Markov processes conditioned on their location at large exponential times

Steven N. Evans and Alexandru Hening

Stochastic Processes and their Applications, 2019, vol. 129, issue 5, 1622-1658

Abstract: Suppose that (Xt)t≥0 is a one-dimensional Brownian motion with negative drift −μ. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like X conditioned to hit 0, after which time it behaves as X killed at the last time X visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift +μ when it is negative, like Brownian motion with negative drift −μ when it is positive, and is killed according to the local time spent at 0.

Keywords: Excursion; Local time; Doob h-transform; Bang–bang Brownian motion; Campbell measure; Diffusion; Resurrection (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.spa.2018.05.013

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