A numerical method for SDEs with discontinuous drift

We consider an Euler-Maruyama type approximation method for a stochastic differential equation (SDE) with a non-regular drift and regular diffusion coefficient. The method regularizes the drift coefficient within a certain class of functions and then the Euler-Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors. The first one is the error of regularization of the drift coefficient within a given class of parametrized functions. The second one is the error of the regularized Euler-Maruyama scheme. After an optimization procedure with respect to the parameters we obtain various rates, which improve other known results.

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“…[34,38,39,41,42], ...). Among them, let us note [28] which uses a related approach to control the error on the densities.…”

Section: Numerical Resultsmentioning

confidence: 99%

Weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with non-regular drift

Kohatsu‐Higa

Lejay

Yasuda

2017

Journal of Computational and Applied Mathematics

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“…Recently several contributions for strong approximation have been given in a series of articles of Ngo and Taguchi [32,33,31] and Leobacher and Szölgyenyi [21,22,24]. For the multidimensional SDE (1) these works provide (i) the L 1 -convergence order 1/2 for the equidistant Euler-Maruyama scheme, if µ is one-sided Lipschitz and an appropriate limit of smooth functions, and σ is bounded and uniformly non-degenerate, see [31], (ii) the L 2 -convergence order 1/4 − for arbitrarily small > 0 for the equidistant Euler-Maruyama scheme under Assumption 2.1 and additionally the boundedness of µ and σ, see [24], (iii) the L 2 -convergence order 1/2 for a transformation based Euler-Maruyama method under Assumption 2.1, see [22].…”

Section: Numerical Methods For Sdes With Discontinuous Coefficientsmentioning

confidence: 99%

An Adaptive Euler--Maruyama Scheme for Stochastic Differential Equations with Discontinuous Drift and its Convergence Analysis

Neuenkirch¹,

Szölgyenyi²,

Szpruch³

2019

SIAM J. Numer. Anal.

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We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an adaptive step sizing strategy for the explicit Euler-Maruyama scheme. As a result, we obtain a numerical method which has -up to logarithmic terms -strong convergence order 1/2 with respect to the average computational cost. We support our theoretical findings with several numerical examples.

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“…, s n ∈ [0, T ]. Clearly, there exist 0 ≤ t 0 < t 1 ≤ T such that (24) [t 0 , t 1 ] ⊂ [0, τ 1 /2], (t 0 , t 1 ) ∩ {s 1 , . .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The first two statements in (24) imply that there exist measurable functions Φ 1 , ϕ : We may thus apply Lemma 3 with…”

Section: Define Processesmentioning

confidence: 99%

A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives

Müller-Gronbach

Yaroslavtseva

2018

Journal of Mathematical Analysis and Applications

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Recently, it has been shown in [Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system of autonomous stochastic differential equations (SDE) on the time interval [0, T ] with infinitely differentiable and bounded coefficients such that no strong approximation method based on evaluation of the driving Brownian motion at finitely many fixed times in [0, T ], e.g. on an equidistant grid, can converge in absolute mean to the solution at the final time with a polynomial rate in terms of the number of Brownian motion values that are used. In the literature on strong approximation of SDEs, polynomial error rate results are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition. This assumption is violated for the pathological SDEs from the above mentioned negative result. However, in the present article we construct an SDE with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can not be approximated with a polynomial rate, whatever method based on observations of the driving Brownian motion at finitely many fixed times is used. Most interestingly, it turns out that using a method that adjusts the number of evaluations of the driving Brownian motion to its actual path, the latter SDE can be approximated with rate 1 in terms of the average number of evaluations that are used. To the best of our knowledge, this is only the second example in the literature of an SDE for which there exist adaptive methods that perform superior to non-adaptive ones with respect to the convergence rate.

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A numerical method for SDEs with discontinuous drift

Cited by 60 publications

References 38 publications

Weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with non-regular drift

Weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with non-regular drift

An Adaptive Euler--Maruyama Scheme for Stochastic Differential Equations with Discontinuous Drift and its Convergence Analysis

A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives

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