Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method

Stochastic optimal control problems typically involve a dynamical system described by a stochastic differential equation (SDE) dx t = b (t, x t, u t)dt + σ (t, x t) dB t, t [0, T], (1.1) in Stratonovich or Itˆ o form, where x t is the state of the system at time t, u t is the control input, b is the drift, σ is the diffusion, B is a Brownian motion, T is the final time, and consist of optimizing an objective E[null T 0 f ( t, x t, u t)dt + g (x T)] over a set of control input trajectories subject to state and control constraints. By now, a rich literature on stochastic optimal control is available, with optimality conditions characterized by the dynamic programming principle as Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) [6-8], and by the Pontryagin Maximum Principle (PMP) as forward-backward stochastic differential equations (FBSDEs) [8-11]. For problems with linear dynamics and linear-quadratic costs, both approaches lead to tractable solutions characterized by stochastic Riccati equations [7,12,13]. However, for general nonlinear problems, solving HJB-PDEs or FBSDEs remains computationally challenging for high-dimensional state spaces, despite recent progress [14-17]. In practice, an effective approach consists of optimizing over a class of solutions u θ t parameterized by finitely-many parameters θ R k [18,19] (see [20,21] for machine learning applications). However, restricting solutions to a finite-dimensional space may obscure the structure of solutions and lead to suboptimality.