Optimal Resource Extraction under Distribution Learning and Infinite-Horizon Stochastic Hamilton--Jacobi Equations
Abstract
We study an infinite-horizon stochastic control problem for the optimal exploitation of an exhaustible resource with unknown total reserves. Information is generated both endogenously through continued extraction without depletion and exogenously through an external information flow. This interaction makes the natural problem non-Markovian and time-inconsistent. We show that it nevertheless admits an equivalent time-consistent formulation with the same optimal controls. The associated value function is characterized as the unique viscosity solution of a stochastic Hamilton--Jacobi equation with random coefficients. We prove comparison on the infinite horizon through a Snell-envelope-based strict-contact argument and establish uniqueness by an independent Brownian regularization and a BSDE correction, avoiding piecewise Markovian approximations. Finally, we identify the deterministic benchmark and show that, under persistent reserve uncertainty, the rescaled stochastic value function converges to a pathwise deterministic control problem, with the long-run optimal extraction rate determined by the asymptotic hazard rate of the limiting reserve distribution.
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BibTeX
@misc{horst2026optimal,
title = {{Optimal Resource Extraction under Distribution Learning and Infinite-Horizon Stochastic Hamilton--Jacobi Equations}},
author = {Ulrich Horst and Jinniao Qiu and Yang Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17717},
eprint = {2609.17717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17717},
}
RIS
TY - UNPB
TI - Optimal Resource Extraction under Distribution Learning and Infinite-Horizon Stochastic Hamilton--Jacobi Equations
AU - Ulrich Horst
AU - Jinniao Qiu
AU - Yang Yang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17717
UR - https://arxiv.org/abs/2609.17717
AB - We study an infinite-horizon stochastic control problem for the optimal exploitation of an exhaustible resource with unknown total reserves. Information is generated both endogenously through continued extraction without depletion and exogenously through an external information flow. This interaction makes the natural problem non-Markovian and time-inconsistent. We show that it nevertheless admits an equivalent time-consistent formulation with the same optimal controls. The associated value function is characterized as the unique viscosity solution of a stochastic Hamilton--Jacobi equation with random coefficients. We prove comparison on the infinite horizon through a Snell-envelope-based strict-contact argument and establish uniqueness by an independent Brownian regularization and a BSDE correction, avoiding piecewise Markovian approximations. Finally, we identify the deterministic benchmark and show that, under persistent reserve uncertainty, the rescaled stochastic value function converges to a pathwise deterministic control problem, with the long-run optimal extraction rate determined by the asymptotic hazard rate of the limiting reserve distribution.
ER -