A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations

This article introduces and analyzes multilevel Monte Carlo schemes for the evaluation of the expectation E [ f ( Y ) ] , where Y = ( Y t ) t ∈ [ 0 , 1 ] is a solution of a stochastic differential equation driven by a Lévy process. Upper bounds are provided for the worst case error over the class of all path dependent measurable functions f , which are Lipschitz continuous with respect to the supremum norm. In the case where the Blumenthal–Getoor index of the driving process is smaller than one, one obtains convergence rates of order 1 / n , when the computational cost n tends to infinity. This rate is optimal up to logarithms in the case where Y is itself a Lévy process. Furthermore, an error estimate for Blumenthal–Getoor indices larger than one is included together with results of numerical experiments.