On the process of the eigenvalues of a Hermitian Lévy process

Victor Pérez-Abreu; Alfonso Rocha-Arteaga

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arXiv · 1505.05125

On the process of the eigenvalues of a Hermitian Lévy process

Victor Pérez-Abreu·Alfonso Rocha-Arteaga

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Abstract

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.

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BibTeX RIS

Victor Pérez-Abreu, Alfonso Rocha-Arteaga. 2015-06-25. On the process of the eigenvalues of a Hermitian Lévy process. https://arxiv.org/abs/1505.05125

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