A note on the strong formulation of stochastic control problems with model uncertainty

TL;DR

This paper analyzes a stochastic control problem with model uncertainty, demonstrating that under certain conditions, the value can be characterized as a symmetric game with feedback strategies, independent of the adverse player's information.

Contribution

It establishes the equivalence of a two-step optimization problem with a symmetric game formulation under model uncertainty, clarifying the value function's independence from the adverse player's filtration.

Findings

Value function matches the symmetric game formulation.

Independence of the value from the adverse player's filtration.

Provides a technical foundation for related modeling issues.

Abstract

We consider a Markovian stochastic control problem with model uncertainty. The controller (intelligent player) observes only the state, and, therefore, uses feed-back (closed-loop) strategies. The adverse player (nature) who does not have a direct interest in the pay-off, chooses open-loop controls that parametrize Knightian uncertainty. This creates a two-step optimization problem (like half of a game) over feed-back strategies and open-loop controls. The main result is to show that, under some assumptions, this provides the same value as the (half of) the zero-sum symmetric game where the adverse player also plays feed-back strategies and actively tries to minimize the pay-off. The value function is independent of the filtration accessible to the adverse player. Aside from the modeling issue, the present note is a technical companion to [S\^I3b].