A framework for adaptive Monte-Carlo procedures

Bernard Lapeyre (1) , Jérôme Lelong (2, 3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHFI - Financial mathematics
3 MATHFI - Mathématiques financières

Adaptive Monte Carlo methods are recent 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 Vazquez-Abad and Dufresne, Fu and Su, and Arouna. 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 some examples of valuation of financial derivatives.

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. Monte Carlo Methods and Applications, 2011, 17 (1), pp.77-98. ⟨10.1515/MCMA.2011.002⟩. ⟨hal-00448864v2⟩