Journal of Applied Probability

The resolvent and expected local times for Markov-modulated Brownian motion with phase-dependent termination rates

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


We consider a Markov-modulated Brownian motion (MMBM) with phase-dependent termination rates, i.e. while in a phase i the process terminates with a constant hazard rate ri ≥ 0. For such a process, we determine the matrix of expected local times (at zero) before termination and hence the resolvent. The results are applied to some recent questions arising in the framework of insurance risk. We further provide expressions for the resolvent and the local times at zero of an MMBM reflected at its infimum.

Article information

J. Appl. Probab., Volume 50, Number 2 (2013), 430-438.

First available in Project Euclid: 19 June 2013

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 local time resolvent Markov-additive process


Breuer, Lothar. The resolvent and expected local times for Markov-modulated Brownian motion with phase-dependent termination rates. J. Appl. Probab. 50 (2013), no. 2, 430--438. doi:10.1239/jap/1371648951.

Export citation


  • Albrecher, H., Cheung, E. and Thonhauser, S. (2011). Randomized observation times for the compound Poisson risk model: Dividends. ASTIN Bull. 41, 645–672.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.
  • Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.
  • Breuer, L. (2008). First passage times for Markov additive processes with positive jumps of phase type. J. Appl. Prob. 45, 779–799.
  • Breuer, L. (2012). Exit problems for reflected Markov-modulated Brownian motion. J. Appl. Prob. 49, 697–709.
  • Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. Doctoral Thesis, Universiteit van Amsterdam.
  • Jiang, Z. and Pistorius, M. (2008). On perpetual American put valuation and first-passage in a regime-switching model with jumps. Finance Stoch. 12, 331–355.