A General Framework for Importance Sampling with Markov Random Walks

ArXiv ID: 2311.12330 “View on arXiv”

Authors: Unknown

Abstract

Although stochastic models driven by latent Markov processes are widely used, the classical importance sampling methods based on the exponential tilting for these models suffers from the difficulties in computing the eigenvalues and associated eigenfunctions and the plausibility of the indirect asymptotic large deviation regime for the variance of the estimator. We propose a general importance sampling framework that twists the observable and latent processes separately using a link function that directly minimizes the estimator’s variance. An optimal choice of the link function is chosen within the locally asymptotically normal family. We show the logarithmic efficiency of the proposed estimator. As applications, we estimate an overflow probability under a pandemic model and the CoVaR, a measurement of the co-dependent financial systemic risk. Both applications are beyond the scope of traditional importance sampling methods due to their nonlinear features.

Keywords: Importance sampling, Markov processes, CoVaR, Systemic risk, Variance estimation, Multi-asset (Systemic Risk)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced theoretical concepts such as locally asymptotically normal families and logarithmic efficiency, which require sophisticated mathematical tools; while it includes numerical examples and simulation applications, it lacks the backtest-ready, code-heavy implementation typical of high empirical rigor.

  flowchart TD
    A["Research Goal: Develop efficient IS<br/>for latent Markov models"] --> B{"Key Methodology"}
    B --> C["Separate twisting of observable<br/>and latent processes"]
    B --> D["Optimal link function via<br/>locally asymptotically normal family"]
    C & D --> E["Computational Process:<br/>Minimize estimator variance"]
    E --> F["Data/Inputs:<br/>Nonlinear Markov processes"]
    E --> G["Prove Logarithmic Efficiency"]
    F --> H["Applications & Outcomes"]
    G --> H
    subgraph H ["Applications & Outcomes"]
        H1["Pandemic Model:<br/>Overflow probability estimation"]
        H2["Finance (CoVaR):<br/>Systemic risk measurement"]
    end
    H --> I["Key Outcome:<br/>General framework beyond<br/>traditional exponential tilting"]