A fixed-point policy-iteration-type algorithm for symmetric nonzero-sum stochastic impulse control games

TL;DR

This paper introduces a new fixed-point policy-iteration algorithm for symmetric nonzero-sum stochastic impulse control games, improving efficiency and convergence over previous heuristic methods, and providing high-precision solutions.

Contribution

It presents a simpler, more precise, and efficient fixed-point policy-iteration algorithm for symmetric games, removing dependence on initial guesses and relaxation schemes.

Findings

Algorithm achieves high-precision equilibrium payoffs.

Convergence analysis based on graph-theoretic interpretations.

Successfully solves challenging problems beyond current theoretical scope.

Abstract

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author's best knowledge, the only numerical method in the literature is the heuristic one we put forward to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory.