We prove simple general formulas for expectations of functions of a random walk and its running extremum. Under additional conditions, we derive analytical formulas using the inverse $Z$-transform, the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use a new efficient numerical realization of the inverse $Z$-transform, the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-10 and better in several dozen of milliseconds, and E-14 - in a fraction of a isecond.
翻译:我们证明随机行走及其运行的极限功能的预期的简单通用公式。 在其他条件下,我们用反Z$转换法、Fleier/Laplace反转法和Wiener-Hopf 系数化法来得出分析公式,并讨论实现这些公式的有效数字方法。作为应用,计算了过程及其运行最大值的累积概率分布功能以及将股权转换为最大值的选项的价格。最有效的数字方法使用新的高效数字实现逆Z$转换法、正辛醇加速技术以及简化的陷阱型规则。在有中度特性的Mac上运行的 Matlab 程序在几十毫秒内实现了精度E-10且更好, E-14 以一秒的一小部分实现。