  We consider stochastic differential equations driven by a general L\'evy
processes (SDEs) with infinite activity and the related, via the Feynman-Kac
formula, Dirichlet problem for parabolic integro-differential equation (PIDE).
We approximate the solution of PIDE using a numerical method for the SDEs. The
method is based on three ingredients: (i) we approximate small jumps by a
diffusion; (ii) we use restricted jump-adaptive time-stepping; and (iii)
between the jumps we exploit a weak Euler approximation. We prove weak
convergence of the considered algorithm and present an in-depth analysis of how
its error and computational cost depend on the jump activity level. Results of
some numerical experiments, including pricing of barrier basket currency
options, are presented.

Numerical Analysis;Numerical Analysis;Probability