Optimal extraction of exhaustible resources with stochastic price

In this paper, a stochastic optimal model of the exploration of exhaustible resources has been established with the aim to maximize the profit of extraction and by the assumption that both the price and the reserve of resource following the geometric Brownian motion. The HJB equation of the model and its economics explanation are deduced. Moreover, it is shown that the price of the exhaustible resource is equal to the sum of the extracted cost and its shadow rent.