Duality methods in stochastic optimal control

We prove two duality descriptions of the value function for a generic stochastic optimal problem. These descriptions also hold when the diffusion is controlled, a case left open by the literature so far.

💡 Research Summary

The paper addresses a fundamental gap in the theory of stochastic optimal control: the existence of dual representations for problems where the diffusion coefficient depends on the control. Classical duality results have been limited to cases with uncontrolled diffusion, and pathwise (or “scenario‑wise”) dual formulations often break down when the diffusion term is controlled. The authors consider a general continuous‑time control problem driven by a d‑dimensional state process X satisfying

 dXₛ = b(s, Xₛ, πₛ) ds + σ(s, Xₛ, πₛ) dWₛ, Xₜ = x,

with admissible controls π taking values in a measurable set U. The performance functional is

 J(t, x, π) = 𝔼