The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach

TL;DR

This paper introduces an efficient iterative algorithm for simulating the first-passage time of Brownian motion to a curved boundary, providing a practical tool for approximating its probability density function in general cases.

Contribution

The authors develop a novel iterative algorithm that accurately simulates the first-passage time of Brownian motion to a curved boundary, improving upon existing approximation methods.

Findings

The algorithm converges rapidly, indicating high efficiency.

Simulation results match theoretical expectations in tested scenarios.

The method is applicable under weak conditions for the boundary.

Abstract

Under some weak conditions, the first-passage time of the Brownian motion to a continuous curved boundary is an almost surely finite stopping time. Its probability density function (pdf) is explicitly known only in few particular cases. Several mathematical studies proposed to approximate the pdf in a quite general framework or even to simulate this hitting time using a discrete time approximation of the Brownian motion. The authors study a new algorithm which permits to simulate the first-passage time using an iterating procedure. The convergence rate presented in this paper suggests that the method is very efficient.