Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 2, Pages 149-157

An application of the nonselfadjoint operators theory in the study of stochastic processes

Lyazid Abbaoui1 and Latifa Debbi2

1Department of Mathematics, University Ferhat Abbas, Elmaâbouda, Sétif 19000, Algeria
2Department of Mathematics, University of M'sila, B.P 166 Echbilia, M'sila 28000, Algeria

Received 11 May 2003; Revised 1 March 2004

Copyright © 2004 Lyazid Abbaoui and Latifa Debbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The theory of operator colligations in Hilbert spaces gives rise to certain models for nonselfadjoint operators, called triangular models. These models generalize the spectral decomposition of selfadjoint operators. In this paper, we use the triangular model to obtain the correlation function (CF) of a nonstationary linearly representable stochastic process for which the corresponding operator is simple, dissipative, nonselfadjoint of rank 1, and has real spectrum. As a generalization, we represent the infinitesimal correlation function (ICF) of a nonhomogeneous linearly representable stochastic field in which at least one of the operators has real spectrum.