MARKOV PROCESSES CONDITIONED ON THEIR LOCATION AT LARGE EXPONENTIAL TIMES.
Steven N Evans1, Alexandru Hening1
1Department of Statistics #3860, 367 Evans Hall, University of California, Berkeley, CA 94720-3860, USADepartment of Mathematics, Tufts University, Bromfield-Pearson Hall, 503 Boston Avenue, Medford, MA 02155, United States.
Stochastic Processes and Their Applications
|November 5, 2019
Summary
This study introduces a novel Markov process by conditioning Brownian motion to hit zero at an exponential time. The resulting killed process exhibits unique dynamics, akin to a "bang-bang" Brownian motion.
Area of Science:
- Stochastic Processes
- Probability Theory
- Mathematical Finance
Background:
- One-dimensional Brownian motion with negative drift is analyzed.
- Conditioning processes to reach a specific state at random times is a complex problem.
Purpose of the Study:
- To define and analyze a Markov process by conditioning a Brownian motion to hit zero at an exponential random time.
- To investigate the limiting behavior of this killed conditioned process as the exponential time's rate parameter approaches zero.
- To generalize this construction for arbitrary Borel right processes.
Main Methods:
- Utilizing Campbell measures associated with local times.
- Applying excursion theory for process analysis.
- Developing a generalized "bang-bang" construction for Markov processes.
Main Results:
- The limit process behaves like Brownian motion conditioned to hit zero, then killed at its last visit to zero.
- This limiting process is equivalent to a killed "bang-bang" Brownian motion with state-dependent drift.
- The results are extended to general Borel right processes conditioned and killed at an exponential random time.
Conclusions:
- The study establishes a novel class of Markov processes with unique dynamics.
- The infinitesimal generator of the limiting process provides insights into h-transforms of diffusion processes.
- The findings offer a new perspective on conditioned stochastic processes and their applications.
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