A Linear-Quadratic Optimal Control Problem of Forward-Backward Stochastic Differential Equations With Partial Information

Abstract

This paper studies a linear-quadratic optimal control problem derived by forward-backward stochastic differential equations, where the drift coefficient of the observation equation is linear with respect to the state x, and the observation noise is correlated with the state noise, in the sense that the cross-variation of the state and the observation is nonzero. A backward separation approach is introduced. Combining it with variational method and stochastic filtering, two optimality conditions and a feedback representation of optimal control are derived. Closed-form optimal solutions are obtained in some particular cases. As an application of the optimality conditions, a generalized recursive utility problem from financial markets is solved explicitly.Index terms-Linear-quadratic optimal control, forwardbackward stochastic differential equation, partial information, correlated state and observation noises, closed-form solution, recursive utility.

Linear‐Quadratic Optimal Control Problem for Partially Observed Forward‐Backward Stochastic Differential Equations of Mean‐Field Type

“…In addition, the cross-variation of x v and Y v is ∫ t 0 c s h s ds, not zero. Meanwhile, our work is a great continuation of the results of [24] in the context of linear-quadratic control problem. Therefore, a new approach is desired to solve these problems.…”

“…Furthermore, we prove that the optimal cost functional can be explicitly expressed by the solution of the Riccati equations for the special case. Meanwhile, our work is a great continuation of the results of [24] in the context of linear-quadratic control problem. This paper is organized as follows.…”

“…One is that it is natural to determine the control v by the observation Y v , while Y v dependents on the control v by the state x v , which makes it different from the setups of [16,21] and [22]. Wang, Wu and Xiong [24] also adopted a backward separation technique to solve a linear-quadratic optimal control problem under the condition of convex domain. In addition, the cross-variation of x v and Y v is ∫ t 0 c s h s ds, not zero.…”

Exponential stabilisation of continuous‐time periodic stochastic systems by feedback control based on periodic discrete‐time observations

“…3, 9.95), the system is set to be observed 1375 times with interval 0.0012. The highest observation frequency is required for t ∈ [11,12). Moreover, existing theory yields the constant observation interval τ ≤ 0.00026, calculated with the same controller and same Lyapunov function, according to [23] with observation of system mode discretised.…”

Partially observed linear quadratic control problem with delay via backward separation method

“…Moreover, backward separation method is applicable to a broad class of nonlinear control systems, for example, controlled forward–backward stochastic differential equations (FBSDEs). Interested readers may refer to and for recent advancements on this topic.…”