[工程科技]AMarkovpropertyforset-indexedprocesses.pdf

4 0 A Markov Property For Set-Indexed Processes 0 2 c R.M. Balan∗† B.G. Ivanoff‡§ e D February 1, 2008 7 1 ] Abstract R P We consider a type of Markov property for set-indexed processes which . is satisfied by all processes with independent increments and which allows h us to introduce a transition system theory leading to the construction of t a the process. A set-indexed generator is defined such that it completely m characterizes the distribution of the process. [ Keywords: set-indexed process; Markov property; transition system; generator. 1 v 1 Introduction 0 5 3 The Markov property is without doubt one of the most appealing notions that 2 exists in the classical theory of stochastic processes and many processes mod- 1 elling physical phenomena enjoy it. Attempts have been made to generalize this 4 concept to processes where the index set is not totally ordered. 0 / However, to define a Markov property for processes indexed by an uncount- h able partially ordered set is not a straightforward task for someone who has t a as declared goals to prove that: (i) all processes with independent increments m possess this property; (ii) there exists a systematic procedure which allows us to : construct a general process which enjoys this property; and (iii) we can define a v i