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Markov processes generated by generalized convolutions
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DOI：10.1007/s10986-026-09719-0.png)
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Abstract
En 中文
The generalized convolution is certain operation on the set of probability measures. It is defined by K. Urbanik (see [K. Urbanik, Generalized convolutions, Stud. Math., 23:217-245 1964], [J.K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Classes of measures closed under mixing and convolution. Weak stability, Stud. Math., 167(3):195-213, 2005], and [B.H. Jasiulis, Limit property for regular and weak generalized convolutions, J. Theor. Probab., 23(1):315-327, 2010]). They allow us to define and study the infinite divisibility of probability measures and to construct L & eacute;vy processes in the sense of generalized convolutions. This approach was undertaken in [M. Borowiecka-Olszewska, B.H. Jasiulis-Go & lstrok;dyn, J.K. Misiewicz, and J. Rosi & nacute;ski, L & eacute;vy Processes and stochastic integrals in the sense of generalized convolutions, Bernoulli, 21(4):2513-2551, 2015]. It turned out that L & eacute;vy processes with respect to generalized convolutions are Markov in the classical sense. Of course, Markov processes are not necessarily L & eacute;vy. In this paper, we investigate the converse: when Markov processes are L & eacute;vy in the sense of some generalized convolution?
Keywords:
weakly stable distribution
symmetric stable distribution
scale mixture
L & eacute
vy process
subordinated processes
stochastic integral
Journal
L
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0.7
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27
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0