基于鞅驱动的时滞倒向随机微分方程最优控制

Optimal Control for Backward Stochastic Differential Equations with Time Delayed Generator Driven by Martingale

讨论了一类倒向随机微分方程(BSDEs)的最优控制问题,方程是由连续鞅驱动并具有时滞结构.并研究了解的存在唯一性问题,通过引入与方程对偶的超前随机微分方程(ASDEs),构造压缩映射,利用不动点定理证明了具有连续鞅的时滞倒向随机微分方程及其对偶方程解的存在唯一性,利用方程的对偶性,实现最优控制.本文提出的最优控制方法可应用于具有延迟盈余的养老金、具有时滞效应的股票期权定价与原保险费率等计算.

In this paper,we study the optimal control problem for a class of backward stochastic differential equations,which are driven by continuous martingales and have time delayed structures.By introducing an anticipated stochastic differential equation dual to the given equation,constructing a contraction mapping,we prove the existence and uniqueness of the solution for the backward stochastic differential equation with time delayed generator driven by continuous martingales by fixed point theorem,the same result is also obtained for its dual equations.By using the duality of these two equations,the optimal control is derived,which can be applied to the calculation of pension fund with delayed surplus,stock option pricing and original premium rate with time delay in finance.