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

Abstract:

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
\[
  X_t = x_0 + \int_0^t k(t-s)(b+\beta X_s)ds + \int_0^t
k(t-s)\sigma(X_s)dB_s
\]
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.
\vskip2mm
This talk is partially based on joint works with Mohamed Ben Alaya and
Pen Jin.