A Mixed Value and Policy Iteration Method for Stochastic Control with Universally Measurable Policies

TL;DR

This paper introduces a mixed value and policy iteration method for stochastic control models with complex measurability issues, ensuring convergence in various cost scenarios and addressing limitations of standard policy iteration.

Contribution

The paper proposes a novel mixed iteration method that overcomes measurability challenges in stochastic control, with proven convergence for infinite horizon problems and new insights into value iteration.

Findings

Convergence of the proposed method in discounted and undiscounted cases.

A new convergence theorem for value iteration with bounded initial functions.

The method effectively handles models with Borel spaces and measurable policies.

Abstract

We consider stochastic control models with Borel spaces and universally measurable policies. For such models the standard policy iteration is known to have difficult measurability issues and cannot be carried out in general. We present a mixed value and policy iteration method that circumvents this difficulty. The method allows the use of stationary policies in computing the optimal cost function, in a manner that resembles policy iteration. It can also be used to address similar difficulties of policy iteration in the context of upper and lower semicontinuous models. We analyze the convergence of the method in infinite horizon total cost problems, for the discounted case where the one-stage costs are bounded, and for the undiscounted case where the one-stage costs are nonpositive or nonnegative. For undiscounted total cost problems with nonnegative one-stage costs, we also give a new convergence theorem for value iteration, which shows that value iteration converges whenever it is initialized with a function that is above the optimal cost function and yet bounded by a multiple of the optimal cost function. This condition resembles Whittle's bridging condition and is partly motivated by it. The theorem is also partly motivated by a result of Maitra and Sudderth, which showed that value iteration, when initialized with the constant function zero, could require a transfinite number of iterations to converge. We use the new convergence theorem for value iteration to establish the convergence of our mixed value and policy iteration method for the nonnegative cost case.