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

  • David Landriault (a1), Bin Li (a1) and Hongzhong Zhang (a2)


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.


Corresponding author

* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
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*** Email address:
**** Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027, USA. Email address:


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

  • David Landriault (a1), Bin Li (a1) and Hongzhong Zhang (a2)


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