Stochastic Approximation with Averaging Innovation Applied to Finance

Sophie Laruelle (1) , Gilles Pagès (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: Stochastic Approximation; sequence with low discrepancy; quasi-Monte Carlo; $\alpha$-mixing process; Gál-Koksma theorem; stationary process; ergodic control; two-armed bandit algorithm; calibration; optimal asset allocation; Value-at-Risk; Conditional Value-at-Risk

Domaines: Probabilités [math.PR]

Sophie Laruelle, Gilles Pagès. Stochastic Approximation with Averaging Innovation Applied to Finance. Monte Carlo Methods and Applications, 2012, 18 (1), pp.1-52. ⟨hal-00504644v4⟩