Levy Systems and Path Decompositions

Abstract

Itô [21] introduced the idea of a point process attached to a Markov process X, and subsequent work of Weil [42], Getoor [11], [12] and Maisonneuve [29] has shown that the existence of a suitably Markovian Lévy system for such a point process can be instrumental in establishing path decompositions of the Markov process. A path decomposition, or splitting time theorem, is a result to the effect that some fragment of the trajectory of X is conditionally independent of some other fragment given suitable conditioning variables, usually with one or more of the fragments being conditionally Markovian. Millar [32] gives a survey of such results, and more recent work may be found in the papers of Getoor, Pittenger, and Sharpe: [12], [14], [15], [16], [17], [18], [36], [37], [40]. Lévy systems suitable for deriving path decompositions were constructed in varying degrees of generality by Watanabe [41] and Benveniste and Jacod [2] for the point process of jumps, and by Itô [21], Dynkin [10] and Maisonneuve [28] for point processes of excursions.

Research supported in part by NSF Grant 78-25301