Non-Markovian Impulse Control Under Nonlinear Expectation

TL;DR

This paper establishes a dynamic programming principle for non-Markovian impulse control problems under nonlinear expectations, proving the existence of a game value and conditions for saddle points in a path-dependent stochastic differential game.

Contribution

It introduces a novel approach to non-Markovian impulse control under adverse nonlinear expectations, proving the DPP and game value existence in this complex setting.

Findings

DPP holds for the non-Markovian impulse control problem

The upper and lower value functions coincide, establishing a game value

Conditions for the existence of saddle points are provided

Abstract

We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle-point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).