The maximum principle for partially observed optimal control of forward-backward stochastic systems with random jumps

Abstract

This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backward stochastic differential equation theory with certain classical convex variational techniques, the necessary maximum principle is proved for the partially observed optimal control, where the control domain is a nonempty convex set. Under certain convexity assumptions, the author also gives the sufficient conditions of an optimal control for the aforementioned optimal optimal problem. To illustrate the theoretical result, the author also works out an example of partial information linear-quadratic optimal control, and finds an explicit expression of the corresponding optimal control by applying the necessary and sufficient maximum principle.

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

“…Furthermore, some filter estimates of adjoint processes were obtained, which were used to describe optimal control. Along this line, there are a few works including Wang and Wu [17], Shi and Wu [14], Xiao [23]. Concerning optimal control of stochastic differential equations (SDEs, for short) with partial observation, see e.g.…”

“…Note that the drift coefficient h of the observation equations in [30], [31], [15], [22], [17], [14], [23], [20] is uniformly bounded with respect to (t, x, v). The assumption simplifies the computations of this class of problems.…”

“…Because the cross-variation of x v t and Y v t is t 0 c s h s ds, and the drift coefficient of the observation equation is linear with respect to the state x, the LQ problem is different from the setups of [1], [2], [30], [31], [15], [22], [17], [14], [23], [18] and the references therein. In addition, it seems that the approximation method with Girsanov's transformation introduced in [18] is not available to deal with the LQ problem.…”

“…Since G t is independent of the state and the control, Girsanov's transformation as used in Problem (NLC) is unnecessary. Thus, the current work as well as [22], [17], [14], [23], [18] is distinguished from [7], [9], [12], [13], [19].…”

A Maximum Principle via Malliavin calculus for combined stochastic control and impulse control of forward‐backward systems

“…Similarly, by (14), the duality formulae of Malliavin derivatives and the Fubini Theorem (see theorem 64 in [34]) we get:…”