Unified Optimality Conditions for Stochastic Optimal Control in the Rough Path and It\^o Frameworks

Abstract

Stochastic differential equations (SDEs) can be studied via It\^{o} calculus and rough path theory. For stochastic optimal control, these two frameworks give distinct Pontryagin Maximum Principle (PMP) optimality conditions with forward-backward SDEs (FBSDEs) or rough differential equations. We show that the adjoint equations of the It\^{o} and rough PMPs are connected via the conditional expectation $p_t^{\text{It\^{o}}}=\mathbb{E}[p_t^{\text{rough}} \mid \mathcal{F}_t]$, where $\mathcal{F}_t$ represents information available at time $t$. First, we derive a rough stochastic PMP for problems with adapted controls that does not use FBSDEs. Its proof extends the rough stochastic PMP over deterministic controls by considering stochastic needle variations. Second, we derive a unified PMP connecting the It\^{o} and rough PMPs, using It\^{o}-Stratonovich conversion formulas and duality identities between the forward tangent and backward adjoint SDEs. As a first application, we rederive the adjoint matching method for fine-tuning generative models. As a second application, we propose an indirect shooting method for a class of feedback problems. Overall, these results give a new conditional bridge connecting two popular frameworks for stochastic optimal control.