Course No.：S070102ZY005
Course Category：Professional Course
Period/Credits：40/2
Prerequisites：stochastic analysis, numerical methods for ordinary differential equations.
Aims & Requirements：
In the past decades, there has been an increasing interest in numerical solution of stochastic differential equations in the fields of mathematics, engineering technology and physics etc. This course is one of speciality courses for master/doctor degree candidates majoring in computational mathematics. It can also be taken as an elective course for those majoring in related fields. Participants of the course are expected to have a command of the basic theories of numerical analysis and numerical methods of stochastic differential equations, and lay the foundation for future work and research.
Primary Coverage：
Chapter 1 Preliminary knowledge for stochastic differential equations (SODEs)
Brownian motion and white noises; stochastic integrals; Itô’s formula; definition, existence and uniqueness of solution, and examples of SODEs; properties of solutions; stochastic Taylor expansion.
Chapter 2 Stochastic strong approximation
Euler’s methods; Milstein’s methods; Itô-Taylor strong approximation; Stratonovich-Taylor strong approximation; multi-step methods; Runge-Kutta strong approximation; asymptotic stability and convergence of strong approximations.
Chapter 3 Stochastic weak approximation
Weak Taylor approximation; extrapolation; Predictor-corrector schemes; convergence of weak approximation.
Chapter 4 Stochastic structure-preserving algorithms
Strong and weak symplectic methods for stochastic Hamiltonian systems; Liouville methods for stochastic volume-preserving systems; strong and weak quasi-symplectic methods for equations of Langevin type.
Chapter 5 Implementation of stochastic numerical methods
Modeling of Itô integrals; Monte-Carlo realization of weak approximation methods.
References：
[1] P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992.
[2] G.N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, 1995.
[3] G.N. Milstein, M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer-Verlag Berlin Heidelberg, 2004.
Author：Lijin Wang (School of Mathematical Sciences, GUCAS)
Date：June, 2010
