Revisiting the separation principle in stochastic control

The separation principle is the statement that under suitable conditions the design of stochastic control can be divided into two separate problems, one of optimal control with state information and one of filtering. The literature over the past 50 years contains several derivations where subtle difficulties are overlooked and inadmissible shortcuts taken. Other contributions that have established the separation principle under various hypotheses require considerable mathematical sophistication, which makes the ideas difficult to include in standard textbooks. The contribution of the present work is a new set of conditions that are in line with basic engineering thinking and ensure that the separation principle holds. The feedback system is required to be well-posed in the sense that it defines a map between sample paths, representing signals rather than stochastic processes per se. This approach allows certain generalizations of the separation theorem to a wide class of feedback laws, models and stochastic noise, including martingales with possible jumps.