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The Resolvent and Expected Local Times for Markov-Modulated Brownian Motion with Phase-Dependent Termination Rates

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Lothar Breuer
Lothar Breuer*
Affiliation:
University of Kent
*
∗ Postal address: Institute of Mathematics and Statistics, University of Kent, Canterbury CT2 7NF, UK. Email address: [email protected]
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Abstract

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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.

Keywords

Markov-modulated Brownian motionlocal timeresolventMarkov-additive process

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; L'evy processes 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 430 - 438
Copyright
© Applied Probability Trust 

References

Albrecher, H., Cheung, E. and Thonhauser, S. (2011). Randomized observation times for the compound Poisson risk model: Dividends. ASTIN Bull. 41, 645–672.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Breuer, L. (2008). First passage times for Markov additive processes with positive Jumps of phase type. J. Appl. Prob. 45, 779–799.Google Scholar
Breuer, L. (2012). Exit problems for reflected Markov-modulated Brownian motion. J. Appl. Prob. 49, 697–709.Google Scholar
Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. Doctoral Thesis, Universiteit van Amsterdam.Google Scholar
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.CrossRefGoogle Scholar