On the heat equation with a moving boundary and applications to hitting times for Brownian motion

Gerardo Hernández-del-Valle and Wincy A. Guerra-Polania ()
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Wincy A. Guerra-Polania: Universidad EIA

No 05/2023, CEMLA Working Paper Series from CEMLA

Abstract: In this paper we provide conditions under which the hitting-time problem for Brownian motion is equivalent to solving a heat equation with moving boundary and distributional initial conditions. Motivated by the hitting time problem, we study the heat equation with absorbing moving boundaries. Using Fourier analysis we develop a procedure to solve this problem for a family of curves that includes the square root, quadratic, and cubic boundaries. As an application of our results, and using Sturm-Liouvile theory, we compute the density of the hitting time of a Brownian motion to a family of quadratic boundaries.

Keywords: Heat equation; Brownian motion; Hitting times; Sturm-Liouville theory. (search for similar items in EconPapers)
Pages: 22
Date: 2023-04
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