The efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods. Among these methods, we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter θ. The optimal choice of θ is approximated using Robbins-Monro procedures, provided that a nonexplosion condition is satisfied. Otherwise, one can use either a constrained Robbins-Monro algorithm (see, e.g., Arouna (Monte Carlo Methods Appl. 10 (2004) 1-24) and Lelong (Statist. Probab. Lett. 78 (2008) 2632-2636)) or a more astute procedure based on an unconstrained approach recently introduced by Lemaire and Pagès in (Ann. Appl. Probab. 20 (2010) 1029-1067). In this article, we develop a new algorithm based on a combination of the statistical Romberg method and the importance sampling technique. The statistical Romberg method introduced by Kebaier in (Ann. Appl. Probab. 15 (2005) 2681-2705) is known for reducing efficiently the complexity compared to the classical Monte Carlo one. In the setting of discritized diffusions, we prove the almost sure convergence of the constrained and unconstrained versions of the Robbins-Monro routine, towards the optimal shift θ * that minimizes the variance associated to the statistical Romberg method. Then, we prove a central limit theorem for the new algorithm that we called adaptive statistical Romberg method. Finally, we illustrate by numerical simulation the efficiency of our method through applications in option pricing for the Heston model.Eψ(X T ) = Eg(θ, X T ).
An important family of stochastic processes arising in many areas of applied probability is the class of Lévy processes. Generally, such processes are not simulatable especially for those with infinite activity. In practice, it is common to approximate them by truncating the jumps at some cut-off size ε (ε ց 0). This procedure leads us to consider a simulatable compound Poisson process. This paper first introduces, for this setting, the statistical Romberg method to improve the complexity of the classical Monte Carlo one. Roughly speaking, we use many sample paths with a coarse cut-off ε β , β ∈ (0, 1), and few additional sample paths with a fine cut-off ε. Central limit theorems of LindebergFeller type for both Monte Carlo and statistical Romberg method for the inferred errors depending on the parameter ε are proved. This leads to an accurate description of the optimal choice of parameters with explicit limit variances. Afterwards, the authors propose a stochastic approximation method of finding the optimal measure change by Esscher transform for Lévy processes with Monte Carlo and statistical Romberg importance sampling variance reduction. Furthermore, we develop new adaptive Monte Carlo and statistical Romberg algorithms and prove the associated central limit theorems. Finally, numerical simulations are processed to illustrate the efficiency of the adaptive statistical Romberg method that reduces at the same time the variance and the computational effort associated to the effective computation of option prices when the underlying asset process follows an exponential pure jump CGMY model. MSC 2010: 60E07, 60G51, 60F05, 62L20, 65C05, 60H35.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.