Non-asymptotic estimation of risk measures using stochastic gradient Langevin dynamics

TL;DR

This paper develops a non-asymptotic method for estimating law invariant risk measures using stochastic gradient Langevin dynamics, providing theoretical convergence rates and numerical validation.

Contribution

It introduces a novel non-asymptotic approximation approach for general law invariant risk measures based on stochastic gradient Langevin dynamics.

Findings

The method achieves explicit convergence rates.

Numerical simulations validate theoretical results.

Extension from AVaR to general risk measures using Kusuoka's representation.

Abstract

In this paper we will study the approximation of arbitrary law invariant risk measures. As a starting point, we approximate the average value at risk using stochastic gradient Langevin dynamics, which can be seen as a variant of the stochastic gradient descent algorithm. Further, the Kusuoka's spectral representation allows us to bootstrap the estimation of the average value at risk to extend the algorithm to general law invariant risk measures. We will present both theoretical, non-asymptotic convergence rates of the approximation algorithm and numerical simulations.