MARKOV PROCESSES CONDITIONED ON THEIR LOCATION AT LARGE EXPONENTIAL TIMES
| Autor: | Alexandru Hening, Steven N. Evans |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Local time Statistics & Probability Excursion Markov process bang–bang Brownian motion 01 natural sciences Article 010104 statistics & probability symbols.namesake 60J25 60J35 60J55 60J60 Killed process FOS: Mathematics Limit (mathematics) Infinitesimal generator 0101 mathematics Brownian motion Mathematics Doob h-transform Bang-bang Brownian motion Borel right process Applied Mathematics 010102 general mathematics Mathematical analysis Probability (math.PR) diffusion Statistics resurrection Banking Exponential function Modeling and Simulation Doob h–transform symbols Finance and Investment Campbell measure Mathematics - Probability |
| Zdroj: | Stochastic processes and their applications, vol 129, iss 5 Evans, SN; & Hening, A. (2018). Markov processes conditioned on their location at large exponential times. Stochastic Processes and their Applications. doi: 10.1016/j.spa.2018.05.013. UC Berkeley: Retrieved from: http://www.escholarship.org/uc/item/5s93h4v3 Stoch Process Their Appl |
| DOI: | 10.1016/j.spa.2018.05.013. |
| Popis: | Suppose that $(X_t)_{t \ge 0}$ 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 Borel right processes conditioned to be in some state $a$ at an exponential random time, at which time they are 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 general Markov processes. 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. Comment: 39 pages |
| Databáze: | OpenAIRE |
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