Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB

Abstract

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 strong numerical methods with orders of accuracy 1.0, 1.5, 2.0, 2.5, and 3.0 for SDEs. The Euler (Euler-Maruyama) method for Ito SDEs is also considered. Moreover, weak numerical methods for Ito SDEs are presented. This book contains 20 chapters divided into 4 parts. This book has many overlaps with the monograph: Dmitriy F. Kuznetsov, Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs, 2026, 1248 pp., https://arxiv.org/abs/2003.14184v82. Thus, both monographs are placed within a single submission, and their Internet links will differ only by the version numbers within https://arxiv.org/abs/2003.14184

1599 pp. Some inaccuracies have been corrected

References in corpus (5)

• First order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems
• Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function
• 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
• 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 Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series

Cited by in corpus (3)

• 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
• 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
• The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series