We consider stochastic differential equations driven by a general Lévy 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: (1) we approximate small jumps by a diffusion; (2) we use restricted jump-adaptive time-stepping; and (3) 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.

我们考虑由具有无限活动性的一般Lévy过程（SDE）驱动的随机微分方程，以及通过Feynman-Kac公式针对抛物线积分微分方程（PIDE）的Dirichlet问题。我们使用数值方法对SDE近似估计PIDE的解。该方法基于三个要素：（1）通过扩散近似小跳跃；（2）我们使用受限的跳跃自适应时间步长；（3）在跳跃之间，我们利用弱欧拉近似。我们证明了所考虑算法的弱收敛性，并对其误差和计算成本如何取决于跳跃活动水平进行了深入分析。提出了一些数值实验的结果，包括障碍货币选择权的定价。