AI 中文总结
本文研究总储量未知的耗竭性资源在无限时域随机控制下的最优开采，提出时间一致等价形式，以随机Hamilton--Jacobi方程粘性解刻画值函数，并证明收敛至确定性基准，长期开采率由极限分布风险率决定。
AI 中文摘要
我们研究了一个无限时域随机控制问题，该问题涉及对总储量未知的耗竭性资源的最优开采。信息既通过持续开采而未耗竭内生地产生，也通过外部信息流外生地产生。这种相互作用使得自然问题非马尔可夫且时间不一致。我们证明，尽管如此，它仍可等价地转化为一个具有相同最优控制的时间一致形式。相应的值函数被刻画为具有随机系数的随机Hamilton--Jacobi方程的唯一的粘性解。我们通过基于Snell包络的严格接触论证证明了无限时域上的比较原理，并通过独立的布朗正则化和BSDE修正建立了唯一性，避免了分段马尔可夫近似。最后，我们确定了确定性基准，并表明在持续储量不确定性下，重标度的随机值函数收敛到一个逐路径的确定性控制问题，长期最优开采率由极限储量分布的渐近风险率决定。
英文摘要
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.
Comments25 pages
