A Policy Iteration Scheme for Semilinear Stochastic Hamilton-Jacobi-Bellman Equations with Exponential Convergence

Hasib Uddin Molla; Jinniao Qiu

A Policy Iteration Scheme for Semilinear Stochastic Hamilton-Jacobi-Bellman Equations with Exponential Convergence

Optimization and Control 2026-07-31 v1 Mathematical Finance

Authors: Hasib Uddin Molla , Jinniao Qiu

Abstract

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.

Keywords

non-Markovian stochastic optimal control SHJB equation semilinear form policy iteration successive linearization mean-square convergence exponential rate

Cite

@article{arxiv.2607.29024,
  title  = {A Policy Iteration Scheme for Semilinear Stochastic Hamilton-Jacobi-Bellman Equations with Exponential Convergence},
  author = {Hasib Uddin Molla and Jinniao Qiu},
  journal= {arXiv preprint arXiv:2607.29024},
  year   = {2026}
}