Noname manuscript No. (will be inserted by the editor) Regulating Industries under Exogenous Uncertainty

Abstract We present a quantitative method to find jointly optimal strategies for an industry regulator and a firm, who operate under exogenous uncertainty. The firm controls its operating policy in order to maximize its expected future profits, whilst taking account of regulatory fines. The regulator aims to control the probability that the firm enters a given undesirable state, such as ceasing production, by imposing a fine which is as low as possible, while achieving the required reduction in probabil-ity. The exogenous uncertainty is modeled using a stochastic differential equation, and we show this implies that the firm’s behavior can be solved via the Hamilton-Jacobi-Bellman equation, and the regulatory fine can be obtained via the Feynman-Kac formula. We discuss both analytic and numerical solution methods. Our results are illustrated for a security of supply problem for vaccine production where future production costs are uncertain and, using empirical data, for an abandonment prob-lem in a gold mining operation where future commodity prices are uncertain. The method determines the level of fine which establishes a Nash equilibrium in these nonzero-sum games, under uncertainty