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Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise

28 citations0/0 files storedarXiv (Cornell University)

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Stochastic processes and financial applications Mathematical functions and polynomials Numerical methods for differential equations

Abstract

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on the generalized iterated Fourier series converging pointwise. The case of Fourier-Legendre series as well as the case of trigonometric Fourier series are considered in details. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree $2n$ $(n\in \mathbb{N})$ of the expansion is proved. Some recent results on the expansion of iterated Stratonovich stochastic integrals of multiplicities 3 to 6 are given. The results of the article can be applied to the numerical solution of Ito stochastic differential equations.

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References

2016Numerical solution of stochastic differential equations 1983Stochastic Differential Equations and Diffusion Processes 1965On the relation between ordinary and stochastic differential equations 1965On the Convergence of Ordinary Integrals to Stochastic Integrals 2010Numerical Solution of Stochastic Differential Equations with Jumps in Finance 1994Numerical solution of SDE through computer experiments 2001Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions 2006Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations

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