Control Problems Of Several Stochastic Systems

It is well known that the dynamic evolutions of some natural and social phenomena can be described by stochastic differential equations(SDEs).In some cases,the processes not only depend on the current states but also relate to their past situations.Thus,the SDEs about these evolutions will contain time delays and be called stochastic differential delay equations(SDDEs).Today,SDDEs have been widely used in the fields of biology,finance and physics.Moreover,the research of the theory of SDDEs also reaches a new stage.Recently,a new type of SDDEs is introduced and getting many attentions.Here we call them backward stochastic differential delay equations(BSDDEs).Because of containing time delays,these equations can be regarded as a generalization of classical backward stochastic differential equations(BSDEs).Besides the developments on application and theory,some optimal control problems about those equations have become a rising concern of researchers.In our paper,we study several optimal control problems of forward and backward stochastic delay systems.The main conclusions are listed below:1.We study a a new type of differential game problems of backward stochastic differential delay equations.A three-coupled system of adjoint equations is given by two backward stochastic differential equations and one forward stochastic differential equation.By means of duality and convex variation technique,we give necessary and sufficient conditions for a Nash equilibrium point of non-zero sum games.As an application,a consumption choice problem is embedded in the theoretic frame to illuminate the main results.In terms of the theoretical guidelines,an equilibrium point is obtained.Moreover,some characteristics of BSDDEs are also presented as auxiliary result.2.An optimal control problem of a kind of general BSDEs is considered.The control domain is supposed to be non-convex and the control variable enters the diffusion term.The second-order variable equation and adjoint equation are introduced to deal with the above difficulties.By means of spike variation and estimation technique of BSDE,we establish the maximum principle which is the necessary condition for the optimal control.3.Combining classical variation and adjoint technique,we introduce the anticipated stochastic differential equation as the adjoint equation and obtain the maximum of BSDDEs under partial information.The main results are used to study linear quadratic optimal control problem.During looking for the optimal control,we encounter a new kind of forward and backward stochastic differential equations,which contain time-advanced,time-delayed terms and filtering estimation and can be regarded as the generalization of classical FBSDEs.Here we called them general stochastic differential filtering equations and study the existence of the solutions of those equations.Based on those equations,we get the optimal control.4.We study the partially-observed optimal control problem of SDDEs because sometimes the controllers don't observe complete systems' states.By Girsanov's theorem,the original problem can be converted to the case of full observation.After using the estimation technique of SDDEs to deal with the time-delayed terms and with the help of anticipated SDEs,we deduce the maximum principle for those stochastic delay systems with partial observation.5.Linear quadratic optimal control problem for SDDEs under partial observation is included in the scope of our research.We use the so-called backward separation method to get the optimal control in the form of feedback with the observation equation containing time delay.Moreover,initial efforts are made to explore the filtering for anticipated BSDEs.The obtained results can be viewed as some beneficial supplements of classical filtering theory.