Continuous and Impulse Controls Differential Game in Finite Horizon with Nash-Equilibrium and Application

TL;DR

This paper introduces a new class of finite-horizon differential games with mixed control types, characterizes the value function via viscosity solutions, and provides a verification theorem for Nash-equilibrium, with applications to portfolio optimization.

Contribution

It develops a novel framework for differential games with continuous and impulse controls, including existence, approximation, and verification of Nash-equilibrium solutions.

Findings

Existence of an approximate value function converging to the true value

Characterization of the value function as a viscosity solution of a double-obstacle HJBI equation

Derivation of a new portfolio optimization model with computational algorithms

Abstract

This paper considers a new class of deterministic finite-time horizon, two-player, zero-sum differential games (DGs) in which the maximizing player is allowed to take continuous and impulse controls whereas the minimizing player is allowed to take impulse control only. We seek to approximate the value function, and to provide a verification theorem for this class of DGs. We first, by means of dynamic programming principle (DPP) in viscosity solution (VS) framework, characterize the value function as the unique VS to the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) double-obstacle equation. Next, we prove that an approximate value function exists, that it is the unique solution to an approximate HJBI double-obstacle equation, and converges locally uniformly towards the value function of each player when the time discretization step goes to zero. Moreover, we provide a verification theorem which characterizes a Nash-equilibrium for the DG control problem considered. Finally, by applying our results, we derive a new continuous-time portfolio optimization model, and we provide related computational algorithms.