Linear Quadratic Optimal Control Under Backward Stochastic Differential Equations And Stochastic Stackelberg Differential Game Problems

Stochastic optimal control is one of the central subjects of modern control science.Stochastic control systems driven by backward stochastic differential equations are widely used in mathematical economics and mathematical finance.Among them,the linear quadratic optimal control problem occupies an important position in the stochastic control system as an approximation of various control problems.At the same time,the stochastic Stackelberg differential game problem under the framework of backward stochastic differential equation also has broad application space and research significance because of its good structure and economic background.Based on this,we have carried out the following three researches and made the following progress:(1)First,we study a kind of linear quadratic optimal control problem driven by forward-backward stochastic differential equations(FBSDEs in short)with deterministic coefficients.The cost functional is defined by the solution of FBSDEs.By means of the Girsanov transformation,the original issue is turned equivalently into the well-known classical LQ problem.By functional analysis approach,some necessary and sufficient conditions for the existence of optimal controls have been obtained.Moreover,we investigate the relationship between two groups of first-order and secondorder adjoint equations.A new stochastic Riccati equation is derived,which leads to the state feedback form of optimal control.By introducing a new Hamiltonian function with an exponential factor,we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward-backward stochastic system with nonconvex control domain using first-order adjoint equation.An illustrative example is given as well.(2)Next,we concern with a Stackelberg stochastic differential game,where the systems are driven by stochastic differential equation(SDE for short),in which the control enters the randomly disturbed coefficients(drift and diffusion).The control region is postulated to be convex.By making use of the first-order adjoint equation(backward stochastic differential equation,BSDE for short),we are able to establish the Pontryagin’s maximum principle for the leader’s global Stackelberg solution,within adapted open-loop structure(AOL for short)and adapted closed-loop memoryless information one(ACLM for short),respectively,where the term global indicates that the leader’s domination over the entire game duration.Since the follower’s adjoint equation turns out to be a BSDE,the leader will be confronted with a control problem where the state equation is a kind of fully coupled forward-backward stochastic differential equation.As an application,we study a class of linear-quadratic(LQ for short)Stackelberg games in which the control process is constrained in a closed convex subset Γ of full space Rm.The state equations are represented by a class of fully coupled FBSDEs with projection operators on Γ.By means of monotonicity condition method,the existence and uniqueness of such FBSDEs are obtained.When the control domain is full space,we derive the resulting backward stochastic Riccati equations.(3)Based on(2),the Stackelberg stochastic differential game with infinite interval random coefficients under convex control constraints is further studied.By using the first-order adjoint equation(BSDE),we can also obtain the Pontryagin maximum principle of the leader’s global Stackelberg solution in the adaptive open-loop(abbreviated as AOL)structure in the global sense.Leaders will face a state equation that is an infinite interval fully coupled FBSDEs control problem.An application of LQ Stackelberg game in infinite interval is studied.With the help of homotopy method,we prove the existence and uniqueness of this kind of FBSDEs.When the control domain is assumed to be full space,we will deduce the corresponding infinite horizon backward stochastic Riccati equation with asymmetric coefficients.