Exploration and exhaustibility in dynamic Cournot games

We study the stochastic effect of resource exploration in dynamic Cournot models of exhaustible resources, such as oil. We firstly treat the case of a monopolist who may undertake costly exploration to replenish his diminishing reserves. We then consider a stochastic game between such an exhaustible producer and a 'green' producer that has access to an inexhaustible but relatively expensive source, such as solar power. The effort control variable is taken to be either continuous or discrete (switching control). In both settings, we assume that new discoveries occur according to a jump process with intensity given by the exploration effort. This leads to a study of systems of non-linear first-order delay ordinary differential equations (ODEs). We derive asymptotic expansions for the case of a small-exploration success rate and present some numerical investigations.