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：我们构建了一个快速精确算法，用于模拟受控稳定子穿越任意绝对连续非增函数的首次通过时间、欠冲与过冲的联合分布。我们证明该算法的运行时间具有有限指数矩，并给出其期望运行时间的显式界限（依赖于过程特征与函数初始值）。期望运行时间随稳定性参数（趋近0或1时）至多呈三次增长，随温度参数与函数初始值呈线性增长。基于专用GitHub存储库实现的数值性能与理论界限吻合良好。我们提供数值示例展示该算法在蒙特卡洛估计中的表现。