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.
