Construction of a Third-Order K-Scheme and Its Application to Financial Models

Shinozaki, Yuji

doi:10.1137/16m1067986

Cited by 6 publications

(6 citation statements)

References 29 publications

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“…The iterated integrals appearing in (2) have a rich algebraic structure (see Sect. 4.1), so algebraic approaches have been adopted in the literature [8,19,21,22,24,29]. However, solving complicated equations of Lie algebra is required in those approaches, and constructions of the formula are limited to a small range (1)…”

Section: Cubature On Wiener Spacementioning

confidence: 99%

Monte Carlo construction of cubature on Wiener space

Hayakawa

Tanaka

2021

Japan J. Indust. Appl. Math.

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In this paper, we investigate application of mathematical optimization to construction of a cubature formula on Wiener space, which is a weak approximation method of stochastic differential equations introduced by Lyons and Victoir (Proc R Soc Lond A 460:169–198, 2004). After giving a brief review on the cubature theory on Wiener space, we show that a cubature formula of general dimension and degree can be obtained through a Monte Carlo sampling and linear programming. This paper also includes an extension of stochastic Tchakaloff’s theorem, which technically yields the proof of our primary result.

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Section: Cubature On Wiener Spacementioning

confidence: 99%

Monte Carlo construction of cubature on Wiener space

Hayakawa

Tanaka

2021

Japan J. Indust. Appl. Math.

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“…We should mention the Kusuoka approximation (Kusuoka 2001, Kusuoka 2004, Shinozaki 2017, which is closely related to cubature on Wiener space. However, our objective in this paper is limited to the construction of the cubature on Wiener space, and the optimization-based approach to the general Kusuoka approximation is deferred for future work.…”

Section: Cubature On Wiener Spacementioning

confidence: 99%

Monte Carlo construction of cubature on Wiener space

Hayakawa

Tanaka²

2020

Preprint

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In this paper, we investigate application of mathematical optimization to construction of a cubature formula on Wiener space, which is a weak approximation method of stochastic differential equations introduced by Lyons and Victoir (Cubature on Wiener Space, Proc. R. Soc. Lond. A 460,. After giving a brief review of the cubature theory on Wiener space, we show that a cubature formula of general dimension and degree can be obtained through a Monte Carlo sampling and linear programming. This paper also includes an extension of stochastic Tchakaloff's theorem, which technically yields the proof of our main result.

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“…Next, we introduce the discretization methods that follow the KLNV-scheme, particularly the Gaussian type KLNV-scheme from (Ninomiya and Victoir 2008;Ninomiya and Ninomiya 2009;Kusuoka 2013) and (Shinozaki 2017). For a comparison of these methods from the viewpoint of implementation and performance, refer to Section 4 of (Shinozaki 2017).…”

Section: Klnv-scheme For Forward Sdementioning

confidence: 99%

“…Remark 3 (Constructions of operators). As described in (Shinozaki 2017), the Q ð7;2Þ ðsÞ method is constructed based on a moment similar family (Kusuoka 2001) and the formula derived in (Takanobu 1995). On the other hand, the N-V and N-N methods are construed based on other ideas such as the splitting method of ordinary differential equations (ODEs)…”

Section: Klnv-scheme For Forward Sdementioning

confidence: 99%

“…To overcome this problem, the TBBA (Crisan and Lyons 2002) was applied to these methods in (Ninomiya 2003b;Ninomiya and Mitsuzono 2004;Ninomiya 2010;Crisan and Ortiz-Latorre 2013). On the other hand, using the framework of KLNVscheme and the Gaussian random variables, the Ninomiya-Victoir (N-V) (Ninomiya and Victoir 2008) and Ninomiya-Ninomiya (N-N) (Ninomiya and Ninomiya 2009) methods have been proposed as practically feasible second-order methods, and the Q ð7;2Þ ðsÞ method (Shinozaki 2016(Shinozaki , 2017 as a third order method. Note that the N-V and N-N methods are implemented by using only substitutions and linear combinations in the same way as explicit Runge-Kutta methods are and are free from symbolic calculations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Higher-order Discretization Methods of Forward-backward SDEs Using KLNV-scheme and Their Applications to XVA Pricing

Ninomiya

Shinozaki

2019

Applied Mathematical Finance

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This study proposes new higher-order discretization methods of forward-backward stochastic differential equations. In the proposed methods, the forward component is discretized using the Kusuoka-Lyons-Ninomiya-Victoir scheme with discrete random variables and the backward component using a higher-order numerical integration method consistent with the discretization method of the forward component, by use of the tree based branching algorithm. The proposed methods are applied to the XVA pricing, in particular to the credit valuation adjustment. The numerical results show that the expected theoretical order and computational efficiency could be achieved.

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Construction of a Third-Order K-Scheme and Its Application to Financial Models

Cited by 6 publications

References 29 publications

Monte Carlo construction of cubature on Wiener space

Monte Carlo construction of cubature on Wiener space

Monte Carlo construction of cubature on Wiener space

Higher-order Discretization Methods of Forward-backward SDEs Using KLNV-scheme and Their Applications to XVA Pricing

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