Adaptive Multilevel Stochastic Approximation of the Value-at-Risk

TL;DR

This paper introduces an adaptive multilevel stochastic approximation method to efficiently compute the value-at-risk in finance, improving complexity over previous approaches by adaptively selecting inner sample sizes.

Contribution

It proposes a novel adaptive multilevel stochastic approximation algorithm that reduces complexity from suboptimal to near-optimal for value-at-risk estimation.

Findings

The new algorithm achieves a complexity of O(ε^{-2} |ln ε|^{5/2})

Numerical experiments confirm theoretical complexity improvements

Addresses discontinuity issues in stochastic gradient evaluation

Abstract

Cr\'epey, Frikha, and Louzi (2025) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The best complexity of the scheme is in O($\varepsilon^{-\frac52}$), $\varepsilon>0$ being a prescribed accuracy, which is suboptimal compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity ofthe Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its best complexity is in O($\varepsilon^{-2}|\ln{\varepsilon}|^\frac52$). Our theoretical analysis is exemplified through numerical experiments.