This book presents the latest developments and progress in the numerical study of the stochastic functional differential equation, with a particular emphasis on the longtime asymptotics and probabilistic characteristics of numerical methods used to solve such equation. The longtime asymptotics under investigation include the time-independent convergence analysis in both the strong and weak senses, the numerical invariant measure, and the ergodicity of numerical methods. Additionally, the probabilistic characteristics of numerical solutions explored in this book encompass the density function, limit theorems, and the Freidlin–Wentzell type large deviation principle. The topics presented here lie at the intersection of several fascinating areas: numerical analysis, stochastic analysis, ergodicity theory, Malliavin calculus, large deviation theory, and probability theory, providing a rich framework to deepen our understanding of stochastic functional differential equations from both theoretical and numerical perspectives. This book will appeal to researchers interested in these topics.

Chuchu Chen, Associate Professor, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China/School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Tonghe Dang, Postdoctor, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Jialin Hong, Professor, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China/School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Guoting Song, Lecturer, School of Mathematics and Statistics, Changchun University, Changchun 130022, China