Markov processes conditioned on their location at large exponential times

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Evans, Steven N.¹  Hening, Alexandru² 

[1] Univ Calif Berkeley, Dept Stat, 3860,367 Evans Hall, Berkeley, CA 94720 USA

[2] Tufts Univ, Dept Math, Bromfield Pearson Hall,503 Boston Ave, Medford, MA 02155 USA

关键词: Excursion;    Local time;    Doob h-transform;    Bang-bang Brownian motion;    Campbell measure;    Diffusion;    Resurrection;   

DOI  :  10.1016/j.spa.2018.05.013

来源: Elsevier

【 摘 要 】

Suppose that (X-t)(t >= 0) is a one-dimensional Brownian motion with negative drift -mu 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 +mu when it is negative, like Brownian motion with negative drift -mu when it is positive, and is killed according to the local time spent at 0. An extension of this result holds in great generality for a Borel right process conditioned to be in some state a at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the bang-bang construction for a general Markov process. As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the h-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free