Improved convergence rate for the simulation of Lévy processes of type G

MAGNUS WIKTORSSON

Centre for Mathematical Sciences, Lund University Box 118 221 00 Lund, Sweden

Abstract

 A random variable is said to be of type G if it is a Gaussian
variance mixture with the mixing distribution being in nitely
divisible. A LÈvy process is said to be of type G if its increments is
of type G. Every such LÈvy process on [0,1] can be represented as an
infinite series which converges uniformly a.s. In practice, however,
we must truncate this infinite series. The question is now how to
approximate the neglected terms in the series, the tail-sum process,
with a simpler stochastic process such that  a good convergence rate
is achieved. In order to do this we note that both the original
process and the tail-sum process can be represented as subordinated
Wiener processes. The main idea is then to approximate the
subordinator (i.e. a non-negative increasing Lévy process) by its mean
function. This leads to an approximation of the original process which
has a better integrated mean square convergence rate compared to that
obtained using the truncated series representation only. We also show
that this approach is generalisable to arbitrary real-valued
subordinated LÈvy processes provided that the subordinand has two
nite moments and that it is possible to simulate the subordinand
exactly.