An Asymptotic CVaR Measure of Risk for Markov Chains

TL;DR

This paper introduces Asymptotic CVaR (ACVaR), a novel risk measure for Markov chains that captures long-term risk behavior, with a simulation-based computation method supported by theoretical guarantees.

Contribution

It proposes a new asymptotic risk measure, ACVaR, and develops a simulation algorithm using large deviations and stochastic approximation with proven theoretical guarantees.

Findings

ACVaR effectively captures long-term risk in Markov chains.

The simulation algorithm converges with theoretical guarantees.

Numerical experiments demonstrate the method's performance.

Abstract

Risk sensitive decision making finds important applications in current day use cases. Existing risk measures consider a single or finite collection of random variables, which do not account for the asymptotic behaviour of underlying systems. Conditional Value at Risk (CVaR) is the most commonly used risk measure, and has been extensively utilized for modelling rare events in finite horizon scenarios. Naive extension of existing risk criteria to asymptotic regimes faces fundamental challenges, where basic assumptions of existing risk measures fail. We present a complete simulation based approach for sequentially computing Asymptotic CVaR (ACVaR), a risk measure we define on limiting empirical averages of markovian rewards. Large deviations theory, density estimation, and two-time scale stochastic approximation are utilized to define a 'tilted' probability kernel on the underlying state space to facilitate ACVaR simulation. Our algorithm enjoys theoretical guarantees, and we numerically evaluate its performance over a variety of test cases.