Abstract We show that the First-Passage-Time probability distribution of a Levy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show that it preserves the structural properties (non-negativity, boundedness, time monotonicity) of the theoretical solution, having to be a probability distribution. Numerical experiments confirming such findings are reported. Simulations of the sample paths of the considered process are also performed and used to both provide suitable boundary conditions and to validate the numerical results.

中文翻译：

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时变布朗运动首次通过时间的分数阶偏微分方程及其数值解

摘要 我们证明了具有漂移的 Levy 时变布朗运动的首次通过时间概率分布是受初始和边界条件约束的时间分数阶对流扩散方程的解；考虑相对于时间的 Caputo 分数阶导数。我们提出了一种高阶紧致隐式离散化方案来解决这个分数 PDE 问题，我们表明它保留了理论解的结构特性（非负性、有界性、时间单调性），必须是概率分布。报告了证实这些发现的数值实验。所考虑过程的样本路径的模拟也被执行并用于提供合适的边界条件和验证数值结果。