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A unified approach for drawdown (drawup) of time-homogeneous Markov processes

Part of: Stochastic processes

Published online by Cambridge University Press:  22 June 2017

David Landriault ,
Bin Li  and
Hongzhong Zhang
David Landriault*
Affiliation:
University of Waterloo
Bin Li*
Affiliation:
University of Waterloo
Hongzhong Zhang*
Affiliation:
Columbia University
*
* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
**** Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027, USA. Email address: [email protected]
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Abstract

Drawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

Keywords

Drawdownintegral equationreflected processtime-homogeneous Markov process

MSC classification

Primary: 60G07: General theory of processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 603 - 626
Copyright
Copyright © Applied Probability Trust 2017 

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