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

Author

Listed:
  • 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.

Suggested Citation

  • Hasib Uddin Molla & Jinniao Qiu, 2026. "A Policy Iteration Scheme for Semilinear Stochastic Hamilton-Jacobi-Bellman Equations with Exponential Convergence," Papers 2607.29024, arXiv.org.
    • Handle: RePEc:arx:papers:2607.29024
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    References listed on IDEAS

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    1. Christian Bayer & Eric Joseph Hall & Raãšl Tempone, 2022. "Weak Error Rates For Option Pricing Under Linear Rough Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 25(07n08), pages 1-47, November.
    2. Qiu, Jinniao, 2017. "Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1926-1959.
    3. Du, Kai & Zhang, Qi, 2013. "Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1616-1637.
    4. Ma, Jin & Yin, Hong & Zhang, Jianfeng, 2012. "On non-Markovian forward–backward SDEs and backward stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3980-4004.
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