Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Abstract

Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [11] proposed and analyzed a MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational 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 TOL 2 . 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 single level algorithm to essentially O TOL −2 log TOL −1 2 for the adaptive MLMC.

Multilevel ensemble Kalman filtering

“…If the dynamics (1) is given by the SDE in Remark 1 and Ψ N n denotes the corresponding Euler-Maruyama numerical solution, then Assumption 2 holds with β = 1, cf. [41,21,22,19].…”

Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations

“…The most significant prior research on adaptive timestepping in MLMC has been by Hoel, von Schwerin, Szepessy and Tempone [9] and [10]. In their research, they construct a multilevel adaptive timestepping discretisation in which the timesteps used on level are a subdivision of those used on level −1, which in turn are a subdivision of those on level −2, and so on.…”

Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

“…Their development of weak error adaptivity took inspiration from Talay and Tubaro's seminal work [33], where an error expansion for the weak error was derived for the Euler-Maruyama algorithm when uniform time steps were used. In [16], Szepessy et al's weak error adaptive algorithm was used in the construction of a weak error adaptive MLCM algorithm. To the best of our knowledge, the present work is the first on MSE a posteriori adaptive algorithms for SDE both in the MC-and MLMC setting.…”

“…The first component of the vector coincides with equation (12), whereas the second one is the first variation of the path from equation (16). The last three components can be understood as the second, third and fourth variations of the path, respectively.…”