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On operator fractional Lévy motion: integral representations and time-reversibility

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  06 June 2022

B. Cooper Boniece  and
Gustavo Didier
B. Cooper Boniece*
Affiliation:
University of Utah
Gustavo Didier*
Affiliation:
Tulane University
*
*Postal address: 155 South 1400 East, Salt Lake City, UT 84112, USA. Email address: [email protected]
**Postal address: 6823 St. Charles Avenue, New Orleans, LA 70118, USA. Email address: [email protected]
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Abstract

In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).

Keywords

Infinite divisibilityLévy processesoperator self-similarity

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G51: Processes with independent increments; L'evy processes 60H05: Stochastic integrals
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 493 - 535
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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