An Optimal Extraction Problem with Price Impact

A price-maker company extracts an exhaustible commodity from a reservoir, and sells it in the spot market. In absence of any actions of the company, the commodity’s spot price evolves as an Ornstein–Uhlenbeck process. While extracting, the company’s actions have an impact on the commodity’s spot price. The company aims at maximizing the total expected profits from selling the commodity, net of the total expected proportional costs of extraction. We model this problem as a two-dimensional degenerate singular stochastic control problem with finite fuel. The optimal extraction rule is triggered by a strictly decreasing smooth curve that depends on the current level of the reservoir, and for which we provide an explicit expression. Finally, our study is complemented by a theoretical and numerical analysis of the dependency of the optimal extraction strategy and value function on the model’s parameters.