In this paper, we study an optimal control problem of linear backward stochastic differential equation (BSDE) with quadratic cost functional under partial information. This problem is solved completely and explicitly by using a stochastic maximum principle and a decoupling technique. By using the maximum principle, a stochastic Hamiltonian system, which is a forward-backward stochastic differential equation (FBSDE) with filtering, is obtained. By decoupling the stochastic Hamiltonian system, three Riccati equations, a BSDE with filtering, and a stochastic differential equation (SDE) with filtering are derived. We then get an optimal control with a feedback representation. An explicit formula for the corresponding optimal cost is also established. As illustrative examples, we consider two special scalar-valued control problems and give some numerical simulations.
This paper is concerned with a linear quadratic optimal control problem of delayed backward stochastic differential equations. An explicit representation is derived for the optimal control, which is a linear feedback of the entire past history and the future state trajectory in a short period of time. This is one of the major distinctive features of the delayed backward stochastic linear quadratic optimal control problem. To obtain the optimal feedback, a new class of delayed Riccati equations is introduced and the unique solvability of their solutions are discussed in detail.
This paper is concerned with linear quadratic optimal control problems for mean-field backward stochastic differential equations (MF-BSDEs, for short) with deterministic coefficients. The optimality system, which is a linear mean-field forward-backward stochastic differential equation with constraint, is obtained by a variational method. By decoupling the optimality system, two coupled Riccati equations and an MF-BSDE are derived. It turns out that the coupled two Riccati equations are uniquely solvable. Then a complete and explicit representation is obtained for the optimal control.
This paper focuses on a kind of linear quadratic non-zero sum differential game driven by backward stochastic differential equation with asymmetric information, which is a natural continuation of Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352]. Different from Wang and Yu [IEEE TAC (2010) 55: 1742-1747, Automatica (2012) 48: 342-352], novel motivations for studying this kind of game are provided. Some feedback Nash equilibrium points are uniquely obtained by forward-backward stochastic differential equations, their filters and the corresponding Riccati equations with …
This paper is concerned with a linear-quadratic (LQ, for short) optimal control problem for backward stochastic differential equations (BSDEs, for short), where the coefficients of the backward control system and the weighting matrices in the cost functional are allowed to be random. By a variational method, the optimality system, which is a coupled linear forward-backward stochastic differential equation (FBSDE, for short), is derived, and by a Hilbert space method, the unique solvability of the optimality system is obtained. In order to construct the optimal control, a new stochastic Riccati…
A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and the cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite and cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approac…
This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are present in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished und…
This paper is concerned with optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs. In order to obtain the state feedback representation of the optimal control, we propose a new decoupling technique and obtain one kind of non-Riccati-type ordinary differential equations (ODEs). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs under some assumptions and derive the state feedback form of the optimal control. For this FBLQ problem, the optimal control depends on …
This paper is concerned with a linear quadratic (LQ, for short) optimal control problem for mean-field backward stochastic differential equations (MF-BSDE, for short) driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is established. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a …
We discuss an open-loop backward Stackelberg differential game involving single leader and single follower. Unlike most Stackelberg game literature, the state to be controlled is characterized by a backward stochastic differential equation (BSDE) for which the terminal- instead initial-condition is specified as a priori; the decisions of leader consist of a static terminal-perturbation and a dynamic linear-quadratic control. In addition, the terminal control is subject to (convex-closed) pointwise and (affine) expectation constraints. Both constraints are arising from real applications such as…
In this paper, we investigate infinite horizon jump-diffusion forward-backward stochastic differential equations under some monotonicity conditions. We establish an existence and uniqueness theorem, two stability results and a comparison theorem for solutions to such kind of equations. Then the theoretical results are applied to study a kind of infinite horizon backward stochastic linear-quadratic optimal control problems, and then differential game problems. The unique optimal controls for the control problems and the unique Nash equilibrium points for the game problems are obtained in closed…
In this paper, we investigate the solvability of matrix valued Backward stochastic Riccati equations with jumps (BSREJ), which is associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and Poisson jumps. By dynamic programming principle, Doob-Meyer decomposition and inverse flow technique, the existence and uniqueness of the solution for the BSREJ is established. The difficulties addressed to this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brown…
We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.
In this paper, we consider a partial information two-person zero-sum stochastic differential game problem where the system is governed by a backward stochastic differential equation driven by Teugels martingales associated with a Lévy process and an independent Brownian motion. One sufficient (a verification theorem) and one necessary conditions for the existence of optimal controls are proved. To illustrate the general results, a linear quadratic stochastic differential game problem is discussed.
This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. We first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then we establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, we a…
This paper is concerned with the existence of optimal controls for backward stochastic partial differential equations with random coefficients, in which the control systems are represented in an abstract evolution form, i.e. backward stochastic evolution equations. Under some growth and monotonicity conditions on the coefficients and suitable assumptions on the Hamiltonian function, the existence of the optimal control boils down to proving the uniqueness and existence of a solution to the stochastic Hamiltonian system, i.e. a fully coupled forward-backward stochastic evolution equation. Using…
This paper is concerned with 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 uniquely solvability of infinite horizon forward (backward) stochastic differential equation with jumps is obtained and more extended analysis, especially for the backward case, is made. Some new estimates are first given and proved for the critical variational inequality. Then an ergodic maximum principle is obtained by introducing some infi…
This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward-backward stochastic differential equations with jumps (FBSDEJs). A general sufficient maximum principle for optimal control for a system driven by a Markov regime-switching forward and backward jump-diffusion model is developed. After, an equivalent maximum principle is proved. Malliavin calculus is also employed to derive a general stochastic maximum principle. The latter does not require concavity of Hamiltonian. Applications of the stochastic maximum principle …
We introduce a time-implicit, finite-element based space-time discretization scheme for the backward stochastic heat equation, and for the forward-backward stochastic heat equation from stochastic optimal control, and prove strong rates of convergence. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise.
This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$σ$-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate fe…
We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear FBSDEs. Inspired by Hu (Hu, Probability, Uncertainty and Quantitative Risk, 2(1) (2017):pp 1-20), we develop a new decoupling approach by introducing an adjoint equation which is a quadratic BSDE. By revealing the relations among the terms of the first-order Taylor's expansions, we estimate the orders of them and derive a global stochastic maximum principle whi…
This paper first makes an attempt to investigate the partial information near optimal control of systems governed by forward-backward stochastic differential equations with observation noise under the assumption of a convex control domain. By Ekeland's variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for any $\varepsilon $-near optimal control in a local form with an error order of exact $\varepsilon ^{% \frac{1}{2}}.$ Moreover, under additional convexity conditions on Hamiltonian function, we prove that an $\vareps…
This paper is concerned with a non-zero sum differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information. We establish a necessary maximum principle and sufficient verification theorem of the game system by virtue of the duality and convex variational method. We apply the theoretical results and stochastic filtering theory to study a linear-quadratic game system and derive the explicit form of the Nash equilibrium point and discuss the existence and uniqueness in particular cases. As an application, we consider a time-delayed p…
IIn this paper, we study 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 enter 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. For preparation, we first derive the existence and uniqueness of the solutions to the fully coupled forward-backward stochastic system with random jumps and its e…
Memory-limited partially observable stochastic control (ML-POSC) is the stochastic optimal control problem under incomplete information and memory limitation. In order to obtain the optimal control function of ML-POSC, a system of the forward Fokker-Planck (FP) equation and the backward Hamilton-Jacobi-Bellman (HJB) equation needs to be solved. In this work, we firstly show that the system of HJB-FP equations can be interpreted via the Pontryagin's minimum principle on the probability density function space. Based on this interpretation, we then propose the forward-backward sweep method (FBSM)…
It is a longstanding unsolved problem to characterize the optimal feedbacks for general SLQs (i.e., stochastic linear quadratic control problems) with random coefficients in infinite dimensions; while the same problem but in finite dimensions was just addressed in a recent work [36]. This paper is devoted to giving a solution to this problem under some assumptions which can be verified for several interesting concrete models. More precisely, under these assumptions, we establish the equivalence between the existence of optimal feedback operator for infinite dimensional SLQs and the solvability…
In this article, we concern a kind of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding open-loop Nash equilibrium control is required to adapted to the filtration that the observation process generated. To find this open-loop Nash equilibrium point, we prove the maximum principle as a necessary condition of the existence of this point, and give a verification theorem as a sufficient condition to verify it is the real open-…
In this paper, we consider the Forward--Backward proximal splitting algorithm to minimize the sum of two proper convex functions, one of which having a Lipschitz continuous gradient and the other being partly smooth relative to an active manifold $\mathcal{M}$. We propose a generic framework under which we show that the Forward--Backward (i) correctly identifies the active manifold $\mathcal{M}$ in a finite number of iterations, and then (ii) enters a local linear convergence regime that we characterize precisely. This gives a grounded and unified explanation to the typical behaviour that has …
In this paper, we consider a class of stochastic control problems for stochastic differential equations with random coefficients. The control domain need not to be convex but the control process is not allowed to enter in diffusion term. Moreover, the terminal cost involves a non linear term of the expected value of terminal state. Our purpose is to derive a new version of the Pontryagin's stochastic maximum principle by adopting an idea inspired from the work of Peng [S. Peng, Maximum Principle for Stochastic Optimal Control with Nonconvex Control Domain, Lecture Notes in Control & Informatio…
A linear-quadratic partially observed Stackelberg stochastic differential game with multiple followers and its application to multi-agent formation control
East China University of Science and Technology（华东理工大学）；University of Alberta（阿尔伯塔大学）；Shandong University（山东大学）；Fudan University（复旦大学）
In this paper, a leader-follower stochastic differential game is studied for a linear stochastic differential equation with a quadratic cost functional. The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. The follower first solves a stochastic linear quadratic optimal control problem, and his optimal closed-loop strategy is characterized by a Riccati equation, together with a…
We study a linear quadratic optimal control problem with stochastic coefficients and a terminal state constraint, which may be in force merely on a set with positive, but not necessarily full probability. Under such a partial terminal constraint, the usual approach via a coupled system of a backward stochastic Riccati equation and a linear backward equation breaks down. As a remedy, we introduce a family of auxiliary problems parametrized by the supersolutions to this Riccati equation alone. The target functional of these problems dominates the original constrained one and allows for an explic…
We study methods for solving stochastic control problems of systems of forward-backward mean-field equations with delay, in finite or infinite horizon. Necessary and sufficient maximum principles under partial information are given. The results are applied to solve a recursive utility optimal problem
This paper is concerned with the partial information optimal control problem of mean-field type under partial observation, where the system is given by a controlled mean-field forward-backward stochastic differential equation with correlated noises between the system and the observation, moreover the observation coefficients may depend not only on the control process and but also on its probability distribution. Under standard assumptions on the coefficients, necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin's maximum principles are established…
Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes
We extend the Malitsky-Tam forward-reflected-backward (FRB) splitting method for inclusion problems of monotone operators to nonconvex minimization problems. By assuming the generalized concave Kurdyka-Łojasiewicz (KL) property of a quadratic regularization of the objective, we show that the FRB method converges globally to a stationary point of the objective and enjoys finite length property. The sharpness of our approach is guaranteed by virtue of the exact modulus associated with the generalized concave KL property. Numerical experiments suggest that FRB is competitive compared to the Dougl…
