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 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 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 un-evenly evolving over time, re ecting 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 tted to actual practice and we must take into account the random factors that a ect 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 signi cant results, especially, results related to the stability of stochastic dynamic equations and stochastic dynamic delay equations Some results in this eld 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 r- stochastic dynamic delay equations: giving the de nition 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 r-stochastic dynamic equations and rstochastic 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 di cult 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 di cult 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 nd new suitable techniques to approach the problem ii List of Notations A B C C C rd ld 1;2 d (Ta R ; R) F t+ ( ; F; P; fFtgt2Ta )Stochastic basis; ft = f(t ) I1 I2 I Kt Kt b L2(M) = f tgt2Ta satisfying L 2((a; b]; M) d L1((a; T ]; R ) Ta R L ; loc( L1 loc d (Tt0 ; R ) loc (T t ; R L LV loc M2 M2 r hMi iii R Mt c n R; Z; N; N0 R R+ T Ta T k T k (t) (t) (t) (t) b t [ a; b] iv (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) (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 < : B where A and are motion on time scale de ned as in [23] It is known from [24, Theorem 2.1, pp 1678] Kt = 1: By using V (t; x) = kxk and (3.22) we have > > > > > LV (t; x; y) = x (A + A + A A (t))x + y B BKty: > > Suppose that the spectral abscissa of the matrix A + A + A A (t) is uniformly bounded by a negative constant From the equation (3.24) we see that 2 LV (t; x; y) 1kxk + kBk kyk : It is easy to see that 1(t e s) 6e (t; s) t Ts; (see [41] for details) Suppose that there exists a positive constant r that < and kBk e obtain such LV (t; x; y) Therefore, assumptions of Theorem 3.2.2 are satis ed 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 de ned by T=P ;1 90 Let r(t) be a delay function satisfying r = supt2T(t r(t)) = Consider the stochastic dynamic delay equation on time scale T r 8d X(t) = AX(t ) + > > > > > LV (t; x; y) = x 2A+A A (t) +B BKt x+x (I +A (t))y + y y: > H := 2A + A A 73 36 + B>BKt = 17 6 36 Further, (H) = > LV (t; x; y) x Hx + kI + A Setting > kkyk + := 1+ Combining these estimations and (3.27 2 LV (t; x; y) 1kxk + 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 De nition 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 ln kX(t; s; )k lim sup t t!1 < a:s: (3.28) d holds for any C( s; R ): Theorem 3.3.2 Let 1; 2; p; c1 be positive numbers with > Let be a positive number satisfying ld-continuous function de ned on Tt0 such that Z t0 e( ; t0) trt < a:s:: Suppose that there exists a positive de nite function V C R+) satisfying and for all t > t0; x R 1;2 d (Tt0 R ; d r V t (t; x) + AV (t; x; y) 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 Z h t e (t; t0)V (t; X(t)) = V (t0; (t0)) + t0 e ( ; t0) V ( ; X( )) + (1 + ( )) V r ( ; X( )) + AV ( ; X( ); X( ( ))) r + Z V (t0 ; (t0)) + +(1+ ( )) 92 Using the inequality < 1+ gets (t) e (t; t0)V (t; X(t)) V (t0; (t0)) + Ft + Gt; where Zt Ft = t0 Zt (1 + ( ))e ( r ; Gt = ; t0) t0 e ( ; t0)rH : In view of the hypotheses we see that F1 = limt!1 Ft < 1: De ne Yt = V (t0; (t0)) + Ft + Gt for all t Tt0 : Then Y is a nonnegative special semimartingale By Theorem on page 139 in [39], one sees that fF1 < 1g f lim Yt exists and niteg a:s:: t!1 By P fF1 < 1g = So we must have P f lim Yt exists and niteg = 1: t!1 Note that e (t; t0)V (t; X(t)) Yt for all t > t0 a:s:: It then follows that P flim sup e (t; t0)V (t; X(t)) < 1g = 1: t!1 So lim sup [e (t; t0)V (t; X(t))] < a:s:: t!1 Consequently, there exists a pair of random variables > t and > such that e (t; t0)V (t; X(t)) for all t > a:s:: Using (3.29), we have p c1e (t; t0)kX(t)k 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 t ) for any t T Therefore, + p lim t!1 Thus, lim t!1 The proof is completed 93 3.4 Conclusion of Chapter In this chapter, the thesis has resolved to the problems: - De ned delay functions, given some basic notions of stochastic dy-namic delay equations on time scales - Stated and proved the theorems of existence and uniqueness of so-lutions 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 su cient 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 su cient 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 stochas-tic 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 sta-bility of stochastic dynamic equations on time scales Introduced concepts and theorems, examples of exponential pmoment stability, stochastic stability, exponential almost sure stability of stochas-tic dynamic equations on time scales De ned delay function and stochastic dynamic delay equations on time scales Given the theorems of existence and uniqueness of solutions for stochas-tic 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 di erent 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 r-dynamic equations on disconnected sets, Electron J Di 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 Mathemat-ica 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 oor, 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, Applica-tion and Teaching", DaNang, Vietnam, 23{25, May, 2015 Scienti c Conference, Faculty of Mathematics, Mechanics and Infor-matics, 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 Ra oul (2006), Boundedness in Functional Dynamic Systems on Time scales, Advances in Di erence the equa-tions, 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 Di erence the equations: Theory and Applications, John Wiley and Sons [4] V B Bajic, D LJ 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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. .. stochastic dynamic equations on time scales: de nition of solutions of stochastic dynamic equations; stating and proving the theorems of existence and uniqueness of solutions for stochastic dynamic. .. stability of r -stochastic dynamic equations and rstochastic 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