Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean

Abstract

The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t,T{]}^k),$ $k\in \mathbb{N}.$ The method of generalized multiple Fourier series for expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity $k$ ($k\in \mathbb{N}$) with respect to components of the multidimensional Wiener process is proposed and developed. The obtained expansions contain only one operation of the limit transition in contrast to its existing analogues. In the article it is also obtained the generalization of the proposed method for an arbitrary complete orthonormal systems of functions in the space $L_2([t,T{]}^k),$ $k\in \mathbb{N}$ as well as for complete orthonormal with weight $r(t_1)\dots r(t_k)$ systems of functions in the space $L_2([t,T{]}^k),$ $k\in \mathbb{N}$. The comparison of the considered method with the well-known expansions of iterated Ito stochastic integrals based on the Ito formula and Hermite polynomials is given. The convergence in the mean of degree $2n$ $(n\in \mathbb{N})$ and with probability 1 of the proposed method is proved.

arXiv comment151 pages. Sect. 6 has been revised and expanded. Bibliography has been updated. arXiv admin note: text overlap with arXiv:1712.09516

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• Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
• Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
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Cited by in corpus (12)

• Strong Numerical Methods of Orders 2.0, 2.5, and 3.0 for Ito Stochastic Differential Equations Based on the Unified Stochastic Taylor Expansions and Multiple Fourier-Legendre Series
• Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich Expansions Using Legendre Polynomials
• Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise
• Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
• Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Integration Order Replacement Technique for Iterated Ito Stochastic Integrals and Iterated Stochastic Integrals With Respect to Martingales
• New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
• Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations
• Curved Schemes for SDEs on Manifolds