- Date: 4 Apr. (Fri.)
- Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
- Time: 16:30–18:00
- Speaker: Dima Ivanenko (Taras Shevchenko National University of Kyiv)
- Title: ON APPROXIMATION OF SOME LÉVY PROCESSES
- Abstract:
A Levy process X(t) has the structure X(t) = at + σW(t) + J(t) where W(t) is standard
Brownian motion (BM) and J(t) an independent pure jump process. This class of processes
has been used in numerous application areas, of which we in particular mention nance
and queueing. Calculations for a Levy process are, however, in general more dicult than
for BM, and an abundance of expressions that are explicit for BM are not so even in the
most popular parametric Levy models. Simulation of X(t) is therefore one of the main
computational tools.
A Levy process has countably many jumps on any interval [0, T] and nitely many jumps
of size bigger than some xed ε > 0. In order to simulate Z, we need to take nitely many
jumps of Z, which gives an adequate description of Z. Apart from some particular cases,
e.g. Brownian motion, Gamma process, α-stable process, simulation of the Levy process
with a given triplet is not an easy task.
Usually, the distribution function of a Levy process is unknown or has a rather compli-
cated form, which makes the simulation rather perplex. For the methods of generating
innitely divisible random variables (r.v.’s) and Levy processes (e.g. methods of Khinchin,
Fergusson-Klass, Bondesson, LePage, Rosinski) we refer to Rosinski (2001) and propose
our own method. We also would like to mention that the Damien-Laud-Smith algorithm
from Damien, Laud, and Smith (1995) gives a way to simulate an (approximation of)
an arbitrary onedimensional innitely divisible r.v., which allows us to simulate a Levy
process. On the other hand, it was observed by Bondesson (1982) and later by Asmussen
and Rosinski (2001), that under some conditions small jumps can be substituted by an
(arithmetic) Brownian motion.
The series of seminars includes a general theory of Levy processes, an overview of known
methods for modeling such processes, and a comparison of these methods with our own.
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