612 research outputs found
Some remarks on first passage of Levy processes, the American put and pasting principles
The purpose of this article is to provide, with the help of a fluctuation
identity, a generic link between a number of known identities for the first
passage time and overshoot above/below a fixed level of a Levy process and the
solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and
Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript
(1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished
manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100
(2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asmussen, Avram and
Pistorius [Stochastic Process. Appl. 109 (2004) 79-111] and Chesney and
Jeanblanc [Appl. Math. Fin. 11 (2004) 207-225] to the American perpetual put
optimal stopping problem. Furthermore, we make folklore precise and give
necessary and sufficient conditions for smooth pasting to occur in the
considered problem.Comment: Published at http://dx.doi.org/10.1214/105051605000000377 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Branching processes in random environment die slowly
Let be a branching process evolving in the random
environment generated by a sequence of iid generating functions and let be the
associated random walk with be
the left-most point of minimum of on the
interval and . Assuming that the
associated random walk satisfies the Doney condition we prove (under the quenched approach) conditional limit
theorems, as , for the distribution of and given . It is shown that
the form of the limit distributions essentially depends on the location of
with respect to the point $nt.
The total mass of super-Brownian motion upon exiting balls and Sheu's compact support condition
We study the total mass of a d-dimensional super-Brownian motion as it first
exits an increasing sequence of balls. The process of the total mass is a
time-inhomogeneous continuous-state branching process, where the increasing
radii of the balls are taken as the time parameter. We are able to characterise
its time-dependent branching mechanism and show that it converges, as time goes
to infinity, towards the branching mechanism of the total mass of a
one-dimensional super-Brownian motion as it first crosses above an increasing
sequence of levels. Our results allow us to identify the compact support
criterion given in Sheu (1994) as a classical Grey condition (1974) for the
aforementioned limiting branching mechanism.Comment: 28 pages, 2 figure
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