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

arXiv: https://arxiv.org/abs/2209.14982 · HTML: https://mlsanity.com/paper/2209.14982/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Average-reward reinforcement learning in semi-Markov decision processes via relative value iteration](https://mlsanity.com/paper/2512.06218/) (2025)
- [Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning](https://mlsanity.com/paper/2502.13486/) (2025)
- [Reinforcement Learning for Discounted and Ergodic Control of Diffusion Processes](https://mlsanity.com/paper/2603.13155/) (2026)

## Abstract (arXiv)

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.