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
Abstract
The atricle is devoted to the new simple method for obtainment an expansion of iterated Ito stochastic integrals of multiplicity 2 based on expansion of the Brownian motion (standard Wiener process) using complete orthonormal systems of functions in the space $L_2([t, T]).$ 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 also has been derived with using the method of expansion of iterated Ito stochastic integrals based on generalized multple Fourier series. The mentioned new 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 $2n$ $(n\in\mathbb{N})$ as well as the convergence with probability 1 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 noncommutative semilinear stochastic partial differential equations by the Milstein type method.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.00409
 Bibcode:
 2018arXiv180700409K
 Keywords:

 Mathematics  Probability
 EPrint:
 20 pages. Minor changes. 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