Duopoly Investment Problems with Minimally Bounded Adjustment Costs

TL;DR

This paper analyzes a duopoly investment game with bounded adjustment costs, deriving equilibrium strategies through stochastic differential game theory and PDE characterizations.

Contribution

It introduces a novel formulation of duopoly investment with bounded costs using impulse controls and provides equilibrium characterizations via quasi-variational inequalities.

Findings

Equilibrium strategies are characterized as solutions to a double obstacle quasi-variational inequality.

The game value is represented by a solution to a PDE (HJBI equation).

Both saddle point and Nash equilibria are explicitly characterized.

Abstract

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.