Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift

Fang, Wei; Giles, Michael B.

doi:10.1214/19-aap1507

Cited by 45 publications

(30 citation statements)

References 41 publications

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“…Rather than using the classical Lyapunov function criterion, it is more convenient to use a modified version of the criterion in [19] to establish this ergodicity. Note that even if an SDE is ergodic, its EM scheme may blow up, see [22,39].…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

An approximation to steady-state of M/Ph/n+M queue

Jin¹,

Pang²,

Xu³

et al. 2021

Preprint

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In this paper, we develop a stochastic algorithm based on Euler-Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of the M/P h/n + M queue. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion of the queueing model. Our result also provides an approximation to the steady-state of the prelimit diffusion-scaled queueing processes in the Halfin-Whitt regime given the well established interchange of limits property.To establish the error bound, we employ the recently developed Stein's equation and Malliavian calculus for multi-dimensional diffusions. The main difficulty lies in the nondifferentiability of the drift in the limiting diffusion, so that the standard approaches in Euler type of schemes for diffusions and Stein's method do not work. We first propose a mollified diffusion which has a sufficiently smooth drift to circumvent the nondifferentiability difficulty. We then provide some insights on the limiting diffusion and approximate diffusion from the algorithm by investigating some useful occupation times, as well as the associated Harnack inequalities, the support of the invariant measures and ergodicity properties. These results are used in analyzing the Stein's equation, which provide useful estimates to bound the differences between the corresponding invariant measures. CONTENTS1. Introduction 1.1. Summary of results and contributions 1.2. Related works 1.3. Organization of the paper 1.4. Notations 2. Proof of the main result 3. Malliavin calculus for SDE (1.6) and Stein's equation 3.1. Malliavin calculus associated with SDE (1.6) 3.2. Stein's equation of SDE (1.6) 4. Estimates for weighted occupation times: important ingredients 5. Proofs of Propositions 2.3 and 2.4 Appendix A. Harnack inequality Appendix B. Proof of Proposition 2.1 Appendix C. Proof of Proposition 2.2 Appendix D. Estimates for Jacobi flow and Malliavin derivative

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

An approximation to steady-state of M/Ph/n+M queue

Jin¹,

Pang²,

Xu³

et al. 2021

Preprint

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show abstract

“…Hence, we deduce from Young's inequality and BDG's inequality that 18) for some constant C(ε) > 0, where in the last display we used (3.11), (3.14) and 19) owing to (2) of (A 1 b ). Via Hölder's inequality, we find from (3.11) and (3.12), together with (3.19), that…”

Section: Lemma 32 Assume (Amentioning

confidence: 97%

“…[16]) is not appropriate in the presence of drift terms with super-linear growth. To overcome this problem, several stable time-discretization methods, including a tamed explicit Euler and Milstein scheme [14,17,18], an explicit adaptive Euler-Maruyama method [19], a truncated Euler method [20] and an implicit Euler scheme [21], have been introduced. In [17], a tamed Milstein scheme is introduced and convergence is proven under a commutativity assumption for the diffusion matrix, such that the Lévy area vanishes.…”

Section: Introductionmentioning

confidence: 99%

First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems

et al. 2021

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In this paper, we derive fully implementable first-order time-stepping schemes for McKean–Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean–Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

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“…The adaptive time-stepping scheme, which adapts the timestep size at each iteration to control the numerical solution from instability, has been deeply studied for stochastic ordinary differential equations with non-globally Lipschitz drift; see e.g., [14], [20], [22], [23] and references therein. Numerically this adaptive scheme is simple to implement and the complexity is similar as that of an Euler scheme, which is a big advantage for high dimensional problems (see [22]).…”

Section: Introductionmentioning

confidence: 99%

An adaptive time-stepping full discretization for stochastic Allen--Cahn equation

Chen,

Dang,

Hong

2021

Preprint

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It is known in [1] that a regular explicit Euler-type scheme with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen-Cahn equation. To overcome the divergence, this paper proposes an adaptive time-stepping full discretization, whose spatial discretization is based on the spectral Galerkin method, and temporal direction is the adaptive exponential integrator scheme. It is proved that the expected number of timesteps is finite if the adaptive timestep function is bounded suitably. Based on the stability analysis in C(O, R)-norm of the numerical solution, it is shown that the strong convergence order of this adaptive time-stepping full discretization is the same as usual, i.e., order β in space and β 2 in time under the correlation assumptionon the noise. Numerical experiments are presented to support the theoretical results.

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Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift

Cited by 45 publications

References 41 publications

An approximation to steady-state of M/Ph/n+M queue

An approximation to steady-state of M/Ph/n+M queue

First-order convergence of Milstein schemes for McKean–Vlasov equations and interacting particle systems

An adaptive time-stepping full discretization for stochastic Allen--Cahn equation

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