The minimal hitting probability of continuous-time controlled Markov systems with countable states

TL;DR

This paper investigates the minimal hitting probability in continuous-time controlled Markov systems with countable states, proving existence of optimal policies, characterizing solutions for controlled branching processes, and proposing a new iterative algorithm.

Contribution

It establishes the existence of optimal policies, characterizes the minimal hitting probability for controlled branching processes, and introduces a novel policy iteration algorithm.

Findings

Existence of optimal policies for CTCMSs proven.

Minimal hitting probability characterized as a unique solution for CBPs.

A new policy iteration algorithm for computing minimal hitting probability introduced.

Abstract

This paper concentrates on the minimal hitting probability of continuous-time controlled Markov systems (CTCMSs) with countable state and finite admissible action spaces. The existence of an optimal policy is first proved. In particular, for a special and important case of controlled branching processes (CBPs), it is proved that the minimal hitting probability is the unique solution to an improved optimal system of equations. Furthermore, a novel and precise improved-policy iteration algorithm of an optimal policy and the minimal hitting probability (minimal extinction probability) is presented for CBPs.