Recurrent extensions and Stochastic Differential equations
In the 70’s Itô settled the excursion theory of Markov processes, which is nowadays a fundamental tool for analyzing path properties of Markov processes. In his theory, Itô also introduced a method for building Markov processes using the excursion data, or by gluing excursions together, the resulting process is known as the recurrent extension of a given process. Since Itô’s pioneering work the method of recurrent extensions has been added to the toolbox for building processes, which of course includes the martingale problem and stochastic differential equations. The latter are among the most popular tools for building and describing stochastic processes, in particular in applied models as they allow to physically describe the infinitesimal variations of the studied phenomena. In this work we answer the following natural question. Assume X is a Markov process taking values in R that dies at the first time it hits a distinguished point of the state space, say 0, which happens in a finite time a.s., that X satisfies a stochastic differential equation, and finally that X admits a recurrent extension, say Z, is a processes that behaves like Z up to the first hitting time of 0, and for which 0 is a recurrent and regular state. If any, what is the SDE satisfied by Z? Our answer to this question allows us to describe the SDE satisfied by many Feller processes. We analyze various particular examples, as for instance the so-called Feller brownian motions and diffusions, which include their sticky and skewed versions, and also self-similar Markov processes, continuous state branching processes and real valued Levy processes.
Date: 5 February 2024, 14:00 (Monday, 4th week, Hilary 2024)
Venue: Mathematical Institute, Woodstock Road OX2 6GG
Speaker: Victor Rivero (CIMAT, Guanajuato)
Organising department: Department of Statistics
Part of: Probability seminar
Booking required?: Not required
Audience: Public
Editors: James Martin, Julien Berestycki