# On the adjoint Markov policies in stochastic differential games

## TL;DR

This paper introduces a method for constructing near-optimal strategies in stochastic differential games using adjoint Markov strategies, which are based on a coupled system of the original and adjoint stochastic equations.

## Contribution

It proposes a novel approach to find $ ext{epsilon}$-optimal strategies via adjoint Markov policies linked to a modified Isaacs equation, expanding the toolkit for stochastic differential games.

## Key findings

- Constructed $ ext{epsilon}$-optimal strategies using adjoint Markov policies.
- Showed solvability of a modified Isaacs equation in Sobolev spaces.
- Provided an example where assumptions fail and $ ext{epsilon}$-optimal strategies may not exist.

## Abstract

We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $\varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $\varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev spaces of not the original Isaacs equation but of its appropriate modification. We also give an example of a uniformly nondegenerate game where our assumptions are not satisfied and where we conjecture that there are no not only optimal Markov but even $\varepsilon$-optimal adjoint (time-homogeneous) Markov strategies for one of the players.