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A jump-type SDE approach to real-valued self-similar Markov processes. (English) Zbl 1326.60109
The author studies self-similar Markov processes in continuous time on the state spaces \([0,\infty[\) and \(\mathbb R\), where sometimes, in addition, a trap at zero is assumed. By some transform, it suffices to study these processes with the self-similarity index 1 only. By a classical result of J. Lamperti [Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 205–225 (1972; Zbl 0274.60052)], all self-similar positive processes can be classified as time-changes of exponentials of Lévy processes. In particular, the only continuous positive self-similar Markov processes are the squared Bessel processes.
The paper under review is devoted to the more complicated case with state space \(\mathbb R\) under additional serious restrictions. Motivated by the recent work of L. Chaumont et al. [Bernoulli 19, No. 5B, 2494–2523 (2013; Zbl 1284.60077)], the author uses jump-type SDEs and derives a representation for symmetric self-similar processes on \(\mathbb R\) whose absolute values admit decreasing jumps only and which leave zero continuously. The representation uses some spectrally negative Lévy process killed at certain rates and with certain Laplace exponents. Prominent examples for the theory of the author are (suitably renormalized) one-dimensional Dunkl processes \((X_t)_t\) whose absolute values \((|X_t|)_t\) form squared Bessel processes.

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G18 Self-similar stochastic processes
60J75 Jump processes (MSC2010)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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