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Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Hongzhong Zhang
Hongzhong Zhang*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, 1255 Amsterdam Avenue, Columbia University, New York, NY 10027, USA. Email address: [email protected]
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Abstract

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The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Keywords

Drawdowndrawupdiffusion processoccupation time

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60G17: Sample path properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 210 - 230
Copyright
© Applied Probability Trust 

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