344 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 approximation rates for boundary crossing probabilities for the multivariate Brownian motion process
Motivated by an approximation problem from mathematical finance, we analyse
the stability of the boundary crossing probability for the multivariate
Brownian motion process, with respect to small changes of the boundary. Under
broad assumptions on the nature of the boundary, including the Lipschitz
condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an
analogue of the Borovkov and Novikov (2005) upper bound for the difference
between boundary hitting probabilities for "close boundaries" in the univariate
case. We also obtained upper bounds for the first boundary crossing time
densities.Comment: 15 page
Explicit Bounds for Approximation Rates for Boundary Crossing Probabilities for the Wiener Process
We give explicit upper bounds for convergence rates when approximating (both one- and two-sided general curvlinear) boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries (of simpler form for which computing the possibility is feasible). In particular, we generalize and improve results obtained by Potzelberger and Wang [13] for the case when approximating boundaries are piecewise linear. Applications to barrier option pricing are discussed as well.wiener process, boundary crossing probabilities; barrier options
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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