THE INVERSE FIRST PASSAGE TIME PROBLEM FOR KILLED BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Ettinger, Boris; Hening, Alexandru; Wong, Tak Kwong
• 署名单位: Tufts University; University of Hong Kong
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1529
• 发表日期: 2020
• 页码: 1251-1275
• 关键词: Levy processes

摘要

The classical inverse first passage time problem asks whether, for a Brownian motion (B-t)(t >= 0) and a positive random variable xi, there exists a barrier b : R+ -> R such that P{B-s > b(s), 0 <= s <= t} = P{xi> t}, for all t >= 0. 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((-infinity,)(0]) is the indicator of the set (-infinity, 0] then, under certain compatibility assumptions, there exists a unique continuous function b : R+ -> R such that E[-lambda integral(t)(0)1((-infinity,0]) (B-s - b(s)) ds] = P {zeta > t} holds for all t >= 0. This is a sig- nificant improvement of a result of the first two authors (Ann. Appl. Probab. 24 (2014) 1-33). The main difficulty arises because 1((-infinity,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.