈ Ta ; R 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 1 (Ta , R) Family of real valued, Ft −adapted processes {f (t)}t∈Ta RT satisfying a |f (τ )|∇τ < +∞ a.

for every T ∈ Ta ; Lloc d 1 (Tt0 ; R ) Set of functions, valued in Rd , Ft -adapted such that RT t0 f (τ )∇τ < +∞ for all T ∈ Ta ; Lloc d 2 (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 2 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 TIEU LUAN MOI download : skknchat@gmail.com P
M ct = Mt − s∈(a,t] Ms − Mρ(s) ; Rn n− dimensional Euclidean space; R, Z, N, N0 Real numbers, the integers, the natural numbers, and the nonnegative integers; R Set of all regressive and rd-continuous functions f ; + R Set of positive regressive element of R(T, R); T Time scale; Ta ={x ∈ T : x > a}, a ∈ T; T \ {M } if T has a right-scattered minimum M min min kT = T otherwise;  T \ {M } if T has a left-scattered maximum M k max max T = T otherwise; ρ(t) Backward operator; σ(t) Forward operator; µ(t) = σ(t) − t (Forward graininess); ν(t) = t − ρ(t) (Backward graininess); Ψ bt Density of jumps of M ct ; [a, b] = {t ∈ T : a 6 t 6 b}; iv TIEU LUAN MOI download : skknchat@gmail.com Contents Page Abstract i List of Notations iii Introduction 1 Chapter 1 Preliminaries 12 1.1 Survey on analysis on time scale .3 Stochastic processes on time scales. Basic notations of probability theory. Stochastic processes on time scales. ∇−stochastic integral with respect to square inte- grable martingale.

27 v TIEU LUAN MOI download : skknchat@gmail. ∇−stochastic integral with respect to locally square integrable martingale. ∇−stochastic integral with respect to semimartingale 31 1. Quadratic co-variation.

Counting processes for discontinuous martingales. Martingale problem formulation. 38 Chapter 2 The stability of ∇-stochastic dynamic equations 40 2.1 Solutions of stochastic dynamic equations .2 Locally Lipschitz condition on existence and uniqueness of solutions .3 Finiteness of moments .4 Exponential p-stability of stochastic dynamic equations .5 Stochastic stability of stochastic dynamic equations .6 Almost sure exponential stability of stochastic dynamic equa- tions .7 Conclusion of Chapter 2. 74 Chapter 3 The stability of ∇−stochastic dynamic delay equations 76 3.1 ∇-stochastic dynamic delay equations.

∇-stochastic dynamic delay equations. Solutions of stochastic dynamic delay equations. 78 vi TIEU LUAN MOI download : skknchat@gmail. Existence and uniqueness of solutions.

Rate of the convergence. Locally Lipschitz condition on existence and unique- ness of solutions .2 Exponential p-stability of stochastic dynamic delay equations 87 3.3 Almost sure exponential stability of dynamic delay equations 92 3.4 Conclusion of Chapter 3. 94 Bibliography 96 TIEU LUAN MOI download : skknchat@gmail.com Introduction Stochastic calculus is one mathematical field studying mathematical analytical calculations (integral, differential equations, continuity, stability of solution.) towards the stochastic processes in order to build mathe- matical models for dynamic systems under effects of random factors. Con- sequently, stochastic analysis has many applications in biology, medicine, physics, economy, social sciences., and it attracts a lot of attention from mathematicians.

So far, stochastic analysis with continuous time and dis- crete time has been well studied. But in practice, most systems are not fully functional and are not completely evenly spaced. Sometimes, obser- vations intermingle both continuous and discrete times. For example, some worms develop only during the summer but in the winter their growth is interrupted.

Therefore, in many cases, differential equations or difference equations are insufficient to describe the required information of the model. Hence, in recent years, one focuses on the so-called analysis on time scales. Time scales The theory of analysis on time scale, which was introduced by S. Hilger in his PhD thesis [26], has been born in order to unify continuous and discrete analysis.

The results of analytical calculations on time scales allow us to construct mathematical models of systems that are unevenly evolving over time, reflecting real models. The theoretical study of analysis on time scales has led to a number of important applications, for example in the study of insect density, nervous system, thermodynamic, quantum mechanics and disease model. We know that there are many results of differential equations that are made quite easily and naturally for difference equations. However, there 1 TIEU LUAN MOI download : skknchat@gmail.com are easy results to show for differential equations, not simply for difference equations and vice versa.

Studying the dynamic equations on time scales gives us a clear view to overcome this inconsistency between discontinuous differential equations and discrete difference equations. In addition, it is also avoided that a result may be proved twice, once for differential equations and another for difference equations. We can take the time scale as the set of real numbers R, the resulting results will be similar to those in ordinary differential equations. If the time scale is the set of integers Z, the resulting general result will be similar to the result in the difference equation.

However, time scales are rich in structure, so the results are generalized and much better than the results on the set of real numbers and on the set of integers. Therefore, the basic characteristic of these time scales is unification and expanded.