A new class of exponential integrators for SDEs with multiplicative noise

Erdoğan, Utku; Lord, Gabriel J.

doi:10.1093/imanum/dry008

Cited by 18 publications

(20 citation statements)

References 20 publications

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“…Using the local linearization method, [14] introduced an exponential scheme for (1.1) with d = 1. The article [50] develops an integrator of Euler-exponential type for multidimensional SDEs with multiplicative noise (see also, e.g., [22,35,37,59,60]), and [32] provides a numerical method based on the computation of the conditional mean and the square root of the conditional covariance matrix of a local linearization approximation to (1.1). Schemes adapted to specific SDEs are given, for instance, in [10,13,15,50].…”

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confidence: 99%

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

Mora¹,

Mardones²,

Jiménez³

et al. 2017

SIAM J. Numer. Anal.

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We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution Xt by integrating the system of coupled SDEs that describes the evolution of the norm of Xt and its projection on the unit sphere. This allows us to develop an explicit scheme for stiff SDEs with multiplicative noise that shows a solid performance in various numerical experiments. Under general conditions, the new integrator preserves the almost sure stability of the solutions for any step-size, as well as the property of being distant from 0. The scheme also has linear rate of weak convergence for a general class of SDEs with locally Lipschitz coefficients, and one-half strong order of convergence.

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confidence: 99%

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

Mora¹,

Mardones²,

Jiménez³

et al. 2017

SIAM J. Numer. Anal.

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“…In the interesting paper [12] a non-linear extension in the case of commuting A and B can be found and applications to SPDEs via space discretizations are discussed, which is the same approach we take in the next subsection with the ME.…”

Section: Examplementioning

confidence: 99%

“…In Fig. 4 we plot one realization of the trajectories of the top-right component (X t ) 12 , computed with all the methods above, up to time t = 10. In this case we did not plot a diagonal component of the solution because the latter are exact for m2 and m3, up to discretization errors of Lebesgue integrals.…”

Section: Stochastic Cauchy Problem and Fundamental Solutionmentioning

confidence: 99%

On the Stochastic Magnus Expansion and Its Application to SPDEs

2021

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We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time $$\tau $$ τ and provide a novel asymptotic estimate of the cumulative distribution function of $$\tau $$ τ . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

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“…Applications satisfying Assumption 1 are, e. g., [9] the FitzHugh-Nagumo equation with multiplicative noise, the Lotka-Volterra system and SDEs resulting from spectral spatial discretization of stochastic partial differential equations (SPDEs) with diagonal noise.…”

Section: Introductionmentioning

confidence: 99%

Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

2022

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In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods.

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A new class of exponential integrators for SDEs with multiplicative noise

Cited by 18 publications

References 20 publications

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

On the Stochastic Magnus Expansion and Its Application to SPDEs

Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

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