Abstract
The Lévy system for a Markov process X provides a convenient description of the distribution of the totally inaccessible jumps of the process. We examine the effect of time change (by the inverse of a not necessarily strictly increasing CAF A) on the Lévy system, in a general context. They key to our time-change theorem is a study of the “irregular” exits from the fine support of A that occur at totally inaccessible times. This permits the construction of a partial predictable exit system (à la Maisonneuve).
The second part of the paper is devoted to some implications of the preceding in a (weak, moderate Markov) duality setting. Fixing an excessive measure m (to serve as duality measure) we obtain formulas relating the “killing” and “jump” measures for the time-changed process to the analogous objects for the original process. These formulas extend, to a very general context, recent work of Chen, Fukushima, and Ying. The key to our development is the Kuznetsov process associated with X and m, and the associated moderate Markov dual process \(\hat X\). Using \(\hat X\) and some excursion theory, we exhibit a general method for constructing excessive measures for X from excessive measures for the time-changed process.
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Fitzsimmons, P., Getoor, R. (2009). Lévy Systems and Time Changes. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLII. Lecture Notes in Mathematics(), vol 1979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01763-6_9
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