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June 2020 The inverse first passage time problem for killed Brownian motion
Boris Ettinger, Alexandru Hening, Tak Kwong Wong
Ann. Appl. Probab. 30(3): 1251-1275 (June 2020). DOI: 10.1214/19-AAP1529

Abstract

The classical inverse first passage time problem asks whether, for a Brownian motion $(B_{t})_{t\geq0}$ and a positive random variable $\xi$, there exists a barrier $b:\mathbb{R}_{+}\to\mathbb{R}$ such that $\mathbb{P}\{B_{s}>b(s),0\leq s\leq t\}=\mathbb{P}\{\xi>t\}$, for all $t\geq0$. We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if $\lambda>0$ is a killing rate parameter and $𝟙_{(-\infty,0]}$ is the indicator of the set $(-\infty,0]$ then, under certain compatibility assumptions, there exists a unique continuous function $b:\mathbb{R}_{+}\to\mathbb{R}$ such that $\mathbb{E}[-\lambda\int_{0}^{t}𝟙_{(-\infty,0]}(B_{s}-b(s))\,ds]=\mathbb{P}\{\zeta>t\}$ holds for all $t\geq0$. This is a significant improvement of a result of the first two authors (Ann. Appl. Probab. 24 (2014) 1–33).

The main difficulty arises because $𝟙_{(-\infty,0]}$ is discontinuous. We associate a semilinear parabolic partial differential equation (PDE) coupled with an integral constraint to this version of the inverse first passage time problem. We prove the existence and uniqueness of weak solutions to this constrained PDE system. In addition, we use the recent Feynman–Kac representation results of Glau (Finance Stoch. 20 (2016) 1021–1059) to prove that the weak solutions give the correct probabilistic interpretation.

Citation

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Boris Ettinger. Alexandru Hening. Tak Kwong Wong. "The inverse first passage time problem for killed Brownian motion." Ann. Appl. Probab. 30 (3) 1251 - 1275, June 2020. https://doi.org/10.1214/19-AAP1529

Information

Received: 1 July 2018; Revised: 1 April 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

MathSciNet: MR4133373
Digital Object Identifier: 10.1214/19-AAP1529

Subjects:
Primary: 35K58, 60J70, 91G40, 91G80

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 3 • June 2020
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