On hitting times, Bessel bridges and Schrödinger’s equation

Abstract

In this paper we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schrödinger's equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov's theorem, which suggests a strategy to extend our ideas to problems in R n and general diffusions. This approach also leads to (c) because we may relate, through a Feynman-Kac representation, functionals of a Bessel bridge with two Schrödinger-type problems. Finally, the relationship between (c) and (d) suggests a possible link between Generalized Airy processes and their hitting times.

On a new class of barrier options

“…In turn, our approach leads to solutions in terms of ODEs. We plot function φ , defined in (5). We observe the function is even and its first zero is at ±2.44197.…”

On the zeros of the Pearcey integral and a Rayleigh-type equation

“…In this section we find the function f for which the Pearcey function is zero for every t ≥ 0. The idea is to exploit, on the one hand, the differential form of the Airy function of order 4, defined in (5), and on the other to use the fact that the Pearcey function solves the heat equation (2).…”

“…In particular, the barrier option is a contract of this type. For a more detailed exposition see for instance [5].…”

On the pricing of defaultable bonds and Hitting times of Ito processes

“…We will use as well the following: Definition 4.2. Given a constant a ∈ R and a real-valued, twice continuously differentiable function f (•)-which we refer to as a "moving boundary"-B is a standard Brownian motion, and X is a solution to (14) we define the following stopping times…”

“…Examples of processes which satisfy equation (14) and that have unbounded domain before hitting the boundary f are:…”