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Construction of a third-order K-scheme and its application to financial models. (English) Zbl 1407.91274

Summary: The author proposes a new discretization method of stochastic differential equations (SDEs) that lies in the framework of K-scheme (Kusuoka approximation, Kusuoka-Lyons-Ninomiya-Victoir method). K-scheme is a higher-order discretization framework for performing weak approximations of SDEs. The Ninomiya-Victoir and Ninomiya-Ninomiya methods are practically feasible discretization methods that belong to the K-scheme class. These are second-order weak discretization methods, and some extrapolations of them have been proposed. The new method proposed herein is a third-order weak discretization method that involves no extrapolation. Polynomials of Gaussian random variables that approximate iterated Wiener integrals play a key role in the new method. In addition, the author discusses the applications of the proposed method to the pricing of derivatives in practically important financial models, achieving the desired theoretical order and computational efficiency.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C20 Probabilistic models, generic numerical methods in probability and statistics
65C30 Numerical solutions to stochastic differential and integral equations
65C05 Monte Carlo methods
62P05 Applications of statistics to actuarial sciences and financial mathematics
60H07 Stochastic calculus of variations and the Malliavin calculus

Software:

TOMS659; sobol.cc
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References:

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