The standard Hotelling model assumes that the stock of an exhaustible resource is known. We expand on the model by Arrow and Chang that introduced stochastic discoveries and for the first time completely solve such a model using impulse control. The model has two state variables: the "proven'' reserves as well as a finite unexplored area available for exploration with constant marginal cost, resulting in a Poisson process of new discoveries. We prove that a frontier of critical levels of "proven'' reserves exists, above which exploration is stopped, and below which it happens at infinite speed. This frontier is increasing in the explored area, and higher "proven'' reserve levels along this critical threshold are indicative of more scarcity, not less. In our stochastic generalization of Hotelling's rule, price expectations conditional on the current state rise at the rate of interest across exploratory episodes. However, the state-dependent conditional expected path of prices realized prior to exhaustion of the exploratory area rises at a rate lower than the rate of interest, consistent with most empirical tests based on observed price histories.

Joint work with Ivar Ekeland, Wolfram Schlenker and Brian Wright.

Bio: Peter Tankov is a professor of quantitative finance at ENSAE, the French National School of Statistics and Economic Administration, now part of Institut Polytechnique de Paris. He earned his doctorate from Ecole Polytechnique in 2004 under the supervision of Rama Cont and was previously a professor at the University of Paris Diderot and lecturer at Ecole Polytechnique. His research focuses on applied probability, quantitative finance, energy finance and green finance. He works on topics related to climate-induced financial risks, electricity markets, energy mix scenarios, forecasting and risk management for the renewable energy industry. He is the author of more than 45 research papers and a reference book on stochastic modelling with jumps.