Continuous affine Volterra processes: Ergodicity, statistics and regularity of the occupation measure

Periodic seminar of the Department of Mathematics
10 November 2023
Start time 
11:00 am
PovoZero - Via Sommarive 14, Povo (Trento)
Seminar Room "-1" (Povo 0) and Zoom (please contact to get the code)
Dipartimento di Matematica
Target audience: 
University community
UniTrento students
Contact person: 
Prof. Luigi Amedeo Bianchi, Prof. Stefano Bonaccorsi, Prof. Michele Coghi
Contact details: 
Università degli Studi Trento 38123 Povo (TN) - Staff Dipartimento di Matematica
+39 04 61/281508-1625-1701-3898-1980-1511
Martin Friesen (Dublin City University)


We study limit distributions, stationary processes, and ergodicity for
continuous affine Volterra processes. Firstly, we prove the existence of
limit distributions and stationary processes for affine Volterra
processes on $\R_+^m$ obtained from
  X_t = x_0 + \int_0^t k(t-s)(b+\beta X_s)ds + \int_0^t
where $\sigma(x) = \mathrm{diag}(\sigma_1 \sqrt{x_1}, \dots, \sigma_m
\sqrt{x_m})$. Although the process is non-markovian, its limit
distribution is independent of the initial state $x_0$ if and only if $k
\not \in L^1(\R_+)$. Afterward, we prove the law-of-large numbers and
deduce that the corresponding stationary process is ergodic and mixing.
As an application we consider the maximum-likelihood estimation of the
drift parameter $b$ for continuous and discrete high-frequency
observations. In the second part of this talk we address the behaviour
of the process at the boundary in terms of regularity of occupation
measures at the boundary of the state space.
This talk is partially based on joint works with Mohamed Ben Alaya and
Pen Jin.