## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 14, Number 2 (2004), 1029-1054.

### Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes

#### Abstract

Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that the hitting probabilities of this additive process at lower levels have a matrix exponential form. We here study the hitting probabilities at upper levels, which do not have a matrix exponential form in general. These probabilities give the ruin probabilities in the terminology of the risk process. Our major interests are in their analytic expressions and their asymptotic behavior when the hitting level goes to infinity under light tail conditions on the jump sizes. To derive those results, we use a certain duality on the hitting probabilities, which may have an independent interest because it does not need any Markovian assumption.

#### Article information

**Source**

Ann. Appl. Probab., Volume 14, Number 2 (2004), 1029-1054.

**Dates**

First available in Project Euclid: 23 April 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1082737121

**Digital Object Identifier**

doi:10.1214/105051604000000206

**Mathematical Reviews number (MathSciNet)**

MR2052912

**Zentralblatt MATH identifier**

1057.60073

**Subjects**

Primary: 90B22: Queues and service [See also 60K25, 68M20] 60K25: Queueing theory [See also 68M20, 90B22]

Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx] 60G55: Point processes

**Keywords**

Risk process Markov additive process hitting probability decay rate Markov renewal theorem stationary marked point process duality

#### Citation

Miyazawa, Masakiyo. Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes. Ann. Appl. Probab. 14 (2004), no. 2, 1029--1054. doi:10.1214/105051604000000206. https://projecteuclid.org/euclid.aoap/1082737121