Prof. Dr. Ludger Overbeck: Rough and path-dependent affine models and their path-valued interpretation

Justus-Liebig-Universität Gießen, Germany

Abstract:

We first extend results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel and inhomogeneous drift and diffusion coefficients and in the case of affine drift and variance we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a time homogeneous kernel of convolution type we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition the coefficients are affine, we prove that the conditional Fourier-Laplace functional is exponential-affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.
Secondly we investigate the equivalence between affine coefficients of a path dependent continuous stochastic differential equations and the affine structure of the log Fourier-transform. Applications in mathematical finance include e.g. delayed Heston model.
In both cases the corresponding path processes are infinite-dimensional affine Markov processes.

This is joint work (partially in progress) with Julia Ackermann (Wuppertal), Boris Günther (Gießen) und Thomas Kruse (Wuppertal).