Journal of Applied Probability

Exit problems for reflected Markov-modulated Brownian motion

Lothar Breuer

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let (X, J) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S. For some a > 0, let σ(a) denote the time when the reflected process Y := S - X first surpasses the level a. Furthermore, let σ-(a) denote the last time before σ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of Sσ(a), σ-(a), and σ(a), where the latter two will be given in terms of their Laplace transforms. We also provide some remarks on scale matrices for MMBMs with strictly positive variation parameters. This extends recent results for spectrally negative Lévy processes to MMBMs. Due to well-known fluid embedding and state-dependent killing techniques, the analysis applies to Markov additive processes with phase-type jumps as well. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.

Article information

J. Appl. Probab., Volume 49, Number 3 (2012), 697-709.

First available in Project Euclid: 6 September 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; Lévy processes 60J55: Local time and additive functionals

Markov-modulated Brownian motion reflection exit problem Markov additive process


Breuer, Lothar. Exit problems for reflected Markov-modulated Brownian motion. J. Appl. Probab. 49 (2012), no. 3, 697--709. doi:10.1239/jap/1346955327.

Export citation


  • Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian motion. Commun. Statist. Stoch. Models 11, 21–49.
  • Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.
  • Asmussen, S. and Kella, O. (2000). A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Prob. 32, 376–393.
  • Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.
  • Asmussen, S., Jobmann, M. and Schwefel, H.-P. (2002). Exact buffer overflow calculations for queues via martingales. Queueing Systems 42, 63–90.
  • Breuer, L. (2008). First passage times for Markov additive processes with positive jumps of phase type. J. Appl. Prob. 45, 779–799.
  • Breuer, L. (2010). A quintuple law for Markov additive processes with phase-type jumps. J. Appl. Prob. 47, 441–458.
  • Breuer, L. (2010). The total overflow during a busy cycle in a Markov-additive finite buffer system. In Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance (Lecture Notes Comput. Sci. 5987), eds B. Müller-Clostermann, K. Echtle and E. Rathgeb, Springer, Berlin, pp. 198–211.
  • Breuer, L. (2012). Occupation times for Markov-modulated Brownian motion. J. Appl. Prob. 49, 549–565.
  • D'Auria, B., Ivanovs, J., Kella, O. and Mandjes, M. (2012). Two-sided reflection of Markov-modulated Brownian motion. Stoch. Models 28, 316–332.
  • Frostig, E. (2008). On risk model with dividends payments perturbed by a Brownian motion–-an algorithmic approach. ASTIN Bull. 38, 183–206.
  • Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–78.
  • Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: analysis with Brownian motion. N. Amer. Actuarial J. 8, 1–20.
  • Ivanovs, J. (2010). Markov-modulated Brownian motion with two reflecting barriers. J. Appl. Prob. 47, 1034–1047.
  • Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. Doctoral Thesis, Universiteit van Amsterdam.
  • 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.
  • Jiang, Z. and Pistorius, M. R. (2008). On perpetual American put valuation and first-passage in a regime-switching model with jumps. Finance Stoch. 12, 331–355.
  • Li, S. and Lu, Y. (2007). Moments of the dividend payments and related problems in a Markov-modulated risk model. N. Amer. Actuarial J. 11, 65–76.
  • Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Prob. 4, 390–413.
  • Zhou, X. (2007). Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. J. Appl. Prob. 44, 1012–1030.