Convergence Rates for Adaptive Weak Approximation of Stochastic Differential Equations

Abstract

to a problem independent factor defined in the algorithm. Numerical examples illustrate the behavior of the adaptive algorithms, motivating when stochastic and deterministic adaptive time steps are more efficient than constant time steps and when adaptive stochastic steps are more efficient than adaptive deterministic steps.

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

“…Theorem 2 and 3, respectively. Analogous theoretical results also hold for the adaptive algorithm with deterministic stepping, but, for the sake of brevity, they are not included here, see [25] for more information on this setting.…”

“…According to Corollary 4.3 and Theorem 4.5 in [25], the error density converges almost surely to a limit density we denoteρ. i.e., ρ →ρ as TOL T ↓ 0.…”

“…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.…”

“…[26]. The idea of extending the MLMC method [11] to hierarchies of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8] was first introduced and tested computationally by the authors in [17].…”

“…In addition, the work [10] shows how to apply the Milstein method for several scalar SDE cases where the Lipschitz condition is not fulfilled and still obtain the cost O TOL −2 . Building on the work on adaptive methods for weak approximation of SDE presented in [25,29] and Giles' work on uniform time stepping MLMC methods [11], the contribution of the present paper is the development and analysis of two novel MLMC algorithms with adaptive, non uniform time stepping: one algorithm that uses adaptive mesh refinements to construct a path dependent mesh for each realization and another algorithm that constructs the meshes adaptively based on sample averaged error densities and then uses the same mesh for all realizations on a given mesh level in the hierarchy. The former algorithm is referred to as the stochastic time stepping algorithm and the latter as the deterministic time stepping algorithm.…”