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.

中文翻译：

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抛物线积分微分方程狄利克雷问题的随机游走算法

我们考虑由具有无限活动的一般 L\'evy 过程 (SDE) 驱动的随机微分方程，以及相关的，通过 Feynman-Kac 公式，抛物线积分微分方程 (PIDE) 的狄利克雷问题。我们使用 SDE 的数值方法来近似 PIDE 的解。该方法基于三个要素：（i）我们通过扩散近似小跳跃；(ii) 我们使用受限跳跃自适应时间步长；(iii) 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。介绍了一些数值实验的结果，包括障碍一揽子货币期权的定价。