Stochastic Approximation with Averaging Innovation Applied to Finance

TL;DR

This paper develops a convergence theorem for multi-dimensional stochastic approximation with innovations that have light averaging properties, unifying various frameworks including simulated and market data, with applications in finance.

Contribution

It introduces a new convergence theorem for stochastic approximation under light averaging assumptions, applicable to diverse innovation types in finance.

Findings

Unified framework for innovations with averaging properties

Applicability to simulated and market data

Validated on five finance-related examples

Abstract

The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the "innovations" satisfy some "light" averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in Quasi-Monte Carlo simulation) or exogenous (like market data) with ergodic properties. We propose several fields of applications and illustrate our results on five examples mainly motivated by Finance.