A coupling technique for stochastic comparison of functions of Markov Processes

The aim of this work is to obtain explicit conditions (i.e., conditions on the transition rates) for the stochastic comparison of Markov Processes. A general coupling technique is used to obtain necessary and sucient conditions for the construction of a coupling Markov Process which stays in a xed set K for all times and with given marginal processes. The strong stochastic comparison|or, more generally, the stochastic comparison through states functions|appears as a particular case. An example in the Reliability Theory is developed and proves the eciency of the method. Systems with multiple component types and redundant units are stochastically compared directly or through particular functions. Stochastic comparison is a useful tool in the study of complex stochastic systems. Stoyan's book (1983) gives a broad survey on the subject. In this paper, the classical stochastic ordering is generalized to the comparison of processes through state functions. For example, in a Stochastic Petri Net (Murata (1989)) where tokens are moved randomly across the places, the total load of the system (sum of all marks) is a quantity of interest for the saturation conditions. Furthermore, a mark in a place can have a cost. The associated linear combination of marks, is a measure of the cost of the corresponding state system. Two processes obtained by changing the transition rates, can be stochastically compared through such state functions. Let X =fXt; t 0g and Y =fYt; t 0g be two Markov processes with values in E and F respectively and '; two functions from E and F respectively into G (ordered by ). X and Y will be stochastically compared by means of ' and using the stochastic ordering in G. We shall call this kind of comparison the ' - comparison. The comparison of functions of Markov processes with a stochastically monotone Markov process has been already studied by Massey (1987). Whitt (1986) compares general non-Markovian processes by assuming that they can be completed by ad- ditional information to become Markovian and always by mean of a stochastically monotone Markov process. We shall use a direct technique developed in particular in Liggett (1985) for interacting particle systems, the comparison by coupling. For