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In this paper, we study two-player investment problems with investment coststhat are bounded below by some fixed positive constant. We seek a descriptionof optimal investment strategies for a duopoly problem in which two firmsinvest in advertising projects to abstract market share from the rival firm. Weshow that the problem can be formulated as a stochastic differential game inwhich players modify a jump-diffusion process using impulse controls. We provethat the value of the game may be represented as a solution to a doubleobstacle quasi-variational inequality and derive a PDE characterisation (HJBIequation) of the value of the game. We characterise both the saddle pointequilibrium and a Nash equilibrium for the zero-sum and non-zero-sum payoffgames.

Duopoly Investment Problems with Minimally Bounded Adjustment Costs