A multi-dimensional martingale for Markov additive processes and its applications. (English) Zbl 0961.60081

Asmussen, Søren; Kella, Offer

Adv. Appl. Probab. 32, No. 2, 376-393 (2000).

This paper is concerned with an extension of martingales connected with Lévy and some associated processes to finite Markov additive processes \((X_t,J_t)\). Here \(J_t\) is a Markov jump process with a finite state space and \(X_t\) is the additive component. For such a process, the matrix with elements \(E_i e^{\alpha X_t}\text{\textbf{1}}_{[J_t= j]}\) has the form \(e^{tF(\alpha)}\) for some matrix \(F(\alpha)\). The paper considers a multidimensional version \(e^{\alpha X_t} e^{-F(\alpha)}\) as well as martingales for \(Z_t= X_t+ Y_t\) and demonstrates the applicability of these martingales with various examples, in particular storage models, queues, Brownian motion and fluid models.

Reviewer: O.K.Zakusilo (Kyïv)

MSC:

60J27	Continuous-time Markov processes on discrete state spaces
60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60G48	Generalizations of martingales

Keywords:

extension of martingales; storage models; queues; Brownian motion; fluid models