# 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.

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This talk is partially based on joint works with Mohamed Ben Alaya and

Pen Jin.