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On the process of the eigenvalues of a Hermitian Lévy process. (English) Zbl 1354.60050

Podolskij, Mark (ed.) et al., The fascination of probability, statistics and their applications. In honour of Ole E. Barndorff-Nielsen. Cham: Springer (ISBN 978-3-319-25824-9/hbk; 978-3-319-25826-3/ebook). 231-249 (2016).
Summary: The dynamics of the eigenvalues (semimartingales) of a Lévy process \(X\) with values in Hermitian matrices is described in terms of Itô stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian motion. The simultaneity of the jumps of the eigenvalues of \(X\) is also studied. If \(X\) has a jump at time \(t\) two different situations are considered, depending on the commutativity of \(X(t)\) and \(X(t-)\). In the commutative case all the eigenvalues jump at time \(t\) only when the jump of \(X\) is of full rank. In the noncommutative case, \(X\) jumps at time \(t\) if and only if all the eigenvalues jump at that time when the jump of \(X\) is of rank one.
For the entire collection see [Zbl 1337.60008].

MSC:

60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
60J65 Brownian motion
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