Non asymptotic controls on a recursive superquantile approximation

Manon Costa,Sébastien Gadat

doi:10.1214/21-ejs1908

Abstract

In this work, we study a new recursive stochastic algorithm for the joint estimation of quantile and superquantile of an unknown distribution. The novelty of this algorithm is to use the Cesaro averaging of the quantile estimation inside the recursive approximation of the superquantile. We provide some sharp non-asymptotic bounds on the quadratic risk of the superquantile estimator for different step size sequences. We also prove new non-asymptotic Lp-controls on the Robbins Monro algorithm for quantile estimation and its averaged version. Finally, we derive a central limit theorem of our joint procedure using the diffusion approximation point of view hidden behind our stochastic algorithm.

Highlights

Let be given a real random variable X that mimics the outcome of a portfolio of some financial assets, [2]
Introduces a set of axioms that shall describe in actuarial science and financial economics a coherent risk measure. One of these key properties is that ρ must bring the diversification principle: the risk of two portfolios together is less than the two individual risks, which means from a mathematical point of view that for two random variables Z1 and Z2, ρ satisfies ρ(Z1 + Z2) ≤ ρ(Z1) + ρ(Z2)
We develop a new method for jointly estimating and obtain some non-asymptotic guarantees for the quadratic loss of estimation, which is a novel type of result for such a kind of estimation problem with stochastic algorithms

Summary

Introduction

Let be given a real random variable X that mimics the outcome of a portfolio of some financial assets, [2]. P. C introduces a set of axioms that shall describe in actuarial science and financial economics a coherent risk measure (denoted by ρ in this introduction). C introduces a set of axioms that shall describe in actuarial science and financial economics a coherent risk measure (denoted by ρ in this introduction) One of these key properties is that ρ must bring the diversification principle: the risk of two portfolios together is less than the two individual risks, which means from a mathematical point of view that for two random variables Z1 and Z2, ρ satisfies ρ(Z1 + Z2) ≤ ρ(Z1) + ρ(Z2). If X refers to the outcome of a portfolio, we define by F the cumulative distribution function:

Objectives

Discussion

Conclusion