Inhomogeneous affine Volterra processes

TL;DR

This paper extends affine Volterra processes to inhomogeneous cases, providing moment bounds, integral equation representations, and applications to financial models like the rough Heston model.

Contribution

It introduces inhomogeneous affine Volterra processes, deriving new moment bounds, integral equations, and applications to finance, expanding the scope of previous homogeneous models.

Findings

Established existence of solutions for inhomogeneous Volterra equations.

Derived exponential-affine representation of the Fourier-Laplace functional.

Applied results to an inhomogeneous rough Heston model.

Abstract

We extend recent 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 $K(t,s)$ and inhomogeneous drift and diffusion coefficients $b(s,X_s)$ and $\sigma(s,X_s)$. In the case of affine $b$ and $\sigma \sigma^T$ we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a kernel of convolution type $K(t,s)=\overline{K}(t-s)$ we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition $b$ and $\sigma \sigma^T$ 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.