On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

Abstract

This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by = + , = , ( > 0, < 0, > 0). We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, we expressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for the distribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

Brownian motion between two random trajectories

“…Another extension is to consider the time-dependent boundary. [14] studied the asymptotic behavior of P(∀ 0≤s≤t |B t | ≤ f (t)), where the boundaries −f (t) and f (t) depend on t. The work [18] considered the Brownian motion with two linear boundaries and calculated the distribution of the Brownian motion hitting the upper boundary before hitting the lower boundary. The model of Brownian motion with two time-dependent boundaries can be applied to many different fields such as finance (see [15]), biophysical models (see [16]) and statistical sequential analysis (see [17]).…”