Two-Sided Modifications of Lévy Processes
Søren Asmussen
Department of Mathematical Statistics
Aarhus University, Denmark
Wednesday, January 12, 2005
4:30 - 5:45 PM
Terman Engineering Center, Room 453
Abstract:
A Lévy process X is an independent sum of a linear drift, a Brownian
motion and a pure jump process, say a stable process, a NIG process or
just a compound Poisson process. We define the two-sided reflected version
V as solution of a Skorokhod problem V(t) = V(0)+X(t)+L^0(t)-L^K(t) where
L^0, L^K are the local times at the two boundaries 0, K. The loss rate l^K
is the stationary expectation of L^K(1); say l^K is the overflow rate of
a dam or the bit loss rate in a data buffer. We give an expression for l^K
in terms of the parameters of the Lévy process and the stationary
distribution; it is notable that this problem is trivial in discrete time,
with one-sided reflection and for certain specific Lévy processes. Also
the asymptotics of l^K as K goes to infinity is determined both with light
and with heavy tails of the Lévy measure. We finally look at the
asymptotics as K goes to infinity of a one-sided reflected version of X
with the Lévy measure truncated at K.
Joint work with Mats Pihlsgård, Lund, Sweden.
Operations Research Colloquia: http://or.stanford.edu/oras_seminars.html