Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing

Ninomiya, Syoiti; Victoir, Nicolas

doi:10.1080/13504860701413958

Cited by 142 publications

(152 citation statements)

References 19 publications

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“…It presents the analysis of the weak error made by Talay and Tubaro [22] and then gives a recursive construction of second order schemes for multidimensional SDEs that extends the results of Ninomiya and Victoir [19]. This method relies on the idea of scheme composition, which dates back to Strang [20] in the field of ODEs.…”

Section: Introductionmentioning

confidence: 99%

“…The QE scheme proposed by Andersen is in fact the only one among those cited that is really well suited for these large values, but no theoretical convergence result is given for this scheme. In another direction, Ninomiya and Victoir [19] have recently proposed a general method to get weak second order discretization schemes for a broad class of multidimensional SDEs. We will present their method in detail in the first part.…”

Section: Introductionmentioning

confidence: 99%

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High order discretization schemes for the CIR process: Application to affine term structure and Heston models

2010

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Abstract. This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya and Victoir. Combine both these results, this allows us to propose a second order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High order discretization schemes for the CIR process: Application to affine term structure and Heston models

2010

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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%

“…In this work, we shall relax the regularity assumptions of the cubature method, similarly as was done in Dörsek [9] and Dörsek and Teichmann [11,12] for the splitting approach of Ninomiya and Victoir [26]. Consider a stochastic differential equation on R n in its Stratonovich form,…”

Section: Introductionmentioning

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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“…See [41] for an overview of this subject. More recently, approximation schemes of higher order (h = 2), based on a cubature method, have been introduced and studied by Kusuoka [48], Lyons [56], Victoir, Ninomiya [58], Alfonsi [1], Kohatsu-Higa and Tankov [45]. Our abstract result intends to cover all these situations.…”

Section: Introductionmentioning

confidence: 99%

Recent advances in various fields of numerical probability

et al. 2015

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Abstract. The goal of this paper is to present a series of recent contributions on some various problems of numerical probability. Beginning with the Richardson-Romberg Multilevel Monte-Carlo method which, among other fields of applications, is a very efficient method for the approximation of diffusion processes, we focus on some adaptive multilevel splitting algorithms for rare event simulation. Then, the third part is devoted to the simulation of McKean-Vlasov forward and decoupled forwardbackward stochastic differential equations by some cubature algorithms. Finally, we tackle the problem of the weak error estimation in total variation norm for a general Markov semi-group.

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Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing

Cited by 142 publications

References 19 publications

High order discretization schemes for the CIR process: Application to affine term structure and Heston models

High order discretization schemes for the CIR process: Application to affine term structure and Heston models

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

Recent advances in various fields of numerical probability

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