Infinite divisibility for stochastic processes and time change. (English) Zbl 1111.60028
The authors present some general results on infinite divisibility, self-decomposability and the class \(L_m\) (the class of measures \(\mu\) such that for any \(c\in (0,1)\) there is a distribution \(\rho^{(c)}\) such that \(\hat{\mu}(z)=\hat{\mu}(cz)\hat{\rho^{(c)}}(z)\), \(z\in \mathbb{R}^d\), and \(\hat{\mu}(z)=\hat{\mu}(cz)\hat{\rho^{(c)}}(z)\) also holds with some \(\rho^{(c)}\in L_{m-1}\)). Several relations between various concepts and the properties of such processes are given. Also, a new concept of temporal self-decomposability is introduced: the process is called temporal self-decomposable if for each \(c\in (0,1)\) there exist independent processes \(X^{(c)}\) and \(U^{(c)}\) such that
\(X\overset{d}{=} X^{(c)}+U^{(c)},\)
and \(X^{(c)}=\{X^{(c)}_t:\,\,t\geq 0\}\overset{d}{=} \{X_{ct}\,\, t\geq 0\}\).
The class of temporal self-decomposable processes is larger than the class of Lévy processes. On the other hand, under some restriction temporary self-decomposable processes are infinitely divisible. The notion of additive processes is also between the notion of Lévy processes and the notion of infinitely divisible processes. It is shown that an additive process in not always temporary self-decomposable, and vice versa. Various examples are given. Further, the time-change of stochastic processes is discussed, in particular, the authors present the results on the inheritance of infinite divisibility under time change when the base process (i.e. the process which is to be time-changed) is a Lévy process.
The class of temporal self-decomposable processes is larger than the class of Lévy processes. On the other hand, under some restriction temporary self-decomposable processes are infinitely divisible. The notion of additive processes is also between the notion of Lévy processes and the notion of infinitely divisible processes. It is shown that an additive process in not always temporary self-decomposable, and vice versa. Various examples are given. Further, the time-change of stochastic processes is discussed, in particular, the authors present the results on the inheritance of infinite divisibility under time change when the base process (i.e. the process which is to be time-changed) is a Lévy process.
Reviewer: Victoria Knopova (Kiev)
MSC:
| 60G51 | Processes with independent increments; Lévy processes |
| 60G18 | Self-similar stochastic processes |
Keywords:
self-decomposability; temporal self-decomposability; class \(L_m\)-property; time change; chronometerReferences:
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