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

Dörsek, Philipp; Teichmann, Josef; Velušček, Dejan

doi:10.1007/s40072-013-0020-4

Cited by 10 publications

(9 citation statements)

References 38 publications

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“…The weak approximation is a numerical calculation of E[f (X(T, x))] for a function f and a diffusion process defined by a stochastic differential equation (SDE) (2). This problems has a great deal in various areas including mathematical finance.…”

Section: Introductionmentioning

confidence: 99%

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Application of the Kusuoka approximation to pricing barrier options

Kusuoka

Ninomiya

2016

Stochastic Systems Theory and its Applications (SSS)

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The main theme of this research is numerical verification of applicability of a higher-order approximation to pricing barrier options, which is a both mathematically and practically important path-dependent type problem in mathematical finance. The authors successfully extend two types of algorithms called NV and NN. Both algorithms are based on a higher-order approximation scheme called Kusuoka approximation and have been shown to attain second-order approximation of stochastic differential equations as long as applied to European-type problems. In extending the algorithms, the authors apply a function representing the probability of hitting a boundary. In the numerical experiments, these two algorithms are compared with the Euler-Maruyama scheme which is one of the most popular first-order approximation schemes. As a result, it is demonstrated that the speed of calculation of these two algorithms could be much higher than that of the Euler-Maruyama scheme extended by the same way. It is concluded that one of the keys to improvement of the results is the construction of the function for calculation of the probability of hitting a boundary.

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Section: Introductionmentioning

confidence: 99%

“…We call this scheme Kusuoka approximation in this paper. This scheme has been recently developed such as in [2,3,5,16]. Two types of algorithms for implementation of the higherorder approximation are successfully constructed in [15,11] which are called NV and NN algorithms in this paper, respectively.…”

Section: Introductionmentioning

confidence: 99%

Application of the Kusuoka approximation to pricing barrier options

Kusuoka

Ninomiya

2016

Stochastic Systems Theory and its Applications (SSS)

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“…Romberg extrapolation is general technique for increasing the order of some methods, it is proved to be applicable to Euler-Maruyama method in [19] and Ninomiya-Victoir and Ninomiya-Ninomiya methods in [11]. Other type of extrapolation methods for Ninomiya-Victoir method which is a kind of classical Richardson extrapolation method is studied in [5,16,3]. However, in these extrapolation methods, negative weighted paths appear, and these methods are not numerically stable in some cases.…”

Section: Introductionmentioning

confidence: 99%

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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“…Ever since, the cubature method has been used to solve the problem of calculating Greeks in finance [Tei06], non-linear filtering problems [CG07], stochastic partial differential equations [BT08], [DTV12] and backward stochastic differential equations [CM10b,CM10a].…”

Section: Introductionmentioning

confidence: 99%