A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

Agarwal, Ankush; Pagliarani, Stefano

doi:10.1080/17442508.2020.1771337

Cited by 8 publications

(4 citation statements)

References 19 publications

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“…Variance reduction technique have been analysed for the class of MV-SDE, namely, importance sampling [20], antithetic multilevel Monte Carlo sampling [4] and antithetic sampling [8]. There also recent progress in the jump-diffusion setting [2,10].…”

Section: Introductionmentioning

confidence: 99%

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

Chen,

Reis

2021

Preprint

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We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation.The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented including a comparative analysis to other known algorithms for this class (taming and adaptive time-stepping) across parallel and non-parallel implementations.

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Section: Introductionmentioning

confidence: 99%

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

Chen,

Reis

2021

Preprint

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“…Theorem 1.1 establishes under suitable conditions that for every δ ∈ (0, ∞) there exists c ∈ R such that the solution X ∈ C([0, T ], R d ) of the differential equation X(t) = ξ + t 0 E[F (X(r), Z 0 )] dr, t ∈ [0, T ], (cf. Lemma 3.3), can be approximated by the recursive MLP approximation schemes in (1) with a root mean square error of size ε ∈ (0, 1] and a computational effort that is bounded by c ε −(2+δ) . The computational effort is quantified by the numbers RV n,m , n, m ∈ N. The function F : R d × S → R d is required to be (B(R d ) ⊗ S)/B(R d ) -measurable and Lipschitz continuous in the first variable, uniformly in the second variable, and we assume that E[ F (ξ, Z 0 ) 2 ] < ∞.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, alternative approximation methods for McKean-Vlasov SDEs relying on cubature formulas [14,15], analytical expansions [22], or tamed Milstein schemes [3,30] have been developed. For further numerical approximation methods for McKean-Vlasov SDEs we also refer, e.g., to [1,13,31]. The problems which are treated in these references are of course far more general and involved than the expectation ODEs which we consider in this article.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

Becker¹,

Braunwarth²,

Hutzenthaler³

et al. 2020

CICP

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We consider ordinary differential equations (ODEs) which involve expectations of a random variable. These ODEs are special cases of McKean-Vlasov stochastic differential equations (SDEs). A plain vanilla Monte Carlo approximation method for such ODEs requires a computational cost of order ε −3 to achieve a root-mean-square error of size ε. In this work we adapt recently introduced full history recursive multilevel Picard (MLP) algorithms to reduce this computational complexity. Our main result shows for every δ > 0 that the proposed MLP approximation algorithm requires only a computational effort of order ε −(2+δ) to achieve a root-mean-square error of size ε.

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“…The details of the computation will be part of the forthcoming paper[AGP17], where an alternative approach is proposed for the re-solution of McKean-Vlasov SDEs with affine coefficients w.r.t. the state variable.…”

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confidence: 99%

Analytical approximations of non-linear SDEs of McKean–Vlasov type

Gobet

Pagliarani²

2018

Journal of Mathematical Analysis and Applications

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We provide analytical approximations for the law of the solutions to a certain class of scalar McKean-Vlasov stochastic differential equations (MKV-SDEs) with random initial datum. "Propagation of chaos" results ([Szn91]) connect this class of SDEs with the macroscopic limiting behavior of a particle, evolving within a mean-field interaction particle system, as the total number of particles tends to infinity. Here we assume the mean-field interaction only acting on the drift of each particle, this giving rise to a MKV-SDE where the drift coefficient depends on the law of the unknown solution. By perturbing the non-linear forward Kolmogorov equation associated to the MKV-SDE, we perform a two-steps approximating procedure that decouples the McKean-Vlasov interaction from the standard dependence on the state-variables. The first step yields an expansion for the marginal distribution at a given time, whereas the second yields an expansion for the transition density. Both the approximating series turn out to be asymptotically convergent in the limit of short times and small noise, the convergence order for the latter expansion being higher than for the former. Concise numerical tests are presented to illustrate the accuracy of the resulting approximation formulas. The latter are expressed in semi-closed form and can be then regarded as a viable alternative to the numerical simulation of the large-particle system, which can be computationally very expensive. Moreover, these results pave the way for further extensions of this approach to more general dynamics and to high-dimensional settings.

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A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

Cited by 8 publications

References 19 publications

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

Analytical approximations of non-linear SDEs of McKean–Vlasov type

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