We construct a fast exact algorithm for the simulation of the first-passage time, jointly with the undershoot and overshoot, of a tempered stable subordinator over an arbitrary non-increasing absolutely continuous function. We prove that the running time of our algorithm has finite exponential moments and provide bounds on its expected running time with explicit dependence on the characteristics of the process and the initial value of the function. The expected running time grows at most cubically in the stability parameter (as it approaches either $0$ or $1$) and is linear in the tempering parameter and the initial value of the function. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds. We provide numerical examples to illustrate the performance of our algorithm in Monte Carlo estimation.

翻译：我们构建了一种快速精确算法，用于模拟 tempered stable 子在任意非增绝对连续函数上的首次通过时间，同时包含下冲量和上冲量。我们证明该算法运行时间具有有限的指数矩，并给出了其期望运行时间的界限，该界限明确依赖于过程特征和函数初始值。期望运行时间在稳定性参数（接近0或1时）最多呈三次增长，且在驯服参数和函数初始值上呈线性增长。基于专用GitHub仓库实现的数值性能与我们的理论界限表现出良好的吻合度。我们提供数值示例以说明该算法在蒙特卡洛估计中的性能。