648 research outputs found
On exit times of Levy-driven Ornstein--Uhlenbeck processes
We prove two martingale identities which involve exit times of Levy-driven
Ornstein--Uhlenbeck processes. Using these identities we find an explicit
formula for the Laplace transform of the exit time under the assumption that
positive jumps of the Levy process are exponentially distributed.Comment: 12 page
On level crossings for a general class of piecewise-deterministic Markov processes
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 upcrossings of a level by the Markov
process. Our main result shows that, under a suitable scaling , the
point process converges, as tends to infinity, weakly 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. This
formula provides an interpretation of the scaling factor . While our
proof of the limit theorem requires additional assumptions, Rice's formula
holds whenever the (stationary) overall intensity of jumps is finite.Comment: 25 page
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