Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series

Abstract

We consider some versions and generalizations of an approach to the expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ based on generalized multiple Fourier series. Expansions of iterated stochastic integrals with respect to martingale Poisson measures and with respect to martingales were obtained. For the iterated stochastic integrals with respect to martingales we have proved theorem, which is a generalization of the expansion for iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series. Also we consider a modification of the mentioned expansion of iterated Ito stochastic integrals for the case of complete orthonormal with weight $r(t_1)\dots r(t_k)\ge 0$ systems of functions in the space $L_2([t,T{]}^k)$. Mean-square convergence of the considered expansions is proved. An example of the expansion of iterated (double) stochastic integrals with respect to martingales using the system of Bessel functions is considered.

40 pages. Some minor changes. arXiv admin note: text overlap with arXiv:1712.09746, arXiv:1801.05654, arXiv:1801.01564, arXiv:1712.08991, arXiv:1801.00231, arXiv:1801.03195, arXiv:1712.09516

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• 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 (8)

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