A higher order weak approximation of McKean–Vlasov type SDEs

Naito, Riu; Yamada, Toshihiro

doi:10.1007/s10543-021-00880-1

Cited by 9 publications

(8 citation statements)

References 19 publications

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“…As in the proof of Theorem 1, Malliavin integration by parts formula plays an important role to prove Corollary 1. The detail proof of the small noise expansion error is shown in [17,18] and the global error analysis of weak approximation error is provided in [19][20][21][22][23]. These results are essential to show the precise error estimate (34) depending on the regularity of the test function, which is an extension of the error estimate of [24].…”

Section: Remarkmentioning

confidence: 97%

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Asymptotic Expansion and Weak Approximation for a Stochastic Control Problem on Path Space

Kannari,

Naito,

Yamada

2024

Entropy

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The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence.

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Section: Remarkmentioning

confidence: 97%

“…There have been extensive studies on asymptotic expansion methods for small noise diffusions with Malliavin calculus (for instance, [16][17][18]). Moreover, by extending these results, high-order discretization methods for SDEs are developed in various papers (for example, [19][20][21][22][23]). In particular, ref.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Expansion and Weak Approximation for a Stochastic Control Problem on Path Space

Kannari,

Naito,

Yamada

2024

Entropy

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“…We consider some auxiliary equations (with N (dw, dz, dr) the Poisson point measure on the state space [0, 1] × R d with intensity measure dwµ(dz)dr defined in Section 2. (40), ( 41) and ( 42)). This is because we take different couplings so η 1 r (w) ̸ = η 3 r (w) and η 2 r (w) ̸ = η 4 r (w).…”

Section: Xpmmentioning

confidence: 98%

“…In this section, we present the integration by parts framework which will be used when we deal with the jump equations ( 51), ( 52) and (40). There are several approaches given in [13], [26], [32], [33], [41], [45] and [52] for example.…”

Section: Malliavin Calculus For the Jump Equationsmentioning

confidence: 99%

“…The coefficients b and c depend on the law of the solution, so our equation is of Mckean-Vlasov type. One can see for example [23] for a mathematical approach to this kind of equation and see [5], [12], [22], [24], [39], [40] and [47] for the approximation schemes of a Mckean-Vlasov equation. Moreover, the intensity of the Poisson point measure N ρt depends on the law of the solution as well, so our equation is also of Boltzmann type.…”

Section: Introductionmentioning

confidence: 99%

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Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analysis in total variation distance)

Qin

2023

Preprint

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We deal with Mckean-Vlasov and Boltzmann type jump equations. This means that the coefficients of the stochastic equation depend on the law of the solution, and the equation is driven by a Poisson point measure with intensity measure which depends on the law of the solution as well. In [3], Alfonsi and Bally have proved that under some suitable conditions, the solution $X_t$ of such equation exists and is unique. One also proves that $X_t$ is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme $X_t^{\mathcal{P}}$ of this equation converges to $X_t$ in Wasserstein distance. In this paper, under more restricted assumptions, we show that the Euler scheme $X_t^{\mathcal{P}}$ converges to $X_t$ in total variation distance and $X_t$ has a smooth density (which is a function solution of the analytical weak equation). On the other hand, in view of simulation, we use a truncated Euler scheme $X^{\mathcal{P},M}_t$ which has a finite numbers of jumps in any compact interval. We prove that $X^{\mathcal{P},M}_{t}$ also converges to $X_t$ in total variation distance. Finally, we give an algorithm based on a particle system associated to $X^{\mathcal{P},M}_t$ in order to approximate the density of the law of $X_t$. Complete estimates of the error are obtained.

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Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

Naito,

Yamada

2023

Comput Econ

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A higher order weak approximation of McKean–Vlasov type SDEs

Cited by 9 publications

References 19 publications

Asymptotic Expansion and Weak Approximation for a Stochastic Control Problem on Path Space

Asymptotic Expansion and Weak Approximation for a Stochastic Control Problem on Path Space

Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analysis in total variation distance)

Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

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