A framework for adaptive Monte Carlo procedures

Abstract

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 in [Arouna, Monte Carlo Methods Appl. 10: 1–24, 2004, Arouna, The Journal of Computational Finance 7: Winter 2003/2004, Su and Fu, Journal of Computational Finance 5: 27–50, 2002, Vázquez-Abad and Dufresne, Accelerated simulation for pricing asian options: 1493–1500, IEEE Computer Society Press, 1998]. 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.

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