Approximation of Infinitely Divisible Random Variables with Application to
the Simulation of Stochastic Processes
Magnus Wiktorsson
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2001
ISBN 916284640X
LUTFMS10142001

Abstract:

This thesis consists of four papers A, B, C and D. Paper A and B treats the
simulation of stochastic differential equations (SDEs). The research presented
therein was triggered by the fact that there were not any efficient
implementations of the higher order methods for simulating SDEs. So in practice
the higher order methods required at least the same amount of work as the
Euler method to obtain a given mean square error. The faster convergence
rate of the higher order methods requires the simulation of the so called
iterated Itô integrals. In (A) we use a

shotnoise type series representation of one iterated Itô integral.
We split the series representation into a sum of n terms and a remainder
term and show that the remainder term is asymptotically Gaussian as n goes
to infinity. We provide an explicit coupling of the remainder a Gaussian
random variable and show that this improves the mean square error by a factor
n^½. In (B) we provide a multidimensional extension of the results
in (A) as well as the not previously known simultaneous characteristic function
of all iterated Itô integrals obtained then pairing m independent Wiener
processes. In (C) we study the simulation of type G Lévy processes.
Recall that random variable is said to be of type G if it is a Gaussian variance
mixture. We note that type G Lévy processes are subordinated Wiener
processes. We use a series representation of the subordinator, a tailsum
approximation and obtain an explicit coupling between type G Lévy
processes and the sum of a compound Poisson process and a scaled Wiener process.
We calculate the mean integrated square error for this approximation. We
examine the

convergence of the scaled tailsum process to its mean value function and
provide a sufficient condition for this convergence. In paper (D) we utilise
the coupling results from paper (C) to obtain approximationsof stochastic
integrals with respect to type G Lévy processes. Depending on the
properties of the integrator we obtain either pointwise mean square error
results or mean integrated square error results for the approximation. We
also show that a stochastic time change representation of stochastic integrals
can be used to obtain useful approximations.


Keywords:

Infinitely divisible distribution, stochastic differential equation, type
G distribution, Lévy process, stochastic integral