Skip to main content Accessibility help
×
Home

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

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

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.

Copyright

Corresponding author

* Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
** Email address: dlandria@uwaterloo.ca
*** Email address: bin.li@uwaterloo.ca
**** Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027, USA. Email address: hz2244@columbia.edu

References

Hide All
[1] Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 13641382.
[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, 79111.
[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, 215238.
[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, 156180.
[5] Baurdoux, E. J. (2009). Some excursion calculations for reflected Lévy processes. ALEA Lat. Amer. J. Prob. Math. Statist. 6, 149162.
[6] Berberian, S. K. (1999). Fundamentals of Real Analysis. Springer, New York.
[7] Carr, P., Zhang, H. and Hadjiliadis, O. (2011). Maximum drawdown insurance. Internat. J. Theoret. Appl. Finance 14, 11951230.
[8] Cherny, V. and Obłoj, J. (2013). Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model. Finance Stoch. 17, 771800.
[9] Douady, R., Shiryaev, A. N. and Yor, M. (2000). On probability characteristics of "drop" variables in standard Brownian motion. Theory Prob. Appl. 44, 2938.
[10] Feinberg, E. A., Kasyanov, P. O. and Zadoianchuk, N. V. (2014). Fatou's lemma for weakly converging probabilities. Theory Prob. Appl. 58, 683689.
[11] Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Springer, Berlin.
[12] Gjessing, H. K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123144.
[13] Grossman, S. J. and Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Math. Finance 3, 241276.
[14] Ivanovs, J. and Palmowski, Z. (2012). Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122, 33423360.
[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, 13301356.
[16] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
[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, 781786.
[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. 97186.
[19] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg.
[20] Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 2444.
[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, 428443.
[22] Kyprianou, A. E. and Zhou, X. (2009). General tax structures and the Lévy insurance risk model. J. Appl. Prob. 46, 11461156.
[23] Landriault, D., Li, B. and Zhang, H. (2017). On magnitude, asymptotics and duration of drawdowns for Lévy models. Bernoulli 23, 432458.
[24] Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601607.
[25] Li, B., Tang, Q. and Zhou, X. (2013). A time-homogeneous diffusion model with tax. J. Appl. Prob. 50, 195207.
[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, 16691680.
[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, 147161.
[28] Mijatovic, A. and Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 38123836.
[29] Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin.
[30] Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183220.
[31] Poor, H. V. and Hadjiliadis, O. (2009). Quickest Detection. Cambridge University Press.
[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, 25632578.
[33] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 1, 2nd edn. Cambridge University Press.
[34] Schuhmacher, F. and Eling, M. (2011). Sufficient conditions for expected utility to imply drawdown-based performance rankings. J. Banking Finance 35, 23112318.
[35] Shepp, L. and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631640.
[36] Taylor, H. M. (1975). A stopped Brownian motion formula. Ann. Prob. 3, 234246.
[37] Tsurui, A. and Osaki, S. (1976). On a first-passage problem for a cumulative process with exponential decay. Stoch. Process. Appl. 4, 7988.
[38] Widder, D. V. (1946). The Laplace Transform. Princeton University Press.
[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, 259274.
[40] Zhang, H. (2015). Occupation time, drawdowns, and drawups for one-dimensional regular diffusions. Adv. Appl. Prob. 47, 210230.
[41] Zhang, H. and Hadjiliadis, O. (2010). Drawdowns and rallies in a finite time-horizon: drawdowns and rallies. Methodology Comput. Appl. Prob. 12, 293308.
[42] Zhang, H., Leung, T. and Hadjiliadis, O. (2013). Stochastic modeling and fair valuation of drawdown insurance. Insurance Math. Econom. 53, 840850.
[43] Zhou, X. (2007). Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. J. Appl. Prob. 44, 10121030.

Keywords

MSC classification

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

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

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed