Numerical approximations of the value of zero-sum stochastic differential impulse controls game in finite horizon

TL;DR

This paper develops a numerical method using policy iteration to approximate the value function and optimal strategies in a finite-horizon zero-sum stochastic differential game with impulse controls, with proven convergence to the viscosity solution.

Contribution

It introduces a novel numerical scheme for solving zero-sum stochastic impulse control games and provides a rigorous convergence analysis of the approximation method.

Findings

The scheme converges to the viscosity solution as discretization improves.

The method successfully computes Nash equilibrium strategies in a simulated exchange rate game.

The approach is applicable to finite-horizon stochastic differential games with impulse controls.

Abstract

In this paper, we consider a differential stochastic zero-sum game in which two players intervene by adopting impulse controls in a finite time horizon. We provide a numerical solution as an approximation of the value function, which turns out to be the same for both players. While one seeks to maximize the value function, the other seeks to minimize it. Thus we find a single numerical solution for the Nash equilibrium as well as the optimal impulse controls strategy pair for both player based on the classical Policy Iteration (PI) algorithm. Then, we perform a rigorous convergence analysis on the approximation scheme where we prove that it converges to its corresponding viscosity solution as the discretization step approaches zero, and under certain conditions. We showcase our algorithm by implementing a two-player almost analytically solvable game in which the players act through impulse control and compete over the exchange rate.