Nonzero-sum stochastic impulse games with an application in competitive retail energy markets

TL;DR

This paper models a nonzero-sum stochastic impulse game in retail energy markets, establishing a connection between Nash equilibria and quasi-variational inequalities, and providing a numerical scheme to analyze competitive behaviors.

Contribution

It develops a novel framework linking Nash equilibria with QVIs in impulse games and introduces a convergent numerical scheme for practical applications.

Findings

Model accurately reproduces electricity retail competition behavior

Proves value functions are constrained viscosity solutions of QVIs

Provides a numerical method for solving complex impulse game systems

Abstract

We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the value function of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth functions. We also introduce a family of value functions converging to our value function of each player, and which is characterized as the unique constrained viscosity solutions of an approximation of our QVIs system. This convergence result is useful for numerical purpose. We apply a probabilistic numerical scheme which approximates the solution of the QVIs system to the case of the competition between two electricity retailers. We show how our model reproduces the qualitative behaviour of electricity retail competition.