Probability Seminar Jonas Kremer (Bergische Universität Wuppertal)

On the long-time behavior of affine processes

In this talk we study stochastic stability properties of affine processes. An affine process with state space $\mathbb{R}_{\geq0}^m\times\mathbb{R}^n$, where $m,\thinspace n\in\mathbb{Z}_{\geq0}$ with $m+n>0$, is a time-homogeneous Markov process $(X_t)_{t\geq0}$ taking values in $\mathbb{R}_{\geq0}^m\times\mathbb{R}^n$, whose $\log$-characteristic function depends in an affine way on the initial value of the process, that is, there exist functions $\phi$ and $\psi=(\psi_1,\ldots,\psi_{m+n})$ such that
\[\mathbb{E}\left.\left[\mathrm{e}^{\langle u,X_t\rangle}\thinspace\right\vert\thinspace X_0=x\right]
= \exp\left(\phi(t,u)+\langle \psi(t,u),x\rangle\right),\] for all $u\in\mathrm{i}\mathbb{R}^{m+n}$, $t\geq0$, and $x\in\mathbb{R}_{\geq0}^m\times\mathbb{R}^n$. We provide sufficient conditions for the existence of limiting distribution of a conservative affine process. Our main theorem extends and unifies some known results for OU-type processes on $\mathbb{R}^{n}$ and one-dimensional CBI processes (with state space $\mathbb{R}_{\geqslant0}$). Finally, we present some results on the (exponential) ergodicity with respect to the Wasserstein distance and total-variation distance for interesting subclasses of affine processes.