Magnus Wiktorsson            
                                        
Department of Statistics and Operations Research    
University of Copenhagen  


Title: Improved convergence rate for the approximation of SDE:s driven
by subordinated Lévy processes

For SDE:s driven by Brownian motion it is possible to use higher order
strong numerical schemes both explicit and implicit. In the
multidimensional case the main problem however is to approximate the
iterated Ito integrals which is needed by the higher order
schemes. The probabilistic difficulties makes it in practise not
efficient to use methods converging faster than O(h), where
 h is the step-size. For Lévy processes with jumps it is even worse.
For a large class of Lévy processes we cannot even simulate the
increments
of the driving process exactly. Due to this we cannot use anything
more advanced than the explicit Euler method to do strong approximation
of the SDE.

We now consider Lévy processes where we cannot simulate the increments
exactly. For the special case where we can simulate the subordinand
but not the subordinator exactly we propose a new approximation of the
increments.  More precisely we approximate the subordinator by its
large jumps plus a deterministic drift. It is then shown that the rate
of convergence for the strong Euler approximation of the SDE is given
by the rate of convergence for the approximated increments to the true
increments. We also have, under some mild conditions, that the rate is
superior to rate accomplished for a compound Poisson approximation of
the increments. Moreover the rescaled error process will converge
weakly in the Skorokhod topology to the solution of an SDE driven by a
Lévy process S(t) independent of the original driving process. We note
that the distribution of S(t) depend for a large class of
subordinators only on the distribution of the subordinand.