Adaptive Euler–Maruyama Method for SDEs with Non-globally Lipschitz Drift

Abstract

This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of the ergodic SDEs with a drift which is not globally Lipschitz over an infinite time interval. If the timestep is bounded appropriately, we show not only the stability of the numerical solution and the standard strong convergence order, but also that the bound for moments and strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo. Numerical experiments support our analysis.Several different methodologies have been developed to estimate it.First, we can compute the probability density function ρ(x) of π by solving the corresponding stationary Fokker-Planck equation, see [14]. However, the stationary Fokker-Planck equation is a partial differential equation (PDE) MSC 2010 subject classifications: 60H10, 60H35, 65C30

Multilevel Monte Carlo method for ergodic SDEs without contractivity

“…The L p norm of the difference between the fine and coarse paths, as we 160 expected, is uniformly bounded since we add enough spring term to recover the contractivity used in [15]. With this result, we can bound the pth-moment of the Radon-Nikodym derivatives and then the MLMC estimator (14).…”

“…One simple way is to use one of the existing numerical methods for finite time SDEs to simulate the SDE for a sufficiently long time T, see [10,11,12,13,14] and references therein. The exponential convergence to the invariant measure [1] is given by…”

“…which means the computational cost of each path using uniform time step 5 h = O(ε) becomes O(ε −1 | log ε|) for numerical schemes with first order weak convergence. Theorem 1 in [15] shows that under the dissipativity condition (2) the p-th moments of the numerical solution are bounded uniformly with respect to T, so the variance of the estimator is bounded by a constant V 0 which does not depend on T. Therefore, the computational cost to achieve ε 2 mean square 10 error (MSE) is O(ε −3 | log ε|).…”

“…for some λ > 0, Theorem 3 in [15] has proved first order strong convergence and that the strong error is uniformly bounded with respect to T. Hence, the variance of the multilevel correction V ℓ on each level ℓ is bounded by Ch 2 ℓ with C > 0 not depending on T. The MLMC computational cost to achieve ε 2 MSE Mathematically, instead of simulating the fine path and coarse path of the original SDEs, that is, in their separated path spaces with different measures:…”

Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients

“…By contrast, as already shown by Higham, Mao and Stuart [12], Mao and Szpruch [31], Andersson and Kruse [1], the backward (implicit) Euler method, computationally much more expensive than the explicit Euler method, can be strongly convergent under certain non-globally Lipschitz conditions. These observations suggest that special care must be taken to construct and analyze convergent numerical schemes in nonglobally Lipschitz setting, and this interesting subject has been investigated in a great portion of the literature [1,3,4,6,8,15,16,17,18,19,20,22,25,26,27,30,31,32,33,37,40,41,42,43,44,45,47,49,50,51]. In 2012, Hutzenthaler, Jentzen and Kloeden [18] introduced an explicit method, called the tamed Euler method, to numerically solve SDEs with super-linearly growing drift coefficients and globally Lipschitz diffusion coefficients.…”