Backward Stochastic Dynamic Equations On Time Scales And Related Optimization Problems

In 1988,Hilger introduced the measure chains theory which was developed into time scales theory in order to unify continuous and discrete analysis.Due to the more appropriate time structure of the real world,time scales theory has received much more focus.The development of time scales is still in its infancy,yet with the deepening of research,people are more and more interested.When the continuous time differential equation and its corresponding discrete difference equation are considered separately,the results will be inconsistent,and people expect that the time scale dynamic equation can give a better explanation.The explanations of these seeming discrepancies are incidentally producing unifying results via time scales methods.The time scales theory contains much more flexible and realistic time structure that are neither classical continuous time nor uniform discrete time.For example,a consumer receives income at one point in time,asset holdings are adjusted at a different point in time,and consumption takes place at yet another point in time.Moreover,consumption and saving decisions can be modeled to occur with arbitrary,time-varying frequency.It is hard to overestimate the advantages of such an approach over the discrete or continuous models used in economics.Time scale calculus would allow exploration of a variety of situations in which timing of the decisions impacts the decisions themselves.The time scales approach is much more flexible and realistic.Because of widely existence of the time scale phenomenon in nature,the results on time scale will give us more inspiration.The time scales theory provides us with a new method to reveal the changes of things with time.Since the 1990s,there have been a lot of research achievements in time scales theory,especially in the fields of deterministic analysis.There are few results on the field of stochastic analysis on time scales.The study of stochastic dynamic equations on time scales is a fairly new subject.This thesis mainly contains four parts.In the first part we study the theory of backward stochastic dynamic equations on time scales.In the second part,we research the existence and uniqueness results of the solutions to fully coupled forward backward stochastic dynamic equations.In the third part,we study the stochastic optimal control problems on time scales with the systems described by partially coupled and fully coupled forward-backward stochastic dynamic equations on time scales and further develop the Maximum Principle on time scales.At last,we apply time scales theory to some applications.This thesis is divided into six chapters.The main framework and results are as follows:In Chapter 1,we introduce the research background,the development of dynamic equations on time scales and illustrate the research contents of this thesis.In Chapter 2,basic results are given on time scales.We introduce and review nabla derivative on time scales.Nabla integral on time scales is constructed by nabla derivative and contribute to the stochastic calculus on time scales.Some basic concepts and important results of stochastic calculus on time scales are introduced such as DoobMeyer decomposition,It? formula,product rule,stochastic dynamic equations.In Chapter 3,we establish the backward stochastic dynamic equations on time scales(BS▽E),and give a condition for the existence and uniqueness of solutions.Finding solutions to the backward stochastic differential equations on continuous time and backward stochastic difference equations on discrete time is equivalent to finding the martingale representation or martingale decomposition property of the terminal random variable.We establish the martingale decomposition theorem on time scales.Based on the martingale decomposition theorem on time scales,we derive the backward stochastic dynamic equations on time scales with explicit form generator.The explicit form generator at time t depends on the solution Yt at time t.The other form is the implicit form,that the generator at time t depends on the solution at time ρ(t).The solution of BS▽E is a triple tuple(Y,Z,N)instead of(Y,Z),in which N is orthogonal to Brownian motion on time scales Wt.BS▽E is more likely backward stochastic difference.We focus on the explicitly depending generator form due to the dual principle of explicit form of forward stochastic dynamic equations and backward stochastic dynamic equations.In Chapter 4,we study the solvability theory of the fully coupled nonlinear forwardbackward stochastic dynamic equations on time scales(FBS▽E).The dual equations of explicit form forward stochastic dynamic equations are the explicit form backward stochastic dynamic equations for simplicity.This is the time scales’ characteristic that is not found in continuous time.The dual principle on time scales combined with the monotone condition make us get the existence and uniqueness theorems for the solutions to FBS▽E through continuation method.FBS▽E can be considered as a unification and generalization of similar results in forward-backward stochastic difference equations and forward-backward stochastic differential equations.In Chapter 5,we study two kinds of FBS▽E optimal control problems on time scales.One is the multidimensional partially coupled FBS▽E optimal control problem.As the Lebesgue ▽-measure at left-scattered point is not zero,the spike variation method is no longer valid.Thanks to the dual principle on time scales,under the convex assumption,we could establish the adjoint equations on time scales.We derive stochastic maximum principle on time scales through calculus of variation.The explicit form forward stochastic dynamic equations with the explicit form backward stochastic dynamic adjoint equations make the stochastic maximum principle on time scales simple.The results degenerated to the continuous and discrete time case are consistent with the existing results.The different forms of stochastic maximum principle on discrete case and Δ on time scales are equivalent.The other is the fully coupled FBS▽E optimal control problem with the same dimension forward and backward variables.We estimate the solutions to the variational equations.We construct the explicit formulation of the adjoint equations by dual principle on time scales.The Maximum Principle for both partially and fully coupled FBS▽E are derived.In Chapter 6,we apply time scales theory to applications.The hedging problem is an important topic in financial mathematics and mathematical economics.The market driven by Brownian Motion on time scales is no longer a complete market unless there is no time gap on time scales.In this chapter,it shows that a unique risk-minimizing hedging strategy exists,and that it can be constructed using the BS▽E on time scales.As time scales is a closed subset of R,the information on time scales can be regarded as partial information compared with the whole information.The work on time scales can be considered as a type partial information theory.In Chapter 7,we summarize the main content and innovation points of this thesis,and put forward the prospects of the work in the future.