Numerical methods for deterministic and stochastic differential equations
Official Course Description
Graduate course on the Numerical solution of Stocahstic Differential Equations
Gustaf Söderlind & Carmen Arévalo
Starting in September (Introduction meeting 2011-09-xx) we plan to give an introduction to the numerical solution of stochastic differential equations (SDEs). SDEs have many applications in physics (the Langevin equation) and have recently become a topic of interest also in economics, in particular in option pricing.
We plan to use S.S. Artemiev and T.A. Averina: Numerical Analysis if Systems of Ordinary and Stochastic Differential Equations (VSP 1997) as the primary course literature. We first start by introducing time stepping methods for deterministic problems to give a framework, and then proceed to SDEs. Various discretizations will be explored, including the Euler-Maruyama and Milstein schemes. Consistency, stability and convergence will be studied, both in the strong and the weak sense. We will also have a look at adaptive schemes, where the time step varies along the solution.
We further plan to combine the lectures with one or two computer projects, where SDEs are solved and the properties of the numerical methods are studied.
Welcome! Gustaf y Carmen
The course is divided into two parts, with the first dealing with the classical theory for deterministic ODEs and the second with the relatively recent theory for stochastic differential equations. The deterministic part reviews the necessary background with a focus on one-step methods such as Runge-Kutta and Rosenbrock methods. In the second part an introduction to SDEs is given, with an aim to establish the basic ideas and techniques used in statistical simulation. This includes root mean square stability and consistency notions, as well as convergence. A few applications will be studied.