Continuity of Cost in Borkar Control Topology and Implications on Discrete Space and Time Approximations for Controlled Diffusions under Several Criteria

TL;DR

This paper proves the continuity of various cost functionals in controlled diffusions under Borkar's topology, and shows that finite action policies can approximate optimal policies, enabling numerical methods for continuous control problems.

Contribution

It establishes the continuity and denseness of finite action policies in the space of stationary controls, facilitating approximation and numerical solution of control problems.

Findings

Continuity of discounted, exit time, and ergodic costs under Borkar topology.

Density of finite action policies in stationary control space.

Finite policies approximate optimal controls with arbitrary precision.

Abstract

We first show that the discounted cost, cost up to an exit time, and ergodic cost involving controlled non-degenerate diffusions are continuous on the space of stationary control policies when the policies are given a topology introduced by Borkar [V. S. Borkar, A topology for Markov controls, Applied Mathematics and Optimization 20 (1989), 55-62]. The same applies for finite horizon problems when the control policies are markov and the topology is revised to include time also as a parameter. We then establish that finite action/piecewise constant stationary policies are dense in the space of stationary Markov policies under this topology. Using the above mentioned continuity and denseness results we establish that finite action/piecewise constant policies approximate optimal stationary policies with arbitrary precision. This gives rise to the applicability of many numerical methods such as policy iteration and stochastic learning methods for discounted cost, cost up to an exit time, and ergodic cost optimal control problems in continuous-time. For the finite-horizon setup, we establish additionally near optimality of time-discretized policies by an analogous argument. We thus present a unified and concise approach for approximations directly applicable under several commonly adopted cost criteria.