Stationarity and limiting distributions of affine Volterra processes

Abstract: An affine process is characterized by the feature that its characteristic function can be expressed in a semi-explicit form in terms of a solution of a Riccati-type equation and includes the Heston model as a particular example. Recent observations on intra-day stock market data suggest that the volatilities are rougher than predicted by existing Markovian models based on the Brownian motion (e.g. the Heston model). To accommodate for this feature, it was proposed in the literature to model these observations by their rough counterparts, the so-called affine Volterra processes. In this talk, we first provide a brief overview of the vastly growing literature on affine Volterra processes. One remarkable feature of affine Volterra processes is that these processes still allow expressing their characteristic function in a semi-explicit form now in terms of a nonlinear Volterra Riccati equation. Based on a detailed study of this Volterra equation, we analyze possible limiting distributions and stationary processes for affine Volterra processes.