# Markovian lifts of positive semidefinite affine Volterra type processes

## TL;DR

This paper introduces Markovian lifts of matrix-valued affine Volterra processes, including Volterra Wishart and pure jump processes, with applications to multivariate rough volatility modeling.

## Contribution

It develops new Markovian representations for affine Volterra processes, including fractional kernels and jump processes, expanding their applicability in multivariate stochastic volatility models.

## Key findings

- Constructed Volterra Wishart processes with fractional kernels.
- Developed multivariate Hawkes type jump processes.
- Applied these processes to multivariate rough volatility modeling.

## Abstract

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein Uhlenbeck processes whose state space are matrix valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston type model.