On an Optimal Extraction Problem with Regime Switching

TL;DR

This paper analyzes an optimal extraction problem for exhaustible resources under regime switching, deriving explicit strategies and value functions for different cost structures using stochastic control and optimal stopping theory.

Contribution

It provides explicit solutions for the optimal extraction policy and value function in a regime-switching setting with finite reserves, considering convex and concave cost functions.

Findings

Optimal extraction involves Skorokhod reflection or instant depletion depending on cost convexity.

Thresholds for extraction are characterized via optimal stopping boundaries.

Explicit formulas for value functions and control strategies are derived.

Abstract

This paper studies a finite-fuel two-dimensional degenerate singular stochastic control problem under regime switching that is motivated by the optimal irreversible extraction problem of an exhaustible commodity. A company extracts a natural resource from a reserve with finite capacity, and sells it in the market at a spot price that evolves according to a Brownian motion with volatility modulated by a two-state Markov chain. In this setting, the company aims at finding the extraction rule that maximizes its expected discounted cash flow, net of the costs of extraction and maintenance of the reserve. We provide expressions both for the value function and for the optimal control. On the one hand, if the running cost for the maintenance of the reserve is a convex function of the reserve level, the optimal extraction rule prescribes a Skorokhod reflection of the (optimally) controlled state process at a certain state and price dependent threshold. On the other hand, in presence of a concave running cost function it is optimal to instantaneously deplete the reserve at the time at which the commodity's price exceeds an endogenously determined critical level. In both cases, the threshold triggering the optimal control is given in terms of the optimal stopping boundary of an auxiliary family of perpetual optimal selling problems with regime switching.