On the Process of the Eigenvalues of a Hermitian Lévy process

Victor Pérez-Abreu () and Alfonso Rocha-Arteaga ()
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Victor Pérez-Abreu: CIMAT, Departamento de Probabilidad y Estadística
Alfonso Rocha-Arteaga: Universidad Autónoma de Sinaloa, Facultad de Ciencias Físico-Matemáticas

A chapter in The Fascination of Probability, Statistics and their Applications, 2016, pp 231-249 from Springer

Abstract: 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 $$X(t-)$$ 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.

Keywords: Dyson–Brownian motion; Infinitely divisible random matrix; Bercovici–Pata bijection; Matrix semimartingale; Simultaneous jumps; Non-colliding process; Rank one perturbation; Stochastic differential equation with jumps (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-25826-3_11

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DOI: 10.1007/978-3-319-25826-3_11

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