JUMPING MARKOV-PROCESSES

This paper is devoted to a systematic study of the basic properties of the so-called Jumping Markov Processes (JMP in short). By this we mea n a Markov process X = (X(t))(t greater than or equal to 0) taking val ues in an arbitrary measurable space (E, epsilon), and which is piecew ise-deterministic in the sense that it follows a ''deterministic'' pat h X(t) = f (t, X(0)) up to some random time tau(1), at which time it ' 'jumps'' to some random value X(tau 1), then it follows the path f (t - tau(1), X(tau 1)) up to another random time tau(2) > tau(1), and so on... Such processes had already been studied by M. H. A. Davis [3] in a particular case, but here the emphasis is on the characterization o f JMPs, in particular in terms of the structure of the martingales, an d on the properties of the basic objects (additive functionals, semima rtingales, semimartingale functions) usually associated with Markov pr ocesses. We also introduce a class of Markov processes which we call ' 'purely discontinuous'' and appear as suitable limits of JMP's.