Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Ahmed Kebaier (1) , Jérôme Lelong (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 DAO - Données, Apprentissage et Optimisation

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

Mots clés: Uniform strong large law of numbers; Central limit theorem; variance reduction; Importance Sampling; Multilevel Monte Carlo

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]; Pricing [q-fin.PR]; Analyse numérique [math.NA]

Ahmed Kebaier, Jérôme Lelong. Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation. Methodology and Computing in Applied Probability, 2018, 20 (2), pp.611-641. ⟨10.1007/s11009-017-9579-y⟩. ⟨hal-01214840v4⟩