A framework for adaptive Monte-Carlo procedures

Bernard Lapeyre (1) , Jérôme Lelong (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 LJK - Laboratoire Jean Kuntzmann

Adaptive Monte Carlo methods are powerful variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Arouna in 2003. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to the valuation of financial derivatives and our numerical experiments show that the computational time needed to achieve a given accuracy is divided by a factor up to $5$.

Mots clés: adaptive Monte Carlo methods; importance sampling; stochastic approximation; variance reduction

Domaines: Probabilités [math.PR]

Bernard Lapeyre, Jérôme Lelong. A framework for adaptive Monte-Carlo procedures. 2010. ⟨hal-00448864v1⟩