Lie symmetries methods in boundary crossing problems for diffusion processes

TL;DR

This paper applies Lie symmetry methods to derive explicit boundary crossing identities for diffusion processes, linking symmetries to first passage time distributions and providing new analytical tools for stochastic boundary crossing problems.

Contribution

It establishes conditions under which Lie symmetries exist for diffusion processes and derives explicit boundary crossing identities, including cases with multiple symmetries, enhancing analytical understanding.

Findings

Boundary crossing identities depend only on process parameters and symmetries.

Explicit symmetry forms are derived for processes satisfying Ricatti equations.

First passage time densities can be expressed via Brownian motion or Bessel process when symmetries are present.

Abstract

This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if Fokker--Planck--Kolmogorov equation has non-trivial Lie symmetry, then boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found necessary and sufficient conditions of symmetries' existence. This paper shows that if drift function satisfy one of a family of Ricatti 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 Brownian motion or Bessel process. Many examples are presented to illustrate the method.