Hostname: page-component-5db58dd55d-f6s65 Total loading time: 0 Render date: 2026-05-31T12:27:01.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

Part of: Stochastic processes

Published online by Cambridge University Press:  30 August 2022

Remco van der Hofstad Open the ORCID record for Remco van der Hofstad [Opens in a new window] ,
Stella Kapodistria ,
Zbigniew Palmowski Open the ORCID record for Zbigniew Palmowski [Opens in a new window]  and
Seva Shneer
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Stella Kapodistria*
Affiliation:
Eindhoven University of Technology
Zbigniew Palmowski*
Affiliation:
Wrocław University of Science and Technology
Seva Shneer*
Affiliation:
Heriot-Watt University
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
****Postal address: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland. Email: zbigniew.palmowski@pwr.edu.pl
*****Postal address: Heriot-Watt University, Edinburgh, UK. Email: v.shneer@hw.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

Keywords

Exit timesfirst passage timesadditive-increase and multiplicative-decrease processgrowth–collapse processLaplace–Stieltjes transformfirst-step analysisqueueing processstorageAIMD algorithm

MSC classification

Primary: 60G07: General theory of processes
Secondary: 60G44: Martingales with continuous parameter 60G55: Point processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 85 - 105
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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.)

Article purchase

Temporarily unavailable

References

Adan, I. J. B. F., Economou, A. and Kapodistria, S. (2009). Synchronized reneging in queueing systems with vacations. Queueing Systems 62, 1–33.CrossRefGoogle Scholar
Artalejo, J. R, Economou, A. and Lopez-Herrero, M. J. (2006). Evaluating growth measures in populations subject to binomial and geometric catastrophes. Math. Biosci. Eng. 4, 573–594.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
Avram, F. Kyprianou, A. E. and Pistorius, M. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.CrossRefGoogle Scholar
Avram, F., Grahovac, D. and Vardar-Acar, C. (2019). The $W, Z/\mu, \delta$ paradigm for the first passage of strong Markov processes without positive jumps. Risks, 7, 18.CrossRefGoogle Scholar
Avram, F. and Pérez, J. L. (2019). A review of first-passage theory for the Segerdahl–Tichy risk process and open problems. Risks 7, 117.CrossRefGoogle Scholar
Avram, F. and Usabel, M. (2004). The Gerber–Shiu expected discounted penalty–reward function under an affine jump-diffusion model. ASTIN Bull. 38, 461–481.CrossRefGoogle Scholar
Bertoin, J. (1996). On the first exit time of a completely asymmetric stable process from a finite interval. Bull. London Math. Soc. 28, 514–520.CrossRefGoogle Scholar
Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.CrossRefGoogle Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.CrossRefGoogle Scholar
Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Cohen, J. W. (1982). The Single Server Queue. North Holland, Amsterdam.Google Scholar
Czarna, I., Pérez, J.-L., Rolski, T. and Yamazaki, K. (2019). Fluctuation theory for level-dependent Lévy risk processes. Stoch. Process. Appl. 129, 5406–5449.CrossRefGoogle Scholar
Darling, D. and Siegert, A. J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
Doney, R. A. (2005). Some excursion calculations for spectrally one-sided Lévy processes. In Séminaire de Probabilités XXXVIII (Lect. Notes Math. 1857), eds M. Émery, M. Ledoux and M. Yor. Springer, Berlin, pp. 5–15.CrossRefGoogle Scholar
Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase, multiplicative-decrease (AIMD) algorithms. Adv. Appl. Prob. 34, 85–111.CrossRefGoogle Scholar
Emery, D. J. (1973). Exit problems for a spectrally positive process. Adv. Appl. Prob. 5, 498–520.CrossRefGoogle Scholar
Gihman, I. I. and Skorohod, A. V. (1972). Stochastic Differential Equations. Springer, New York.CrossRefGoogle Scholar
Gusak, D. and Koralyuk, V. S. (1968). On the first passage time across a given level for processes with independent increments. Theory Prob. Appl. 13, 448–456.CrossRefGoogle Scholar
Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519.CrossRefGoogle Scholar
Feller, W. (1954). The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. Math. 60, 417–436.CrossRefGoogle Scholar
Feller, W. (1955). On second order differential operators. Ann. Math. 61, 90–105.CrossRefGoogle Scholar
Hadjiev, D. J. (1985). The first passage problem for generalized Ornstein–Uhlenbeck processes with nonpositive jumps. In Séminaire de Probabilités XIX (Lect. Notes Math. 1123), eds J. Azéma and M. Yor. Springer, Berlin, pp. 80–90.Google Scholar
Gasper, G. and Rahman, M. (2004). Basic Hypergeometric Series (Encyc. Math. Appl. 96). Cambridge University Press.CrossRefGoogle Scholar
Guillemin, F., Robert, P. and Zwart, A. P. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob., 14, 90–117.CrossRefGoogle Scholar
Ito, K. and McKean, J. P. (1965). Diffusion Processes and their Sample Paths. Springer, New York.Google Scholar
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
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.CrossRefGoogle Scholar
Kella, O. and Yor, M. (2017). Unifying the Dynkin and Lebesgue–Stieltjes formulae. J. Appl. Prob. 54, 252–266.CrossRefGoogle Scholar
Kyprianou, A. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lect. Notes Math. 1857), eds M. Émery, M. Ledoux and M. Yor. Springer, Berlin, pp. 16–29.CrossRefGoogle Scholar
Kyprianou, A. and Palmowski, Z. (2008). Fluctuations of spectrally negative Markov additive processes. In Séminaire de Probabilités XLI (Lect. Notes Math. 1934), eds C. Donati-Martin, M. Émery, A. Rouault, and C. Stricker. Springer, Berlin, pp. 121–135.CrossRefGoogle Scholar
Kyprianou, A. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Lachal, A. (2000). First exit time from a bounded interval for a certain class of additive functionals of Brownian motion. J. Theor. Prob. 13, 733–775.CrossRefGoogle Scholar
Landriault, D., Li, B. and Zhang, H. (2017). A unified approach for drawdown (drawup) of time-homogeneous Markov processes. J. Appl. Prob. 54, 603–626.CrossRefGoogle Scholar
Li, B. and Palmowski, Z. (2018). Fluctuations of Omega-killed spectrally negative Lévy processes. Stoch. Process. Appl. 128, 3273–3299.CrossRefGoogle Scholar
Löpker, A. and Stadje, W. (2011). Hitting times and the running maximum of Markovian growth–collapse processes. J. Appl. Prob. 48, 295–312.CrossRefGoogle Scholar
Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601–607.CrossRefGoogle Scholar
Löpker, A., van Leeuwaarden, J. S. H. and Ott, T. J. (2009). TCP and iso-stationary transformations. Queueing Systems 63, 459–475.Google Scholar
Marciniak, E. and Palmowski, Z. (2016). On the optimal dividend problem for insurance risk models with surplus-dependent premiums. J. Optim. Theor. Appl. 168, 723–742.CrossRefGoogle Scholar
Novikov, A. A. (1981). The martingale approach in problems on the time of the first crossing of nonlinear boundaries. Trudy Mat. Inst. Steklov. 158, 130–152.Google Scholar
Patie, P. (2007). Two-sided exit problem for a spectrally negative $\alpha$ -stable Ornstein–Uhlenbeck process and the Wright’s generalized hypergeometric functions. Electron. Commun. Prob. 12, 146–160.Google Scholar
Paulsen, J. and Gjessing, H. (1997). Ruin theory with stochastic return on investments. Adv. Appl. Prob. 29, 965–985.CrossRefGoogle Scholar
Pistorius, M. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theor. Prob. 17, 183–220.CrossRefGoogle Scholar
Rogers, L. C. G. (1990). The two-sided exit problem for spectrally positive Lévy processes. Adv. Appl. Prob. 22, 486–487.CrossRefGoogle Scholar
Segerdahl, C.-O. (1955). When does ruin occur in the collective theory of risk? Scand. Actuarial J. 38, 22–36.CrossRefGoogle Scholar
Sweet, A. L. and Hardin, J. C. (1970). Solutions for some diffusion processes with two barriers. J. Appl. Prob. 7, 423–431.CrossRefGoogle Scholar
Tichy, R. (1984). Uber eine zahlentheoretische Methode zur numerischen Integration und zur Behandlung von Integralgleichungen. Osterreichische Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse Sitzungsberichte II 193, 329–358.Google Scholar
Zolotarev, V. M. (1964). The first passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theory Prob. Appl. 9, 653–661.CrossRefGoogle Scholar