Abstract
The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum differential games problem.
Highlights
It is well known that optimal control problem is one of the central themes of control science
Peng [20] studied the following type of stochastic optimal control problem
Ji and Zhou [12] obtained a maximum principle for stochastic optimal control of non-fully coupled forward-backward stochastic system with terminal state constraints
Summary
It is well known that optimal control problem is one of the central themes of control science. By spike variational method and the second order adjoint equations, Peng [20] obtained a general stochastic maximum principle for the above optimal control problem. Ji and Zhou [12] obtained a maximum principle for stochastic optimal control of non-fully coupled forward-backward stochastic system with terminal state constraints. The notable difficulties to obtain the maximum principles for the fully coupled forward-backward doubly stochastic control systems within non-convex control domains are how to use the spike variational method to get variational equations with enough high order estimates and how to use the duality technique to obtain the adjoint equations. From the maximum principle for optimal control problems of FBDSDEs obtained in this paper, we can find the equations satisfied by Nash equilibrium points for linear quadratic nonzero sum doubly stochastic differential games problems.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have