Kuznetsov, D. F.
Published in
Computational Mathematics and Mathematical Physics

AbstractThis paper is devoted to the comparative analysis of the efficiency of using the Legendre polynomials and trigonometric functions for the numerical solution of Ito stochastic differential equations under the method of approximating multiple Ito and Stratonovich stochastic integrals based on generalized multiple Fourier series. Using the mul...

Akeb, Tassadit Challali, Nordine Mellah, Omar

The few works devoted to the existence of almost periodicity (or almost automorphy) solutions to stochastic differential equations driven by a fractional Brownian motion impose the condition: the coefficient of fractional stochastic part is deterministic. In this work, without this condition, by using the chaos decomposition approach and the repres...

Borisov, I. S. Khrushchev, S. E.
Published in
Siberian Advances in Mathematics

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some neces...

Borisov, I. S. Khrushchev, S. E.
Published in
Siberian Advances in Mathematics

We study a construction of multiple stochastic integrals of nonrandom functions with respect to the product measures generated by stochastic processes admitting representations as multiple orthogonal random series. This construction is compared with some classical schemes of constructing stochastic integrals of such a kind.

Shen, Guangjun Yan, Litan Cui, Jing
Published in
Journal of Inequalities and Applications

In this paper, using the recent results on Stein’s method combining with Malliavin calculus and the almost sure central limit theorem for sequences of functionals of general Gaussian fields developed by Nourdin and Peccati, we derive the explicit bounds for the Kolmogorov distance in the central limit theorem and obtain the almost sure central limi...

Borisov, I. S. Bystrov, A. A.
Published in
Siberian Mathematical Journal

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert n...

Nourdin, Ivan Tudor, Ciprian

Using the multiple stochastic integrals we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one parameter and two parameter cases. When the drift is zero, we show that in the one-parameter case the solution in an exponential, thus positive,...