A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method

Ninomiya, Mariko; Ninomiya, Syoiti

doi:10.1007/s00780-009-0101-4

Cited by 54 publications

(49 citation statements)

References 33 publications

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“…We remark that it is necessary to solve the ordinary differential equation by a certain order to realize higher order discretization of the SDE, for more details, see Theorem 1.3 of [14]. Here we need to solve the ordinary differential equation by order 7.…”

Section: Implementation Of the New Methodsmentioning

confidence: 99%

“…Here we present numerical results obtained by applying 7th-order Runge-Kutta method. For the implementation of 7th-order Runge-Kutta method, see the appendix of [14]. [8,10], in order to obtain the order 3 discretization method by Q (7,2) (s) , we should take the partitions 0 = t 0 < t 1 · · · < t n = T as follows :…”

Section: Implementation Of the New Methodsmentioning

confidence: 99%

“…Ninomiya-Victoir [15] and Ninomiya-Ninomiya [14] methods are known as a kind of K-scheme. They are practical discretization methods of order 2, and some applications of these methods are studied in [2,1].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Higher order K-scheme and application to derivative pricing

Shinozaki

2016

Stochastic Systems Theory and its Applications (SSS)

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The author presents a new discretization method of stochastic differential equations (SDEs) without proof. The method is a kind of K-scheme (Kusuoka approximation, Kusuoka-Lyons-Ninomiya-Victoir method). K-scheme is a new framework of the higher order discretization methods for approximating SDEs weakly, proposed by Kusuoka. Ninomiya-Victoir and Ninomiya-Ninomiya methods are known as practical realizations of K-scheme, and some extrapolations of these methods are proposed. The new method is higher order than these methods without the use of extrapolations, and the author applies it to a financial problem of derivative pricing.

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Section: Implementation Of the New Methodsmentioning

confidence: 99%

Section: Implementation Of the New Methodsmentioning

confidence: 99%

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Higher order K-scheme and application to derivative pricing

Shinozaki

2016

Stochastic Systems Theory and its Applications (SSS)

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“…An alternative approach can be found in Kusuoka [21,22]. Its implementation as a splitting method is given in Ninomiya and Victoir [26], see also Alfonsi [1], Ninomiya and Ninomiya [25], and Tanaka and Kohatsu-Higa [33]. Our strategy is as follows.…”

Section: Stability Of Cubature Schemesmentioning

confidence: 99%

Cubature methods for stochastic (partial) differential equations in weighted spaces

Dörsek

Teichmann

Velušček

2013

Stoch PDE: Anal Comp

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The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded test functions. We first describe a recently introduced flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da PratoZabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced technical assumption on weak symmetry of the cubature formula. Computational results for a cubature discretization of a spatially extended stochastic FitzHugh-Nagumo model, an SPDE model from mathematical biology, are shown, illustrating the applicability of our theory.

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“…More recently, many discretization schemes of higher weak convergence order have appeared in the literature. Among others, we cite the work of Kusuoka [2001Kusuoka [ , 2004, the Ninomiya and Victoir [2008] scheme which we will use hereafter, the Ninomiya and Ninomiya [2009] scheme and the scheme based on cubature on Wiener spaces of Lyons and Victoir [2004]. Concerning strong approximation, the Milstein scheme has order one of strong convergence.…”

Section: Introductionmentioning

confidence: 99%

High Order Discretization Schemes for Stochastic Volatility Models

Sbai

Jourdain

2011

SSRN Journal

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In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose -a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, -a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price.We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a,b].

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A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method

Cited by 54 publications

References 33 publications

Higher order K-scheme and application to derivative pricing

Higher order K-scheme and application to derivative pricing

Cubature methods for stochastic (partial) differential equations in weighted spaces

High Order Discretization Schemes for Stochastic Volatility Models

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