We derive several sets of sufficient conditions for applicability of the new efficient numerical realization of the inverse $Z$-transform. For large $n$, the complexity of the new scheme is dozens of times smaller than the complexity of the trapezoid rule. As applications, pricing of European options and single barrier options with discrete monitoring are considered; applications to more general options with barrier-lookback features are outlined. In the case of sectorial transition operators, hence, for symmetric L\'evy models, the proof is straightforward. In the case of non-symmetric L\'evy models, we construct a non-linear deformation of the dual space, which makes the transition operator sectorial, with an arbitrary small opening angle, and justify the new realization. We impose mild conditions which are satisfied for wide classes of non-symmetric Stieltjes-L\'evy processes.

翻译：我们推导了几组新的有效数值实现逆Z变换的充分条件。对于大的n，新方案的计算复杂度比梯形法则低数十倍。作为应用，考虑了具有离散监测的欧式期权和单障碍期权的定价；概述了具有障碍-回溯特征的更一般期权的应用。在扇形转移算子的情况下，即对于对称列维模型，证明是直接的。对于非对称列维模型，我们构造了其对偶空间的非线性变形，使转移算子成为具有任意小开角的扇形算子，并论证了该新实现的合理性。我们施加了温和条件，这些条件对广泛类别的非对称施蒂尔杰斯-列维过程均成立。