40 citations · 312 across the 28 of their papers we have counts for
30 papers
A New Modifications of the Milstein and Euler Methods and the Taylor-Ito Expansion for Ito Stochastic Differential Equations with Drift and Diffusion Coefficients Having Non-Differentiable or Singular Time-Dependent Multipliers
Dmitriy F. Kuznetsov, Mikhail D. Kuznetsov
The article presents a new modifications of the Milstein and Euler methods for systems of Ito stochastic differential equations with drift and diffusion coefficients having differe…
New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
Dmitriy F. Kuznetsov
The article is devoted to the systematic derivation of new representations of the Hu-Meyer formulas. The formula expressing a multiple Wiener stochastic integral through the sum of…
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
Dmitriy F. Kuznetsov
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 expansio…
Inflation in simple one-loop effective potentials of perturbative quantum gravity
A. Arbuzov, D. Kuznetsov, B. Latosh +1
We study inflation in scalar-tensor perturbative quantum gravity driven by a one-loop effective potential. We consider effective potentials generated by three models. The first mod…
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
Mikhail D. Kuznetsov, Dmitriy F. Kuznetsov
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 integral…
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
Mikhail D. Kuznetsov, Dmitriy F. Kuznetsov
The article is devoted to the implementation of strong numerical methods with convergence orders and for Ito stochastic differential equati…
The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series
Dmitriy F. Kuznetsov
The article is devoted to the formulation and proof of the theorem on convergence with probability 1 of expansion of iterated Ito stochastic integrals of arbitrary multiplicity bas…
Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
Dmitriy F. Kuznetsov, Mikhail D. Kuznetsov
This monograph is devoted to the problem of numerical integration of stochastic differential equations (SDEs), mainly Ito SDEs. More precisely, the book mainly discusses high-order…
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
Dmitriy F. Kuznetsov
The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of t…
Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
Dmitriy F. Kuznetsov
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 semilin…
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
Dmitriy F. Kuznetsov
We consider a method for the approximation of iterated stochastic integrals of arbitrary multiplicity with respect to the infinite-dimensional -Wiener pr…
Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations
Dmitriy F. Kuznetsov
The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic d…
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
Dmitriy F. Kuznetsov
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…
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
Dmitriy F. Kuznetsov
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 comple…
Numerical Simulation of 2.5-Set of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 5 From the Taylor-Stratonovich Expansion
Dmitriy F. Kuznetsov
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 th…
Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
Dmitriy F. Kuznetsov
The article is devoted to the construction of expansions of iterated Stratonovich stochastic integrals of fifth, sixth, seventh and eighth multiplicities based on the method of gen…
Expansions of Iterated Stratonovich Stochastic Integrals from the Taylor-Stratonovich Expansion, Based on Multiple Trigonometric Fourier Series. Comparison With the Milstein Expansion
Dmitriy F. Kuznetsov
The article is devoted to comparison of the Milstein expansion of iterated Stratonovich stochastic integrals with the method of expansion of iterated stochastic integrals based on…
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2. Combined Approach Based on Generalized Multiple and Iterated Fourier Series
Dmitriy F. Kuznetsov
The article is devoted to the expansion of iterated Stratonovich stochastic integrals of multiplicity 2 on the base of the combined approach of generalized multiple and iterated Fo…
Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
Dmitriy F. Kuznetsov
We consider some versions and generalizations of an approach to the expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on general…
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Iterated Fourier series
Dmitriy F. Kuznetsov
The article is devoted to the expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 on the base of the combined approach of generalized multiple and ite…
Integration Order Replacement Technique for Iterated Ito Stochastic Integrals and Iterated Stochastic Integrals With Respect to Martingales
Dmitriy F. Kuznetsov
The article is devoted to the integration order replacement technique for iterated Ito stochastic integrals and iterated stochastic integrals with respect to martingales. We consid…
The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof
Dmitriy F. Kuznetsov
In this article, we collected more than thirty theorems on expansions of iterated Ito and Stratonovich stochastic integrals, which have been formulated and proved by the author in…
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 2 Based on Double Fourier-Legendre Series Summarized by Pringsheim Method
Dmitriy F. Kuznetsov
The article is devoted to the expansion of iterated Stratonovich stochastic integrals of second multiplicity into the double series of products of standard Gaussian random variable…
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
Dmitriy F. Kuznetsov
The article is devoted to the development of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series…