Stochastic Optimal Control of Hawkes Jump-Diffusion Systems
Abstract
This paper is devoted to developing a framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. Building on the Poisson point process formulation studied in arXiv:2511.13568, we extend the model to disasters driven by a marked Hawkes process, allowing past disasters to temporarily increase the likelihood of subsequent shocks. Our work focuses on a subcritical Markovian Hawkes specification, in which the state space is augmented by the self-excitation component of disaster risk. We establish the well-posedness and nonexplosion of the resulting controlled Hawkes dynamics. Using the Hamilton-Jacobi-Bellman characterization of the corresponding Poisson control problem, we obtain quantitative estimates for the Hawkes excitation process and prove, under a small-excitation scaling, that the Hawkes value function converges to its Poisson counterpart as the magnitude of self-excitation vanishes. This provides a rigorous Poisson approximation of the stochastic control problem and quantifies the effect of self-excitation on optimal growth under environmental disaster risk.
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BibTeX
@misc{sakhanda2026stochastic,
title = {{Stochastic Optimal Control of Hawkes Jump-Diffusion Systems}},
author = {Daria Sakhanda and Joshué Helí Ricalde-Guerrero},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29473},
eprint = {2608.29473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29473},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Stochastic Optimal Control of Hawkes Jump-Diffusion Systems
AU - Daria Sakhanda
AU - Joshué Helí Ricalde-Guerrero
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29473
UR - https://arxiv.org/abs/2608.29473
AB - This paper is devoted to developing a framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. Building on the Poisson point process formulation studied in arXiv:2511.13568, we extend the model to disasters driven by a marked Hawkes process, allowing past disasters to temporarily increase the likelihood of subsequent shocks. Our work focuses on a subcritical Markovian Hawkes specification, in which the state space is augmented by the self-excitation component of disaster risk. We establish the well-posedness and nonexplosion of the resulting controlled Hawkes dynamics. Using the Hamilton-Jacobi-Bellman characterization of the corresponding Poisson control problem, we obtain quantitative estimates for the Hawkes excitation process and prove, under a small-excitation scaling, that the Hawkes value function converges to its Poisson counterpart as the magnitude of self-excitation vanishes. This provides a rigorous Poisson approximation of the stochastic control problem and quantifies the effect of self-excitation on optimal growth under environmental disaster risk.
ER -