Zero-sum risk-sensitive continuous-time stochastic games with unbounded payoff and transition rates and Borel spaces

TL;DR

This paper investigates a complex continuous-time zero-sum stochastic game with unbounded rates and rewards, establishing existence of solutions, Nash equilibria, and developing a convergent value iteration algorithm.

Contribution

It introduces a novel approach to handle unbounded rates and rewards in risk-sensitive stochastic games, proving existence and uniqueness of solutions and equilibria.

Findings

Existence of a solution to the Shapley equation under unbounded conditions

Proof of Nash equilibrium existence and value characterization

Development and convergence proof of a value iteration algorithm

Abstract

We study a finite-horizon two-person zero-sum risk-sensitive stochastic game for continuous-time Markov chains and Borel state and action spaces, in which payoff rates, transition rates and terminal reward functions are allowed to be unbounded from below and from above and the policies can be history-dependent. Under suitable conditions, we establish the existence of a solution to the corresponding Shapley equation (SE) by an approximation technique. Then, by the SE and the extension of the Dynkin's formula, we prove the existence of a Nash equilibrium and verify that the value of the stochastic game is the unique solution to the SE. Moreover, we develop a value iteration-type algorithm for approaching to the value of the stochastic game. The convergence of the algorithm is proved by a special contraction operator in our risk-sensitive stochastic game. Finally, we demonstrate our main results by two examples.