Geodesic
Subordinated Markov Branching Processes and Lévy Processes
Serdica Mathematical Journal, Tome 40 (2014) no. 3-4, pp. 183-208.

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We consider the jump structure of the subordinated Lévy processes and subordinated Markov branching processes. Subordination provides a method of constructing a large subclass of Markov or Lévy processes Y (t) = X(T (t)), where X(t) is a Markov or Lévy process and T (t) is a continuous time subordinator independent of X(t); that is a Lévy process with positive increments and T (0) = 0. Let X(t) be a Lévy process. Then subordination preserves the independence and stationarity of the increments, but it changes their amplitudes and the total mass of the Lévy measure. Let X(t) be a Markov branching process. Then subordination (owing to the independence of X(t) and T (t)) preserves the Markov property, but it disturbs the branching property. The infinitesimal generator of the subordinated process Y (t) involves the total progeny of reproduction. The intensity of the jump times depends on the subordinator’s Bernstein function. 2010 Mathematics Subject Classification: 60J80, 60K05.
Keywords: subordinated Lévy processes, Poisson random measure, pure jump Markov processes, Kolmogorov backward equation 
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Mayster, Penka. Subordinated Markov Branching Processes and Lévy Processes. Serdica Mathematical Journal, Tome 40 (2014) no. 3-4, pp. 183-208. http://geodesic.mathdoc.fr/item/SMJ2_2014_40_3-4_a0/