Stochastic target games with controlled loss

TL;DR

This paper investigates a stochastic game where one player aims to reach a target with controlled loss, deriving a dynamic programming principle and equation for controlled SDEs, with applications to partial hedging under uncertainty.

Contribution

It introduces a relaxed geometric dynamic programming principle and viscosity solutions for stochastic target games with controlled loss, applicable to controlled SDEs.

Findings

Established a general dynamic programming framework for stochastic target games.

Derived viscosity solutions for the dynamic programming equation in controlled SDEs.

Applied the theory to partial hedging problems under Knightian uncertainty.

Abstract

We study a stochastic game where one player tries to find a strategy such that the state process reaches a target of controlled-loss-type, no matter which action is chosen by the other player. We provide, in a general setup, a relaxed geometric dynamic programming principle for this problem and derive, for the case of a controlled SDE, the corresponding dynamic programming equation in the sense of viscosity solutions. As an example, we consider a problem of partial hedging under Knightian uncertainty.