Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T21:03:06.225Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364–1382. Google Scholar
[2] Asmussen, S., Avram, F. and Pistorius, M. R. (2014). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111. Google Scholar
[3] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238. Google Scholar
[4] Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180. Google Scholar
[5] Baurdoux, E. J. (2009). Some excursion calculations for reflected Lévy processes. ALEA Lat. Amer. J. Prob. Math. Statist. 6, 149–162. Google Scholar
[6] Berberian, S. K. (1999). Fundamentals of Real Analysis. Springer, New York. Google Scholar
[7] Carr, P., Zhang, H. and Hadjiliadis, O. (2011). Maximum drawdown insurance. Internat. J. Theoret. Appl. Finance 14, 1195–1230. CrossRefGoogle Scholar
[8] Cherny, V. and Obłoj, J. (2013). Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model. Finance Stoch. 17, 771–800. Google Scholar
[9] Douady, R., Shiryaev, A. N. and Yor, M. (2000). On probability characteristics of "drop" variables in standard Brownian motion. Theory Prob. Appl. 44, 29–38. Google Scholar
[10] Feinberg, E. A., Kasyanov, P. O. and Zadoianchuk, N. V. (2014). Fatou's lemma for weakly converging probabilities. Theory Prob. Appl. 58, 683–689. Google Scholar
[11] Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer, Berlin. Google Scholar
[12] Gjessing, H. K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144. Google Scholar
[13] Grossman, S. J. and Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Math. Finance 3, 241–276. Google Scholar
[14] Ivanovs, J. and Palmowski, Z. (2012). Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122, 3342–3360. CrossRefGoogle Scholar
[15] Jacobsen, M. and Jensen, A. T. (2007). Exit times for a class of piecewise exponential Markov processes with two-sided jumps. Stoch. Process. Appl. 117, 1330–1356. Google Scholar
[16] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. Google Scholar
[17] Kella, O. and Stadje, W. (2001). On hitting times for compound Poisson dams with exponential jumps and linear release rate. J. Appl. Prob. 38, 781–786. CrossRefGoogle Scholar
[18] Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186. Google Scholar
[19] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg. Google Scholar
[20] Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 24–44. Google Scholar
[21] Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428–443. Google Scholar
[22] Kyprianou, A. E. and Zhou, X. (2009). General tax structures and the Lévy insurance risk model. J. Appl. Prob. 46, 1146–1156. Google Scholar
[23] Landriault, D., Li, B. and Zhang, H. (2017). On magnitude, asymptotics and duration of drawdowns for Lévy models. Bernoulli 23, 432–458. Google Scholar
[24] Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601–607. CrossRefGoogle Scholar
[25] Li, B., Tang, Q. and Zhou, X. (2013). A time-homogeneous diffusion model with tax. J. Appl. Prob. 50, 195–207. Google Scholar
[26] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669–1680. Google Scholar
[27] Magdon-Ismail, M., Atiya, A. F., Pratap, A. and Abu-Mostafa, Y. (2004). On the maximum drawdown of a Brownian motion. J. Appl. Prob. 41, 147–161. Google Scholar
[28] Mijatovic, A. and Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 3812–3836. CrossRefGoogle Scholar
[29] Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin. Google Scholar
[30] Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220. Google Scholar
[31] Poor, H. V. and Hadjiliadis, O. (2009). Quickest Detection. Cambridge University Press. Google Scholar
[32] Pospisil, L., Vecer, J. and Hadjiliadis, O. (2009). Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stoch. Process. Appl. 119, 2563–2578. CrossRefGoogle Scholar
[33] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 1, 2nd edn. Cambridge University Press. Google Scholar
[34] Schuhmacher, F. and Eling, M. (2011). Sufficient conditions for expected utility to imply drawdown-based performance rankings. J. Banking Finance 35, 2311–2318. Google Scholar
[35] Shepp, L. and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640. Google Scholar
[36] Taylor, H. M. (1975). A stopped Brownian motion formula. Ann. Prob. 3, 234–246. Google Scholar
[37] Tsurui, A. and Osaki, S. (1976). On a first-passage problem for a cumulative process with exponential decay. Stoch. Process. Appl. 4, 79–88. Google Scholar
[38] Widder, D. V. (1946). The Laplace Transform. Princeton University Press. Google Scholar
[39] Yuen, K. C., Wang, G. and Ng, K. W. (2004). Ruin probabilities for a risk process with stochastic return on investments. Stoch. Process. Appl. 110, 259–274. Google Scholar
[40] Zhang, H. (2015). Occupation time, drawdowns, and drawups for one-dimensional regular diffusions. Adv. Appl. Prob. 47, 210–230. Google Scholar
[41] Zhang, H. and Hadjiliadis, O. (2010). Drawdowns and rallies in a finite time-horizon: drawdowns and rallies. Methodology Comput. Appl. Prob. 12, 293–308. Google Scholar
[42] Zhang, H., Leung, T. and Hadjiliadis, O. (2013). Stochastic modeling and fair valuation of drawdown insurance. Insurance Math. Econom. 53, 840–850. Google Scholar
[43] Zhou, X. (2007). Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. J. Appl. Prob. 44, 1012–1030. Google Scholar