AI 中文总结
针对非马尔可夫随机最优控制问题中的半线性SHJB方程，提出基于逐次线性化的策略迭代算法，证明其近似序列以指数速率单调收敛到值函数。
AI 中文摘要
本文研究值函数为随机场且由随机哈密尔顿-雅可比-贝尔曼（SHJB）方程刻画的非马尔可夫随机最优控制问题。当随机积分系数不受控制时，SHJB方程呈半线性形式，由于存在可测随机性，相比马尔可夫情形面临计算挑战。我们提出基于逐次线性化的策略迭代算法，将非线性SHJB方程转化为一系列线性方程。此外，我们证明所得近似序列在均方意义下以指数速率单调收敛到值函数。
英文摘要
This paper is concerned with the non-Markovian stochastic optimal control problems in which the value function is a random field characterized by a stochastic Hamilton-Jacobi-Bellman (SHJB) equation. When the stochastic integration coefficients are not controlled, the SHJB equation takes a semilinear form, which is subject to computational challenges compared to the Markovian case due to the measurable randomness. We introduce a policy-iteration algorithm based on successive linearization that reduces the nonlinear SHJB equation to a sequence of linear ones. Furthermore, we prove that the resulting approximation sequence converges monotonically to the value function in the mean-square sense with an exponential rate.
