IMPLEMENTATION AND ANALYSIS OF AN ADAPTIVE MULTILEVEL MONTE CARLO ALGORITHM

Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Ito ̂ stochastic differential equations (SDE). The work [11] proposed and analyzed a MLMC method based on a hier-archy of uniform time discretizations and control variates to reduce the compu-tational effort required by a standard, single level, forward Euler Monte Carlo method from O (TOL−3) to O ((TOL−1 log(TOL−1))2) for a mean square error of O (TOL2). Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26, 25, 8], and, as such, it may be considered a generalization of Giles ’ work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diffusion, where the complexity of a uniform single level method is O (TOL−4). For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy O (TOL) from O (TOL−3) for the adaptive sin-gle level algorithm to essentially O TOL−2 lo