An Asymptotic CVaR Measure of Risk for Markov Chains

ArXiv ID: 2405.13513 “View on arXiv”

Authors: Unknown

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.

Keywords: Asymptotic CVaR, Large deviations theory, Stochastic approximation, Risk-sensitive decision making, Markovian rewards, General/Quantitative Strategy

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics including large deviations theory, multiplicative Poisson equations, and two-time scale stochastic approximation, placing it firmly in high math complexity. It also presents a concrete algorithm with simulation-based steps and empirical evaluation, indicating moderate to high empirical rigor.

  flowchart TD
    A["Research Goal: Define & Compute<br>Asymptotic CVaR for Markov Chains"] --> B["Methodology: Theoretical Foundation<br>Large Deviations Theory"]
    B --> C["Data Input: Markovian Reward System"]
    C --> D{"Computational Process:<br>Simulation-based Algorithm"}
    D --> E["Tilted Probability Kernel<br>via Density Estimation"]
    E --> F["Sequential Computation<br>Two-Time Scale Stochastic Approximation"]
    F --> G["Key Finding: Validated ACVaR Measure<br>Numerical Evaluation on Test Cases"]