A class of fractional Ornstein-Uhlenbeck processes mixed with a Gamma distribution

Bianchi Luigi Amedeo, Università di Trento
Bonaccorsi Stefano, Università degli Studi di Trento
Tubaro Luciano, Università di Trento

We consider a sequence of fractional Ornstein-Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely. This is joint work with Stefano Bonaccorsi and Luciano Tubaro:~\cite{BBT2023} \bigskip \begin{thebibliography}{99} \bibitem{BBT2023} L.A.~Bianchi, S.~Bonaccorsi, L.~Tubaro, A Class of Fractional Ornstein\textendash Uhlenbeck Processes Mixed with a Gamma Distribution, Modern Stochastics: Theory and Applications, \textbf{10}(2022), 37--57. \end{thebibliography}

Area: CS16 - Stochastic Evolution Equations (Irene Benedetti)

Keywords: fractional Ornstein-Uhlenbeck processes, empirical means, Gamma mixing, stochastic Volterra equations, generalized Wright function

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