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

Therefore, the transfer of analyticalresults studying determinate models on time scales to stochastic models is as ”Stability of stochastic dynamic equations on time scales”.Thesis is co

VIETNAM NATIONAL UNIVERSITY, HANOIUNIVERSITY 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

This work has been completed at VNU-University of Science under thesupervision of Prof Dr Nguyen Huu Du I declare hereby that the resultspresented in it are new and have never been used in any other thesis.

Author: Le Anh Tuan

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 andadvice 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 HungThang, Doctor Nguyen Thanh Dieu, other members of seminar at Depart-ment of Probability theory and mathematical statistics and all friends inProfessor Nguyen Huu Du’s group seminar for their valuable commentsand 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 tal Science, Hanoi University of Industry, the Dean board as well as to theall my colleagues at Faculty of Fundamental Science for their encourage-ment and support throughout my PhD studies

Fundamen-Finally, during my study, I always get the endless love and tional support from my family: my parents, my parents-in-law, my wife,

uncondi-my little children and uncondi-my dearest aunt I would like to express uncondi-my sinceregratitude to all of them Thank you all

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 andsimultaneously 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 muchattentions from many research groups One of most important problems inanalysis on time scales is to consider the quantity and quality of dynamicequations such as the existence and uniqueness of solutions, numericalmethods for solving these solutions as well the stability theory

However, so far, almost results related to the analysis on time scales aremainly 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 modelsare not fitted to actual practice and we must take into account the randomfactors that affect the environment Therefore, the transfer of analyticalresults studying determinate models on time scales to stochastic models is

as ”Stability of stochastic dynamic equations on time scales”.Thesis is concerned with the following issues:

• Studying the existence and uniqueness of solutions for ∇- stochasticdynamic delay equations: giving the definition of stochastic dynamicdelay equations and the concept of solutions; proving theorems of ex-istence and uniqueness of solutions; estimating the rate of the conver-

gence in Picard approximation for the solutions Proving theorem ofexistence and uniqueness of solutions under locally Lipschitz conditionand estimating moments of solutions for stochastic dynamic equations

on time scales

• Studying the stability of ∇-stochastic dynamic equations and ∇-stochasticdynamic 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 difficulttopics in the probability theory since it relates to many basic knowledgeslike Brownian motions, Markov process and martingale theory Therefore,the theory of stochastic analysis on time scales is much more difficultbecause the structure of time scales is divert That causes very complicatedcalculations when we carry out familiar results from stochastic calculus tosimilar one on time scales Besides, some estimates of stochastic calculusfor stochastic calculus on R are not automatically valid on an arbitrarytime scale Therefore, it requires to reformulate these estimates and tofind new suitable techniques to approach the problem

List of Notations

; R), is called generator;

C1,2(Ta × Rd

; R) Family of all functions V (t, x) defined on Ta × Rd

such that they are continuously ∇−differentiable in tand twice continuously differentiable in x;

t,M = ER(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 kφtk∇t < ∞;

Lloc

1 (Ta, R) Family of real valued, Ft−adapted processes {f (t)}t∈Ta

satisfying RaT |f (τ )|∇τ < +∞ a.s for every T ∈ Ta;

with continuous characteristics;

Mt = Mt− s∈(a,t] Ms− Mρ(s)
;

R, Z, N, N0 Real numbers, the integers, the natural numbers,

and the nonnegative integers;

Page

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 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

1.4 ∇−stochastic integral 27

1.4.1 ∇−stochastic integral with respect to square inte-grable martingale 27

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 Itˆo’s formula 32

1.5.1 Quadratic co-variation 32

1.5.2 Itˆo’s formula 33

1.6 Martingale problem 35

1.6.1 Counting processes for discontinuous martingales 35

1.6.2 Martingale problem formulation 38

Chapter 2 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 equa-tions 71

2.7 Conclusion of Chapter 2 74

Chapter 3 The stability of ∇−stochastic dynamic delay equations 76 3.1 ∇-stochastic dynamic delay equations 77

3.1.1 ∇-stochastic dynamic delay equations 77

3.1.2 Solutions of stochastic dynamic delay equations 78

3.1.3 Existence and uniqueness of solutions 783.1.4 Rate of the convergence 823.1.5 Locally Lipschitz condition on existence and unique-

ness of solutions 833.2 Exponential p-stability of stochastic dynamic delay equations 873.2.1 Sufficient condition 873.2.2 Examples 893.3 Almost sure exponential stability of dynamic delay equations 923.4 Conclusion of Chapter 3 94

mathe-1 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 anddiscrete analysis The results of analytical calculations on time scalesallow us to construct mathematical models of systems that are unevenlyevolving over time, reflecting real models

The theoretical study of analysis on time scales has led to a number ofimportant applications, for example in the study of insect density, nervoussystem, thermodynamic, quantum mechanics and disease model

We know that there are many results of differential equations that aremade quite easily and naturally for difference equations However, there

are easy results to show for differential equations, not simply for differenceequations and vice versa Studying the dynamic equations on time scalesgives us a clear view to overcome this inconsistency between discontinuousdifferential equations and discrete difference equations In addition, it

is also avoided that a result may be proved twice, once for differentialequations and another for difference equations

We can take the time scale as the set of real numbers R, the resultingresults will be similar to those in ordinary differential equations If the timescale 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 instructure, 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 basiccharacteristic of these time scales is unification and expanded That is themain reason there have been dozens of books and thousands of articlesdealing with the analysis on time scales [6, 7, 9, 16] Many familiar results

in the continuous or discrete cases have been ”shifted” to time scales Forexample, on the study of the dynamical system on time scales, there arevery profound results on stability, oscillation, boundary value problems However, as we know, so far the results of the study on time scalesare mainly in deterministic analysis Therefore, these results only describemodels developed in non-perturbed environmental conditions Obviously,the actual models are not so and we must take into account the randomfactors that affect the environment Hence, the transfer of the analyticalresults on time scales of the determinate models to the stochastic model

is an urgent need

2 Stochastic integral on time scales

2.1 Stochastic calculus with continuous time and discrete time

the motion of suspended pollen particles in water is very chaotic By perimenting with particles of inorganic matter, he eliminated the externalcauses of the motion However, the source of the motion has remainedunraveled.

ex-In 1880, Thorvald Nicolai Thiele, a Danish astronomer [63], created theBrownian motion model in mathematics when he analyzed the time series

In 1905, Albert Einstein, the German physicist, described this phenomenonunder the name ”Brownian motion” Although the Brown motion modelinitially proposed by Thorvald Nicolai Thiele but the model was almostunknown

In 1923, N Wiener used measurement theory to construct the Brownianmotion, and then demonstrated its unique existence

Today, in recognition of his contribution, we call the Brownian motion

by Wiener process In his work, N Wiener pointed out that the trajectory

of Brown’s motion has unbounded variations Thus, integration of Wienerprocess can not be constructed in a conventional way as Lebesgue-Stieltjesintegral

b Martingales and semimartingales

In probability theory, a martingale is a stochastic process for which, at

a particular time in the realized sequence, the expectation of a future value

in the process is equal to the present observed value even given knowledge

of all prior observed values

The concept of martingale in probability theory was introduced byPaul L´evy in 1934, though he did not name them The term ”martingale”was introduced later by Ville (1939), who also extended the definition tocontinuous martingales Much of the original development of the theorywas done by Joseph Leo Doob among others Part of the motivation forthat work was to show the impossibility of successful betting strategies

A real valued process X is called a semimartingale if it can be posed as the sum of a local martingale and an adapted finite-variationprocess Semimartingales are good integrators, forming the largest class

decom-of processes with respect to which the Itˆo integral and the Stratonovich

integral can be defined The class of semimartingales is quite large cluding, for example, all continuously differentiable processes, Brownianmotion and Poisson processes) Submartingales and supermartingales to-gether represent a subset of the semimartingales.

(in-c Doob-Meyer expansion

In 1953, J Doob [25] stated and demonstrated the Doob developmenttheorem for submartingale with discrete time and conjecture theorem forsubmartingale with continuous time These theorems were proved in 1962and 1963 by P A Meyer [54] So, the Doob development theorem is calledthe Doob-Meyer expansion theorem

d Stochastic integral

On basis of the Brown motion In 1944, K Itˆo, a Japanese matician built the stochastic integral on the Wiener process [28] ThenJ.L Doob [25] expanded the stochastic integral by the orthogonal incre-ment process Stochastic integral continues to be extended to the square-integrable martingale by P A Meyer [54], by H Kunita and S Watanabe[36]

mathe-In 1970, P A Meyer and C Dol´eans-Dade [55] built a local integrable martingale Also in that year, C Dellacherie and K Bichtelerbuilt stochastic integral by semimartingale Today, stochastic integralmentioned above is called Itˆo’s stochastic integral For stochastic calcu-lations with discrete time, the martingale transformations are consideredItˆo’s stochastic integral

square-e Itˆo’s formula

The Itˆo’s formula for the Wiener process was developed by K Itˆo [29]

in 1951 and is considered as a key tool in stochastic computing In 1967,

H Kunita and S Watanabe [36] extended the Itˆo’s formula to squareintegrable martingale P Meyer [53] extended Itˆo’s formula to martingalewith jump steps

The Itˆo’s formula for semimartingales was developed in 1969 by H P.McKean in [51], expanded by P A Meyer and C Dol´eans-Dade in [55]

For stochastic computing with discrete time, it was formulated in 2002

by D Kannan and B Zhan in [32] Today, the application of the Itˆo’s mula in the study of the qualitative and quantitative properties of stochas-tic dynamic equations has become very familiar in stochastic analysis.2.2 The first attempt on the stochastic calculus on time scales

for-As a natural way, we want to ”shift” and generalize the above notionsand results for stochastic calculus (Brownian motion, stochastic integral,Doob-Mayer expansion ) with continuous or discrete times to time scales.The stochastic analysis on time scales has just been born in some lastyears Since it is quite new subject, there are not too much in mathematicalliterature In [5], the authors developed the theory of Brownian motion

on time scales Base on the concept of infinitesimal operator, they studiedBrownian motions valued in a time scale with continuous time After, S.Suman in his Ph.D Dissertation [60] tried to defined ”stochastic integral

on time scales” but he just deals with time scales consisting only of isolatedpoints

In [13], the authors N H Du and N T Dieu have done the first attempt

to study systematically stochastic calculus on the time scale In that paper,they have dealt with the Doob - Meyer decomposition theorem, stochasticintegration, Itˆo’s formula for stochastic processes indexed by a time scale.When constructing a stochastic integration on a time scale, the mostdifficulty we are faced here is the forward jump operator σ(t) since it canmake an adapted process become non-adapted One can avoid this dis-advantage by using the ∆−integration on the semi-open intervals of theform [ti, ti+1) However, the predictable progressively measurable assump-tion for integrands is important in constructing the stochastic integral Ingeneral, the predictable progressively measurable processes are generated

by simple processes of the form φ1(ti,ti+1], where φ is Fti measurable randomvariable The difference between them makes a wide gap on the stochasticcalculus with the ∆−integration In using ∇−integration for stochasticcalculus on time scale, we can overcome this difficulty Although it makessome inconveniences when we try to define a stochastic dynamic equa-

tions on time scale because the ∆−dynamic equations are more popular inreferences, ∇−dynamic equations are also interesting in both theory andpractice.

3 Stochastic dynamic equations

3.1 Stochastic dynamic equations with continuous time and crete time

dis-The stochastic differential equation with noise to be a Wiener processhas been built in 1951 by K Itˆo [30] and get further studies by H P.McKean [51], I I Gihman and A V Skorohod [20] N Kazamaki [33]

in 1972 has dealt with stochastic differential equations with a square grable martingale After, these results were developed by P E Protter [58]and many other mathematicians [31, 47] X Mao studied the stochasticdifferential equation with interference semimartingale [45, 49]

inte-Besides, the stochastic difference equation was studied by many ematicians [22, 50, 64, 65], since it mights define the simplest dynamicalsystems, but nevertheless, they play an important role in the investigation

math-of a dynamical system The difference equations arise naturally when wewant to study the evolution of biological population or economic models

on a fixed period of time They can also be illustrated as discretization

of continuous time systems in computing process So far, problems forstochastic analysis of discrete and continuous times have been explored bymathematicians for quite a long period

3.2 Stochastic dynamic equations on time scales

With the concept of stochastic integral on time scales, we can considerthe notion of stochastic dynamic equations on time scales Here, we men-tion some of the first attempts on this direction In [60], S Sanyal in hisPh.D Dissertation has tried to define “stochastic integral and stochasticdynamic equations” on time scales with the positive graininess; in [42],authors have prove the existence and uniqueness of solutions for ordinaryrandom dynamic equations; N H Du and N T Dieu [14] have developedthe theory of ∇−stochastic dynamic equations on time scales, and gave theconditions for the existence and uniqueness of solutions They also have

investigated the Markov property of solutions and have concerned with itstime-dependent generator.

4 Stochastic dynamic delay equations

4.1 Stochastic dynamic delay equations with continuous timeand discrete time

In many cases, the future state of the considered system depends notonly on the present but also on the past The system of stochastic dy-namic delay equations is the mathematical formula of those dynamics.The problem of existence and uniqueness and the stability of these systemshave been studied by a number of authors [3, 35, 48] and have obtained

a number of important research results However, for time-varying delaystochastic dynamics, the results are still very limited

4.2 Stochastic dynamic delay equations on time scales

For the time scales, different research teams pay attention to the tative and quantitative properties of deterministic and stochastic dynamicequations [6, 8, 14, 17, ] However, there are little works about determin-istic dynamic delay equations on time scales (and no work with stochasticdelay equation), while the study of dynamic delay equations is importantbecause they are used to describe many systems derived from science andtechnology in which future depends not only on the present but also on itspast The main reason is that the time scale does not preserve addition andsubtraction, so we have difficulty in conceptualizing about the stochasticdynamic delay equations on time scales In [40, 41, 44, ], the authorsexamined the qualitative properties of solution for the deterministic delayequations on time scales, but the assumptions imposed on time scales aretoo strict

quanti-5 Stability

5.1 The stability of stochastic dynamic equations with ous time and discrete time

continu-Stability theory is an extremely important part of the qualitative theory

of dynamic equations It has great practical significance, is applied in manyproblems in physics, mechanics, control

Stability problems were studied in the late nineteenth century andhave been studied and dealt with by many major mathematicians such

as Lagrange, Poincare, Especially since A.M Lyapunov- Great Russianmathematician - published the famous work ”General Problem of Motion-Stability” in 1882, this theory has made great strides and has solved manyproblems of practicality Despite having a history of development for over ahundred years, Lyapunov’s stable theory is not ”old-fashioned”, ”underde-veloped” but on the contrary is still a very exciting development theory ofmathematics Over the past several decades, it has been increasingly used

in various fields such as economics, engineering, control systems, ecology,environmental studies

There are two main methods used in studying the stability of the systemdescribed by the dynamic equations The first method, called characteris-tic exponent method, is to study the stability of linear systems and theirfirst approximation by comparing growth rate of solutions with exponentialfunctions The second one, say the Lyapunov function method, is a verypowerful and especially important tool when studying the stability of non-linear systems, which does not require the investigation of the system TheLyapunov function is considered as the main tool for transferring very com-plex systems into relatively simple systems and we need only to study thesimplified systems For linear systems, especially autonomous systems, theform of the Lyapunov function is often formulated as a whole Hence, thestable condition, the exponential stability equates to the existence of thesolution of the Lyapunov equation inequality Normally, in the linear sys-tems, the Lyapunov equation the solution of the Riccati matrix equation.The Lyapunov function has been exploited at various angles for algebraicdifferential equations, such as [56]; for implicit difference equations, as in[4] Two these methods also have become the most widely used tool forstudying the stability of stochastic differential/difference delay equations.For longterm behavior of stochastic differential equations, we mentionsome of the presentations, interesting books of Khas’minskii [34], Arnold [3]and Kushner [37], in them authors use the Lyapunov functions to study the

stability Later, S Foss and T Konstantopoulos [18] present an overview ofstochastic stability methods, mostly motivated by stochastic network ap-plications; L Socha [61] considers the exponential p-stability of singularlyperturbed stochastic systems for the ”slow” and ”fast” components of thefull-order system; T E Govindan [21] proves the existence and unique-ness of a mild solution under two sets of hypotheses and considers theexponential second moment stability of the solution process for stochas-tic semilinear functional differential equations in a Hilbert space In [47],the author examines stochastic asymptotic stability and boundedness forstochastic differential delay equations However, up until 1989, most ofthese research results refer only to the stability over the class of stochasticItˆo differential equations During the second development period of thestochastic stability theory, Mao published a number of articles such asMao [45, 46, 47], related to the stability of stochastic differential equationsdriven by semimartingales, including continuous and discrete cases andachieved a lot of important results.

5.2 The stability of stochastic dynamic equations on time scales

In deterministic cases, many authors have also exploited the Lyapunovfunction to determine the stability of the dynamic equations on time scales

In [10], the author uses the Lyapunov function in quadratic form to studythe stability of linear dynamic equations; some other papers examine thestability and instability of the equilibrium point of nonlinear dynamic equa-tions [1, 11, 27, 38, 57]

However, while the stability of deterministic dynamic equations on timescales has been studied for a long time; there is no work involving thestability for stochastic dynamic equations, especially for the stability ofstochastic dynamic delay equations, on time scales There are some reasonsfor this phenomena Firstly, the stochastic analysis on time scales is ahard topic and so far people get very few results on studying this topic.Almost results in studying stochastic dynamic equations on time scalesare still stoping at the existence and uniqueness of solutions Secondly, forstochastic dynamic delay equations, since the substitution rule in integral

can not apply in the calculus on time scale, the method used in consideringthe stability of delay difference/differential equations is no longer valid.Therefore, we can not simply unify the presentation of some known resultsand to obtain them, we have to improve techniques.

For the above reasons and for completing the theory of the stochasticdynamic systems, we have chosen the doctoral thesis research topic as

”Stability of stochastic dynamic equations on time scales”

In addition to the introduction and conclusion parts, this thesis consists

of three chapters:

Chapter 1 presents some most basic concepts of time scales theory.For the first part, we introduce the notion of time scales, definitions ofderivative and integral on time scales and exponential functions Next,

we deal with the concept of stochastic analysis on time scales; especiallythe Doob-Meyer expansion theorem for the submartingale; ∇-stochasticintegration with respect to semimartingarle; Itˆo’s stochastic formula for theset of d−semimartingales and the application of Itˆo’s stochastic formula

to develop the martingale problem

In Chapter 2, we have proposed stochastic dynamic equations on timescales: definition of solutions of stochastic dynamic equations; stating andproving the theorems of existence and uniqueness of solutions for stochas-tic dynamic equations; studying the Markov property of solutions; givingmoment estimation formulas Further, we have introduced the concept ofthe exponential p-stability for stochastic dynamic equations; constructedthe Lyapunov function for determining the stability of stochastic dynamicequations; given theorems of necessary and sufficient conditions for theexponential p-stability; dealt with the theorems for the stochastic stability

of stochastic dynamic equations The concept of the almost sure tial stability and the theorems for the almost sure exponential stability ofstochastic dynamic equations are concerned with In the end of this chap-ter, we have provided illustrative examples for the exponential p-stability,stochastic stability, almost sure exponential stability

exponen-Chapter 3 deals with the stochastic dynamic delay equations on time

scales We have given the definition of stochastic dynamic delay equations,definition of solutions The theorems of existence and uniqueness andthe moment estimates of solutions have been proved in this chapter Inparticular, we have studied theorems of the exponential p-moment stability,exponential almost sure stability on time scales via Lyapunov function.Some illustrative examples are also given.

In conclusion, we think that the theory of stochastic dynamic tions, especially stochastic dynamic delay one, on time scales was foundpromising because it demonstrates the interplay between the theories ofcontinuous-time and discrete-time systems and there are many applica-tions to real life problems However, this is rather hard topic since itrelates to many other fields in analysis and probability theory Therefore,

equa-it is meaningful to develop this theory in both theory and practice Inparticular, the theory of almost sure stability or Lyapunov exponents isworth considering in stochastic analysis on time scales

Due to the limited time and ability, this thesis has inevitable comings and imperfections of the proposed problem, although myself havetried a lot in the process of researching and completing the dissertation

short-I would like to welcome all comments and exchanges of mathematicians,readers and those interested in this matter

Chapter 1

Preliminaries

At the beginning of this chapter, we present the most basic concepts oftime scales theory; ∇- derivatives and ∇-integral of a function defined ontime scales Since in the following we are working only with nabla analysis,the results for ∆-derivatives and ∆-integral will not be presented in thethesis If the reader is interested in it, these notions can be found in manydocuments, such as [6]

In the last part of this chapter, we introduce the theory of stochasticanalysis on time scales For premier step, we define stochastic process,predictable process, martingale, semimartingale, stopping time indexed by

a time scale After, we present the Doob-Meyer expansion; ∇-stochasticintegration on time scales; Itˆo’s formula and applying the Itˆo’s formula

to the martingale problem; in particular, we deal with the formulation ofmartingale problem

1.1 Survey on analysis on time scale

A time scale T is an arbitrary nonempty closed subset of the real bers R We assume throughout of this thesis that the time scale T has thetopology inherited from the standard topology of real numbers R

num-On the time scale T, we define the forward jump operator and backwardjump operator as follow

1 Forward jump operator: σ : T −→ T given by σ(t) := inf{s ∈ T : s >

t}, t ∈ T.

2 Backward jump operator: ρ : T −→ T with ρ(t) := sup{s ∈ T : s <t}, t ∈ T

In this definition we put inf ∅ = sup T; sup ∅ = inf T (i.e., σ(t) = t if

t = max T; and ρ(t) = t if t = min T)

A point t ∈ T is said to be right-scattered if σ(t) > t; it is calledright-dense if σ(t) = t and left-scattered if ρ(t) < t, left-dense if ρ(t) = t.Points that are right-dense and left-dense at the same time are calleddense; points that are right-scattered and simultaneously left-scattered arecalled scattered or isolated

The left dense or right dense is illustrated by the Figure 1.1

Figure 1.1: Points of the time scale T.

Denote (a, b)T = {t ∈ T : a < t < b} To simplify notations, from now

on we write (a, b); (a, b]; [a, b); [a, b] instead of (a, b)T; (a, b]T; [a, b)T; [a, b]T ifthere is no confusion For a ∈ T, denote Ta = {x ∈ T : x > a} Let

I1 = {t : t is left-scattered}, I2 = {t : t is right-scattered}, I = I1∪ I2.Proposition 1.1.1 ([6]) The set I of all left-scattered or right-scatteredpoints of T is at most countable

If T has a left-scattered maximum Mmax, then define kT = T \ {Mmax},

otherwise kT = T If T has a right-scattered minimum Mmin, then define

For any t ∈ T we denote fσ(t) = f (σ(t)) and fρ(t) = f (ρ(t))

We give some examples of typical time scales

1.2 Differentiation

1.2.1 Continuous functions

Since T has a topology inherited from the standard topology on the realline, we have the concept of continuous functions by natural way However,there are some kinds of points in a time scale, we introduce some furthernotions of the continuity

Definition 1.2.1

1 A function f : T −→ R is called regulated, provided its right-sidedlimits exist (finite) at all right-dense points in T and its left-sidedlimits exist (finite) at all left-dense points in T

2 A function f : T −→ R is said to be rd-continuous (right-dense tinuous) if it is continuous at right-dense points in T and its left-sidedlimits exist (finite) at all left-dense points in T

con-3 A function f : T −→ R is ld-continuous (left-dense continuous) if it

is continuous at left-dense points in T and its right-sided limits exist(finite) at all right-dense points in T

4 Let A be an m × n−matrix valued function on time scale T We sayA(·) is rd-continuous (resp ld-continuous) if each entry of A is rd-continuous (resp ld-continuous) on T

For two variables functions we have the following definition Let X be

a Banach space

5 A mapping

f : T × X −→ X(t, x) 7−→ f (t, x)

is said to be ld-continuous if it satisfies the following conditions

a f is continuous in (t, x) at each (t, x) where t is left-dense or t =min T,

b The limits

f (t+, x) := lim

(s,y)→(t,x),s>tf (s, y) and lim

y→xf (t, y)exist at each point (t, x) where t is right-dense

con-Theorem 1.2.2 Consider the function f : T −→ R, we have:

i) If f is continuous, then f is ld-continuous

ii) If f is ld-continuous, then f is regulated

iii) The jump operator ρ is ld-continuous

iv) If f is regulated (ld-continuous), then so is fρ

v) Assume that f ld-continuous If g : T −→ R is regulated (ld-continuous)then f ◦ g has that property too

vi) f is a continuous function iff it is ld-continuous and rd-continuous atthe same time

A function f from T to R is nabla regressive (regressive for short)(respectively positively regressive) if 1 + µ(t)f (t) 6= 0 (respectively 1 +µ(t)f (t) > 0) for every t ∈ T

Denote limσ(s)↑tf (s) by f (t−) or ft − if this limit exists It is easy to seethat t is left-scattered then ft − = ftρ

1.2.2 Nabla derivative

Definition 1.2.3 ([7]) Consider the function f : T → R, ∇ (or derivative of f at a point t ∈ kT, denoted by f∇(t) to be a number (provided

Hilger)-it exists) wHilger)-ith the property that given any  > 0, there is neighborhood U

of t such that

kf (ρ(t)) − f (s) − f∇(t)(ρ(t) − s)k 6 |ρ(t) − s|

for all s ∈ U

We say that f is ∇ (or Hilger)-differentiable (or in short differentiable)

on kT if f∇(t) exists for all t ∈ kT.

The ∇- derivative f∇ of a vector function f = (f1, f2, , fd) is stood as f∇ = (f1∇, f2∇, , fd∇)

under-Example 1.2.4 For T = R, we have f∇(t) = f0(t), the usual derivative,and for T = Z we have the backward difference operator, f∇(t) = ∇f (t) :=

f (t) − f (t − 1)

Definition 1.2.5 ([6]) Consider the function f : T −→ R, ∆− derivative

of f at a point t ∈k T, denoted by f∆(t) to be a number (provided it exists)with the property that given any  > 0, there is neighborhood U of t suchthat

kf (σ(t)) − f (s) − f∆(t)(σ(t) − s)k 6 |σ(t) − s|,for all s ∈ U We say that f is ∆−differentiable (or in short differentiable)

on kT if f∆(t) exists for all t ∈k T.

Theorem 1.2.6 ([7]) Consider the function f: T −→ R and t ∈ kT Then

we have the following statements:

1 If f is ∇−differentiable at t then f is continuous at t

2 If f is continuous at t and t is left-scattered then f is ∇−differentiable

f∇(t) = lim

s→t

f (t) − f (s)

t − s .

(t) = f

∇(t)g(t) − f (t)g∇(t)g(t)gρ(t) .1.2.3 Lesbesgue ∇− integral

Next, we give an outline of constructing a Lebesgue-Stieltjes measure

on time scale Let A be an increasing right continuous function defined on

T Denote by M1 = {(a; b] : a, b ∈ T} the family of all left open and rightclosed interval of T It is seen that M1 is semi-ring of subsets of T Let

m1 be the set function defined on M1 by

m1((a, b]) = Ab − Aa

It is easy to show that m1is a countably additive measure on M1 We write

µA∇ for the Carath´eodory extension of the set function m1, associated withthe family M1 and call it the Stieltjes-Lebesgue ∇−measure associatedwith A on T

Let E be a µA∇−measurable subset of kT and f : T → R, be an

µA∇−measurable function The integral of f associated with the sures µA∇ on E, denoted by REf (τ )µA∇(τ ), is called Lebesgue - Stieltjes

mea-∇−integral By the definition of µA

∇ we see that

• For each t ∈ kT, the single-point set {t} is ∇A− measurable, and

µA∇({t}) = At − At−

• For a, b ∈ T and a < b,

µA∇((a, b)) = Ab− − Aa; µA∇([a, b)) = Ab− − Aa−; µA∇([a, b]) = Ab − Aa−

• For a, b ∈ T and a = b, we put µA

∇((a, b]) = 0

All intervals of family M1 including the empty set are ∇-measurable

By definition, it is obvious that the T is also ∇-measurable Suppose that

T has a finite minimum Tmin Obviously, the set X = T \ {Tmin} can

be represented as a finite or countable union of intervals of the family

M1 and, therefore is ∇-measurable Consequently, the single-point set{Tmin} = T \ X is ∇-measurable as the difference of two measurable sets

T and X Evidently, the single-point set {Tmin} does not have a finite

or countable covering by intervals of M1 Therefore, the single-point set{Tmin} and also any ∇-measurable subset of T containing Tmin have ∇-measure infinity For the details, we can refer to [12]

In this thesis, R(a,b]f (τ )∇µA∇(τ ) will be denoted by Rabf (τ )∇µA∇(τ ) Ifthere is no confusion we write simply Rabf (τ )∇Aτ for Rabf (τ )∇µA∇(τ ) Incase At = t we denote R(a,b]f (τ )∇τ

We now present some unusual and useful properties of integral on timescales

Theorem 1.2.8 ([12]) Assume a, b, c ∈ T, α ∈ R, f, g ∈ Cld(T, R) then1

−Pa k=b+1f (k)(Ak− Ak−1) if a > b.

is regulated one has

Z b a

f (s−)∇s =

Z b a

f (s)∆s, ∀ a, b ∈k T. (1.1)

We note that, the inverse transformation of the cylinder transformation

If p is rd-continuous and regressive function, then the exponential tion ep(t, t0) is solution of the initial value problem

func-y∇(t) = p(t−)y(t−), y(t0) = 1, t > t0 (1.3)Indeed, from [6, Theorem 2.33 pp 59] we have

Lemma 1.2.13 ([8]) If p is rd-continuous and regressive, then the nential function has semigroup property

expo-ep(t, r)ep(r, s) = ep(t, s), for all r, s, t ∈ T

Example 1.2.14

1 If T = R then ep(t, s) = e

t R

Define by induction a sequence of functions hk : T × T → R; k ∈ N by

h0(t, s) = 1 and hk+1(t, s) =

Z t s

hk(τ, s)∆τ for k ∈ N

Since hk(t, s) is continuous,

hk+1(t, s) =

Z t s

hk(τ−, s)∇τ for k ∈ N (1.4)The important properties that will be used in the sequel are that on anarbitrary time scale T, for all k ∈ N0 and t > t0, we have

hk(t, s) 6 (t − s)

k

k! .Moreover, follow Theorem 1.113 in [6] about Taylor’s formula, it is easy tosee that

Lemma 1.2.15 (Gronwall-Bellman inequality) Suppose u(t) be a cadlagfunction and ua, p ∈ R+ Then, the inequality

u(t) 6 ua+ p

Z t a

u(τ−)∇τ for all t ∈ Ta,implies

u(t) 6 uaep(t, a) for all t ∈ Ta.1.3 Stochastic processes on time scales

1.3.1 Basic notations of probability theory

Let T be a time scale Let (Ω, F, {Ft}t∈Ta, P) be a complete ability space with filtration This means that {Ft}t∈Ta is an increasingsub-algebras of F (i.e., Fs ⊂ Ft if s 6 t) and Fa is complete with respect

sat-1.3.2 Stochastic processes on time scales

A family {Xt}t∈Ta of Rd-valued random variables is called a stochasticprocess with parameter set (or index set) Ta and state space Rd

Thus, for each fixed t ∈ Ta, Xt(ω) is a random variable valued in Rd.For any ω ∈ Ω, the function Ta 3 t −→ Xt(ω) ∈ Rd is called sample path

of this process, and we shall write X•(ω) for the sample path Sometimes,for the convenience, when we consider the stochastic process as a function

of two variables (t, ω) from Ta⊗Ω to Rd, we write X(t, ω) instead of Xt(ω).Definition 1.3.1 ([47]) Let {Xt}t∈Ta be an Rd -valued stochastic process

1 {Xt}t∈Ta is said to be continuous (resp right continuous, left ous) if for almost all ω ∈ Ω function X•(ω) is continuous (resp right

continu-continuous, left continuous) on t ∈ Ta.

2 {Xt}t∈Ta is said to be cadlag (right continuous and left limit) if itstrajectory X•(ω) is right continuous and there exists the finite left limitlims↑tXs(ω) for all t ∈ Ta for almost all ω ∈ Ω

3 {Xt}t∈Ta is said to be Ft-adapted (or simply, adapted) if for every

t ∈ Ta, Xt is Ft-measurable

4 {Xt}t∈Ta is said to be measurable if the stochastic process regarded as

a function of two variables (t, ω) from Ta ⊗ Ω to Rd

is B(Ta) ⊗ F-measurable, where B(Ta) is the family of all Borel sub-sets of Ta

5 {Xt}t∈Ta is said to be progressively measurable or progressive if forevery T ∈ Ta, {Xt}t∈Ta regarded as a function of (t, ω) from [a; T ] ⊗ Ω

to Rd is B([a; T ]) ⊗ FT -measurable

6 A right continuous real-valued stochastic process {At}t∈Ta is called anincreasing process if for almost all ω ∈ Ω, Aa = 0 and At(ω) is non-negative nondecreasing on t ∈ Ta It is called a process with finitevariation for every ω ∈ Ω, the sample path A•(ω) has finite varia-tions

Denote by L the set of all real - valued stochastic processes φ = {φt}t∈Tadefined on Ta⊗ Ω with the left continuous paths on Ta and Fρ(t)−adapted.Let P be the σ− algebra of the subsets of Ta ⊗ Ω generated by thestochastic processes in L

Definition 1.3.2 ([47]) Every set in the σ−algebra P is called predictable

A process φ is said to be predictable if it is measurable with respect to P

It is easily to see that P is generated by the family of sets {(s, t] ⊗ F :

s, t ∈ Ta, s < t, F ∈ Fs} We note that in general a left continuous process

is not necessarily predictable

Remark 1.3.3

i If T = N then the process φt is predictable if φt is Ft−1−measurable

ii If T = R then the process φt is predictable if it is understood as tic process, measurable with respect to σ−algebra generated by adaptedleft continuous processes.

stochas-Definition 1.3.4 ([47]) A random variable τ : Ω −→ Ta is called an

Ft-stopping time (or simply, stopping time) if {ω : τ (ω) 6 t} ∈ Ft for any

2 E|Xt| < ∞ for all t ∈ Ta;

3 E(Xt|Fs) = Xs a.s for all s < t

A stochastic process X = {Xt}t∈Ta is called a submartingale (resp asupermartingale) if we replace the sign = in the condition (3) with > (resp.6)

Definition 1.3.6 ([47]) Process X = {Xt}t∈Ta is a locally square-integrablemartingale if there is a sequence of stopping times τ

n },τ n %∞ a.s. such that{Xt∧τ n} is a square-integrable martingale

Definition 1.3.7 ([47]) The process {Xt}t∈Ta is called an Ft-semimartingale

if it is adapted and for any t ∈ Ta,

Xt = Mt + Atwhere Mt is a locally square-integrable martingale and At is a right-continuous,adapted process with the finite variation and Aa = 0

The increasing process A = {At}t∈Ta is called integrable if EAt < ∞for all t ∈ Ta

In order to simplify notations, for all function defined on T, we write thevalue ft instead f (t) Remember that ft− = limσ(s)↑tf (s) By convention,

we write fa − = fa for all functions defined on Ta

Proposition 1.3.8 ([13]) If M is a bounded martingale, A is increasing,integrable process, then for any t ∈ Ta, we have

Definition 1.3.10 ([13]) A process X = {Xt}t∈Ta is said to be in theclass (DL) if for any t ∈ Ta the set

{Xτ : τ is a stopping time satisfying a 6 τ 6 t}

is uniformly integrable

Theorem 1.3.11 (Doob-Meyer decomposition, [13]) Let X = {Xt}t∈Ta

be a right continuous submartingale of class (DL) Then, there exist rightcontinuous martingale M = {Mt}t∈Ta and right continuous increasing pro-cess A = {At}t∈Ta such that

Con-Ft− martingale by Doob-Mayer decomposition The natural increasingprocess hM it is called characteristic of the martingale M

Fix a T ∈ Ta For any δ > 0 we consider a partition of [a, T ] inductively

by letting t0 = a and for i = 1, 2, , kδ setting:

where Bi = (ti−1, ti−1 + δ] ∩ [a, T ] (1.6)

Proposition 1.3.12 ([13]) For any T ∈ Ta, let (ti) be defined by (1.6).Suppose that the characteristic of a martingale M is continuous Then,

on (a, b] We endow L2((a, b]; M ) with the norm

kφk2b,M = E

Z

(a,b]

|φτ|2∇hM iτ,and identify φ and φ in L2((a, b]; M ) if kφ−φkb,M = 0 A process φ defined

on [a, b] is called simple if there exist a partition π of [a, b] : a = t0 < t1 <

· · · < tn = b and bounded random variables {fi} such that fi is Ft i−1−measurable for all i = 1, n and

Lemma 1.4.1 ([13]) L0 is dense in L2((a, b]; M ) with respect to the metric

d(φ, ϕ)2 = kφ − ϕk2b,M = E

Z

(a,b]

|φτ − ϕτ|2∇hM iτ

Definition 1.4.2 ([13]) For a simple process φ with the form (1.7) in L0,define

Next, we define the integration for φ ∈ L2((a, b]; M ) By Lemma 1.4.1

we can find {φ(n)} ⊂ L0 such that kφ − φ(n)kb,M → 0 as n → ∞ Since

(a,b]φ(n)(τ )∇Mτ}n∈Nis a Cauchy sequence In consequences, it converges

in L2(Ω, F , P) to a random variables ξ with E|ξ|2 < ∞ This limit doesnot depend on the choice of the sequence φ(n) This leads to the followingdefinition

Definition 1.4.4 ([13]) The ∇− stochastic integration of a process φ ∈

L2((a, b]; M ) with respect to the square integrable martingale M on (a, b],denoted by R(a,b]φτ∇Mτ, is defined by

φτdMτ,where the integral RabφτdMτ is defined in [59].

The stochastic integration has the following usual properties;

Proposition 1.4.6 ([13]) Let φ ∈ L2((a, b]; M ) and let α, β be two realnumbers Then, the following relations hold

(i) R(a,b]φτ∇Mτ is Fb−measurable;

(ii) ER(a,b]φτ∇Mτ = 0;

(iii) E

h

(a,b]φτ∇Mτi2 = ER(a,b]φ2τ∇hM iτ;

(iv) R(a,b][αφτ + βξτ]∇Mτ = αR(a,b]φτ∇Mτ + βR(a,b]ξτ∇Mτ a.s.;

(v) If ξ is a real Fa−measurable bounded random variable, then ξφ ∈

I(a) = 0; I(t) =

Z

(a,b]

χ(a,t]φτ∇Mτ ∀ a < t 6 b,