Minicurso: Numerical methods for stochastic differential equations

Orador: Angel Tocino (Universidade de Salamanca)

Horário:
23 de janeiro, 17h-18h30
24 de janeiro, 17h-18h30
25 de janeiro, 10h-11h30.

Local: Sala de reuniões do Departamento de Matemática.

Resumo do curso:
Stochastic differential equations (SDEs) have become an important tool in many scientific areas owing to their application for modeling dynamical systems. Most of SDEs cannot be solved analytically, and numerical methods are needed for approximating their solutions.

The short course includes three sessions:
Part I: Devoted to present the concepts and main theoretical results on the subject. The definition of the Itô integral and the exposition of stochastic calculus lead to the concept of SDE. As a natural generalization, the study of SDEs runs in parallel with the one of ordinary differential equations (ODEs): Existence and uniqueness, linear equations, stability, etc.

Part II: The Euler-Maruyama method, as a generalization of Euler method is the simplest numerical method for SDEs. As a result of the so called Itô-Taylor expansions, Taylor methods for ODEs can be generalized to the stochastic case. In addition, implicit methods, Runge-Kutta methods, multi-step methods have their stochastic counterpart. As a qualitative study of numerical schemes, two kinds of convergence, weak and strong, are commonly considered. Mean square stability will be presented as the generalization of A-stability.

Part III: Some research results are presented, including truncated Itô-Taylor expansions and their application to the development of Runge-Kutta schemes, the MS-stability analysis of bi-dimensional systems or the proposition of two-step numerical methods.