The Separation Principle in Stochastic Control, Redux

TL;DR

This paper introduces a new framework for the separation principle in stochastic control, emphasizing sample path mappings and extending applicability to systems with jumps, challenging traditional probabilistic assumptions.

Contribution

It proposes a novel approach that views stochastic systems as sample path maps, extending the separation principle to jump-driven systems and aligning with engineering perspectives.

Findings

Framework extends separation principle to martingale-driven systems with jumps

Control laws satisfy feedback equations almost surely, not deterministically

Approach aligns with engineering views on signal flow in feedback loops

Abstract

Over the last 50 years a steady stream of accounts have been written on the separation principle of stochastic control. Even in the context of the linear-quadratic regulator in continuous time with Gaussian white noise, subtle difficulties arise, unexpected by many, that are often overlooked. In this paper we propose a new framework for establishing the separation principle. This approach takes the viewpoint that stochastic systems are well-defined maps between sample paths rather than stochastic processes per se and allows us to extend the separation principle to systems driven by martingales with possible jumps. While the approach is more in line with "real-life" engineering thinking where signals travel around the feedback loop, it is unconventional from a probabilistic point of view in that control laws for which the feedback equations are satisfied almost surely, and not deterministically for every sample path, are excluded.