Duopoly Investment Problems with Minimally Bounded Adjustment Costs

In this paper, we study two-player investment problems with investment costs that are bounded below by some fixed positive constant. We seek a description of optimal investment strategies for a duopoly problem in which two firms invest in advertising projects to abstract market share from the rival firm. We show that the problem can be formulated as a stochastic differential game in which players modify a jump-diffusion process using impulse controls. We prove that the value of the game may be represented as a solution to a double obstacle quasi-variational inequality and derive a PDE characterisation (HJBI equation) of the value of the game. We characterise both the saddle point equilibrium and a Nash equilibrium for the zero-sum and non-zero-sum payoff games.