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

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.

MSC:

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