Abstract
We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtue of Ekeland’s variational principle. Finally, applications to a state constrained stochastic delayed linear-quadratic control model and a production-consumption choice problem are studied to illustrate the main obtained result.
Highlights
In, the nonlinear backward stochastic differential equation (BSDE in short) was introduced by Pardoux and Peng [ ]
A maximum principle of optimal control of SDDEs on infinite horizon was proved in Agram, Haadem and Øksendal [ ]
A stochastic maximum principle is derived, which presents the required condition of the optimal terminal control
Summary
In , the nonlinear backward stochastic differential equation (BSDE in short) was introduced by Pardoux and Peng [ ]. For a stochastic delayed system, Chen and Wu [ ] obtained a stochastic maximum principle by virtue of a duality between stochastic differential delayed equations (SDDEs in short) and anticipated BSDEs. Øksendal, Sulem and Zhang [ ] studied the optimal control problems for SDDEs with jumps. The stochastic control problem of a forward delayed system with terminal state constraint is studied. Some recent developed results on state constraints (see [ , – ]) as well as the duality relation between time-advanced stochastic differential equations (SDEs, for short) and time-delayed BSDEs (see [ ]) may help us to overcome the above mentioned difficulties. In Section , the original control problem of a forward delayed controlled system with terminal state constraint is formulated. A stochastic maximum principle is derived, which presents the required condition of the optimal terminal control.
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