This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker–Planck–Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries’ existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method.

本文使用李对称性方法分析了一大类扩散过程的边界穿越概率。我们表明，如果Fokker-Planck-Kolmogorov方程具有非平凡的Lie对称性，则存在边界身份，并且仅取决于过程和对称性的参数。对于时间均匀扩散过程，我们发现了对称存在的必要条件和充分条件。本文表明，如果漂移函数满足Riccati方程族中的一个，则该问题具有非平凡的Lie对称性。对于每种情况，我们以显式形式表示对称性。根据获得的结果，我们得出两参数边界交叉身份并证明其唯一性。此外，我们提出了不同过程之间的边界交叉身份。我们展示 如果问题具有6组或4组对称性，则可以通过布朗运动或贝塞尔过程根据第一次通过时间明确表示到任何边界的第一次通过时间密度。给出了许多示例来说明该方法。