Improved Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Differential Equations

Abstract

Abstract. An improved stabilized multilevel Monte Carlo (MLMC) method is introduced for stiff stochastic differential equations in the mean square sense. Using S-ROCK2 with weak order 2 on the finest time grid and S-ROCK1 (weak order 1) on the other levels reduces the bias while preserving all the stability features of the stabilized MLMC approach. Numerical experiments illustrate the theoretical findings.

Long Time Accuracy of Lie--Trotter Splitting Methods for Langevin Dynamics

“…one can show that the dynamics generated by (1) are ergodic, i.e. for arbitrary initial conditions q(0), p(0) (assumed deterministic for simplicity), and for all smooth test function φ ∈ C ∞ P (R 2d , R) with derivatives of all orders with polynomial growth, the time average of the trajectories of (1) satisfy with probability 1,…”

“…Indeed, it applies not only to first weak order integrators for SDEs but also to higher orders weak schemes, as shown in [11], where the antithetic MLMC has been introduced. Also in the context of stiff SDEs, in was shown in [1] that applying a weak second order method at the finer level of the multilevel MonteCarlo method permits to significantly improve the error constant. Finally we mention the schemes based on Markov Chain Monte-Carlo methods, see e.g.…”