Abstract
In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.
Highlights
As known to all, stochastic differential equations and stochastic analysis develop rapidly. e theory of stochastic differential equations is widely used in economy, biology, physics, financial mathematics, and other fields
In order to give the probabilistic expression of stochastic partial differential equations, Pardoux and Peng [1] gave a class of double stochastic differential equations
Zhu and Shi [3] discussed the optimal control problem of the backward doubly stochastic system with partial information. They studied a type of forward-backward doubly stochastic differential equations with random jumps and applied their results to related games [4]
Summary
Stochastic differential equations and stochastic analysis develop rapidly. e theory of stochastic differential equations is widely used in economy, biology, physics, financial mathematics, and other fields. Han et al [2] deduced the maximum principle for the backward doubly stochastic control system. Wu and Wang [7] studied the optimal control problem of the backward stochastic differential delay equation under partial information. Lv et al [8] considered the maximum principle for optimal control of the anticipated forward-backward stochastic delayed system with regime switching.
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