arXiv subjects

Dmitriy F. Kuznetsov

Publications and source records attributed to Dmitriy F. Kuznetsov.

30 records · Page 2Linked to original sources

arXiv2026

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

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 $r(t_1)\ldots r(t_k)$ 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.

math.PR↗

arXiv2025

Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series

The article is devoted to the implementation of strong numerical methods with convergence orders $0.5,$ $1.0,$ $1.5,$ $2.0,$ $2.5,$ and $3.0$ for Ito stochastic differential equations with multidimensional non-commutative noise based on the unified Taylor--Ito and Taylor-Stratonovich expansions and multiple Fourier-Legendre series. Algorithms for the implementation of these methods are constructed and a package of programs in the Python programming language is presented. An important part of this software package, concerning the mean-square approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect to components of the multidimensional Wiener process is based on the method of generalized multiple Fourier series. More precisely, we used the multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ $(k=1,\ldots,6)$ for the mean-square approximation of iterated Ito and Stratonovich stochastic integrals.

math.PR↗

arXiv2023

A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality. Namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

math.PR↗

arXiv2023

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

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 $r(t_1)\ldots r(t_k)\ge 0$ 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.

math.PR↗

arXiv2022

To Numerical Modeling With Strong Orders 1.0, 1.5, and 2.0 of Convergence for Multidimensional Dynamical Systems With Random Disturbances

The article is devoted to explicit one-step numerical methods with strong orders 1.0, 1.5, and 2.0 of convergence for Ito stochastic differential equations with multidimensional and non-commutative noise. For numerical modeling of iterated Ito stochastic integrals with multiplicities 1 to 4 we use the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,2,3,4.$ The article is addressed to engineers who use numerical modeling in stochastic control and for solving the nonlinear filtering problem.

math.PR↗

arXiv2022

Explicit One-Step Strong Numerical Methods of Orders 2.0 and 2.5 for Ito Stochastic Differential Equations Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions

The article is devoted to the construction of explicit one-step strong numerical methods with the orders 2.0 and 2.5 of convergence for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the numerical methods based on the unified Taylor-Ito and Taylor-Stratonovich expansions. For the numerical modeling of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 5 we apply the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,\ldots,5$. The article is addressed to engineers who use numerical modeling in stochastic control and for solving the non-linear filtering problem. The article will be interesting to scientists who working in the field of numerical integration of stochastic differential equations.

math.PR↗

arXiv2022

Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion

The article is devoted to construction of effective procedures of the mean-square approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 5. We apply the method of generalized multiple Fourier series for approximation of iterated stochastic integrals. More precisely, we use multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t,T]^k),$ $k\in\mathbb{N}.$ Considered iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. That is why the results of the article can be applied to implementation of numerical methods with the orders 1.0, 1.5, 2.0 and 2.5 of strong convergence for Ito stochastic differential equations with multidimensional non-commutative noise.

math.PR↗

arXiv2022

Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series

The article is devoted to optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5. The mentioned stochastic integrals are part of strong numerical methods with convergence orders 1.0, 1.5, 2.0, and 2.5 for Ito stochastic differential equations with multidimensional non-commutative noise based on the unified Taylor-Ito expansion and multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ $(k=1,\ldots,5).$ In this article we use multiple Fourier-Legendre series within the framework of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series. We show that the lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Ito stochastic integrals of multiplicities 1 to 5 can be significantly reduced without the loss of the mean-square accuracy of approximation for these stochastic integrals.

math.PR↗

arXiv2022

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

The article is devoted to the construction of explicit one-step numerical methods with the strong orders of convergence 2.0, 2,5, and 3.0 for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the numerical methods based on the unified Taylor-Ito and Taylor-Stratonovich expansions. For numerical modeling of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 we appling the method of multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k=1,\ldots,6$. The article is addressed to engineers who use numerical modeling in stochastic control and for solving the non-linear filtering problem. The article can be interesting for the mathematicians who working in the field of high-order strong numerical methods for Ito stochastic differential equations.

math.PR↗

arXiv2022

Integration Order Replacement Technique for Iterated Ito Stochastic Integrals and Iterated Stochastic Integrals With Respect to Martingales

The article is devoted to the integration order replacement technique for iterated Ito stochastic integrals and iterated stochastic integrals with respect to martingales. We consider the class of iterated Ito stochastic integrals, for which with probability 1 the formulas of integration order replacement corresponding to the rules of classical integral calculus are reasonable. The theorems on integration order replacement for the class of iterated Ito stochastic integrals is proven. Many examples of this theorems usage have been considered. These results are generalized for the class of iterated stochastic integrals with respect to martingales.

math.PR↗

arXiv2022

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

We consider a method for the approximation of iterated stochastic integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ with respect to the infinite-dimensional $Q$-Wiener process using the mean-square approximation method of iterated Ito stochastic integrals with respect to the scalar standard Wiener processes based on generalized multiple Fourier series. The case of multiple Fourier-Legendre series is considered in details. The results of the article can be applied to construction of high-order strong numerical methods (with respect to the temporal discretization) for the approximation of mild solution for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise.

math.GM↗

arXiv2022

Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations

The article is devoted to the application of multiple Fourier-Legendre series to implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise. These methods have strong orders of convergence $1.0-\varepsilon$ and $1.5-\varepsilon$ correspondingly (here $\varepsilon$ is an arbitrary small positive real number) with respect to the temporal discretization. The theorem on mean-square convergence of approximations of iterated stochastic integrals of multiplicities 1 to 3 with respect to the infinite-dimensional $Q$-Wiener process is formulated and proved. The results of the article can be applied to implementation of exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise.

math.PR↗