School Seminars and Colloquia

On level crossings by piecewise-deterministic Markov processes

Stochastic Processes and Financial Mathematics

by Kostya Borovkov

Institution: Department of Mathematics and Statistics, The University of Melbourne
Date: Thu 20th August 2009
Time: 4:15 PM
Location: Old Geology Theatre 2, The University of Melbourne

Abstract: We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. We study the point process of the upcrossings of a given level b. Our main result shows that, under suitable scaling, this point process converges weakly, as b tend to infinity, to a geometrically compound Poisson process. We also prove a version of Rice's formula relating the stationary density of the process to level crossing intensities. While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite. [Joint work with G. Last.]

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