On the Strong Approximation of Jump-Di®usion Processes

In Financial modelling, Filtering and other areas
the underlying dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class
of jump-diffusion SDEs that admits explicit solutions is rather
limited. Consequently, there is a need for the systematic use of
discrete time approximations in corresponding simulations. This
paper presents a survey and new results on strong numerical
schemes for SDEs of jump-diffusion type. These are relevant for
scenario analysis, filtering and hedge simulation in finance. It
provides a convergence theorem for the construction of strong
approximations of any given order of convergence for SDEs driven
by Wiener processes and Poisson random measures. The paper
covers also derivative free, drift-implicit and jump adapted strong
approximations. For the commutative case particular schemes are
obtained. Finally, a numerical study on the accuracy of several
strong schemes is presented.