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On the Process of the Eigenvalues of a Hermitian Lévy process

In: The Fascination of Probability, Statistics and their Applications

Author

Listed:
  • Victor Pérez-Abreu

    (CIMAT, Departamento de Probabilidad y Estadística)

  • Alfonso Rocha-Arteaga

    (Universidad Autónoma de Sinaloa, Facultad de Ciencias Físico-Matemáticas)

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.

Suggested Citation

  • Victor Pérez-Abreu & Alfonso Rocha-Arteaga, 2016. "On the Process of the Eigenvalues of a Hermitian Lévy process," Springer Books, in: Mark Podolskij & Robert Stelzer & Steen Thorbjørnsen & Almut E. D. Veraart (ed.), The Fascination of Probability, Statistics and their Applications, pages 231-249, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-25826-3_11
    DOI: 10.1007/978-3-319-25826-3_11
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