Stochastic Optimal Control And Differential Game Problems Of Partially Observed Systems

In this thesis,partially observed stochastic optimal control and Stackelberg differential game problems are studied.In detail,under partially observed forward-backward stochastic systems with random jumps’ framework,we establish the global maximum principle for progressive optimal control,and the maximum principle for discounted optimal control on infinite horizon.Then a linear quadratic Stackelberg stochastic differential game is investigated.Considering the stochastic systems driven by fractional Brownian motion,we obtain its global maximum principle.Moreover,these theoretical results are also applied to some practical application,such as optimal consumption and investment problem and dynamic advertising problem.In most optimal control problems,the players are often assumed to fully observe the state of the controlled stochastic systems.However,it is not practical in reality due to the asymmetric role and the delayed information.Therefore,we aim to study the partially observed stochastic systems and take only partial information available to the players into account.That is,the players can not fully observe the system which,however,can be solved by observing some related observation process.Therefore,it is meaningful to study the partially observed optimal control problem and obtain its maximum principle of optimality.When consider the differential game of multiple players,a typical example among various dynamic games is the Stackelberg(also known as leader-follower)game introduced by Stackelberg[111]whose economic background can be derived form some markets where certain companies have advantages of domination over others.There are usually two players with asymmetric roles,one follower and one leader.The core is that one player must make a decision after the other player’s decision is made,which shows the hierarchical feature.Since its meaningful structure and background,the Stackelberg game has received substantial interest.Especially,due to the asymmetric roles endowed on the follower and the leader which matches the asymmetric known information between them,a partially observed Stackelberg differential game is worthy to be studied where another more practical asymmetric information setting are given motivated by some illustrative applications.Except for the standard Brownian motion(H=1/2),in recent years,there has been considerable research interest in stochastic system driven by fractional Brownian motion(H ∈(0,1)and H≠1/2).The process has been applied to many fields such as climatology,economics,internet traffic analysis and finance.However,the stochastic integral with respect to fractional Brownian motion is usually not a semi-martingale,so some important stochastic analysis theory,similar to that of Brownian motion,fails to obtain,which is necessary for optimal control problem to solve the optimal condition.Therefore,studying the maximum principle of stochastic system driven by fractional Brownian motion,especially for H<1/2,has great significance in theory and practical applications.Now,we give our main research content of this thesis as follows:In Chapter 1,we introduce the research backgrounds,research frontier and purposes and give the main content and novel contributions of the following four chapters,respectively.In Chapter 2,we discuss a partially observed progressive optimal control problem of forward-backward stochastic differential equations with random jumps,where the control domain is not necessarily convex,and the control variable enters into all the coefficients.In our model,the observation equation is not only driven by a Brownian motion but also a Poisson random measure,which also have correlated noises with the state equation.The partially observed global maximum principle is proved.To show its applications,a partially observed linear quadratic progressive optimal control problem of forward-backward stochastic differential equations with random jumps is investigated,by the maximum principle and stochastic filtering.State estimate feedback representation of the optimal control is given in a more explicit form by introducing some ordinary differential equations.In Chapter 3,we investigate a discounted optimal control problem of partially observed forward-backward stochastic systems with jumps on infinite horizon.The control domain is convex and a kind of infinite horizon observation equation is introduced.The unique solvability of infinite horizon forward(backward)stochastic differential equation with jumps is obtained and more extended analyses,especially for the backward case,are made.Some new estimates are first given and proved for the critical variational inequality.Then a maximum principle is obtained by introducing some infinite horizon adjoint equations whose unique solvabilities are guaranteed necessarily.Meanwhile,a sufficient maximum principle is also given.Moreover,a solvable LQ example is shown and an application to optimal consumption and investment problem with recursive utility is given.Finally,some comparisons are made with two kinds of representative infinite horizon stochastic systems and their related optimal control problems.In Chapter 4,we focus on a linear quadratic partially observed Stackelberg stochastic differential game with correlated state and observation noises,where the control domain is not necessarily convex.Both the leader and the follower have their own observation equations,and the information filtration available to the leader is contained in that available to the follower.Necessary and sufficient conditions of the Stackelberg equilibrium points are derived.In the follower’s problem,the state estimation feedback of optimal control can be represented by a forward-backward stochastic differential filtering equation and some Riccati equation.In the leader’s problem,via the innovation process,the state estimation feedback of optimal control is represented by a stochastic differential filtering equation,a semi-martingale process and three high-dimensional Riccati equations.As an application,a dynamic advertising problem with asymmetric information is studied,and the effectiveness and reasonability of the theoretical result are illustrated by numerical simulations.In Chapter 5,we study the stochastic control problem of partially observed(multidimensional)stochastic system driven by both Brownian motions and fractional Brownian motions.In the absence of the powerful tool of Girsanov transformation,we introduce and study new stochastic processes by applying rough path theory which are used to transform the original problem to a "classical one".The adjoint backward stochastic differential equations and the necessary condition satisfied by the optimal control(maximum principle)are obtained.In Chapter 6,we summarize the whole thesis and look into the problems to be solved in this paper.