MATE 4353 Introducción a los Modelos de Ecuaciones Diferenciales Estocásticas y sus Dinámicas

Stochastic differential equations are nowadays the standard class of models for most continuous time phenomena in engineering, the natural sciences, economics and life sciences. One reason for this is that compared to deterministic differential equations they exhibit a very rich (stochastic) dynamics, often also observed in the phenomena of interest. The essential difference consists in the occurrence of a random switching between different domains of attraction of the (deterministic) dynamical system and many associated phenomena.

This summer course provides a well-motivated, comprehensive introduction to stochastic differential equations for Gaussian and non-Gaussian stochastic (Lévy) processes for a mixed audience. The stochastic dynamics of these processes is then studied by means of large deviation principles, first exit times and metastability.

The course is focussed in the profound understanding of three major applications. First we will understand how different types of randomness can be used for a powerful stochastic optimization procedure known as simulated annealing. As a second application we will analyze the optimal tuning of the noise intensity in order to observe stochastic resonance, that is a synchronization in a periodically changing potential. Finally we will obtain the highly useful approximation of stochastic differential equations by means of ordinary differential equations known as Wong-Zakai procedure.



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