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
arXiv:1807.00409
Abstract
The article is devoted to the expansion of iterated Ito stochastic integrals of second multiplicity based on expansion of the Brownian motion (standard Wiener process) using complete orthonormal systems of functions in the space The cases of Legendre polynomials and trigonometric functions are considered in details. We obtained a new representation of the Levy stochastic area based on the Legendre polynomials. This representation was first derived in the author's work (1997). In this article, we obtain the mentioned representation by a simpler method compared to the author's work (1997). Also, we get the polynomial representation of the Levy stochastic area using the method of expansion of iterated Ito stochastic integrals based on generalized multple Fourier series. The polynomial representation of the Levy stochastic area has more simple form in comparison with the classical trigonometric representation of the Levy stochastic area. The convergence in the mean of degree as well as the convergence with probability 1 for approximations of the Levy stochastic area are proved. The results of the article can be applied to the numerical solution of Ito stochastic differential equations as well as to the numerical approximation of mild solution for non-commutative semilinear stochastic partial differential equations.
arXiv comment23 pages. Minor changes in Sect. 2, 3, 6. arXiv admin note: text overlap with arXiv:1806.10705, arXiv:1712.09516, arXiv:1801.01564, arXiv:1801.06501, arXiv:1801.03195, arXiv:1802.00643, arXiv:1712.09746, arXiv:1802.00888, arXiv:1801.01962, arXiv:1801.00784
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Cited by in corpus (5)
- Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean
- 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
- 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