Affine Volterra processes

TL;DR

This paper introduces affine Volterra processes, a class of stochastic convolution equations with affine coefficients, extending classical affine diffusions to non-semimartingale, non-Markovian settings, with explicit representations and existence results.

Contribution

It defines affine Volterra processes, provides explicit exponential-affine formulas for their Fourier-Laplace transforms, and proves existence and uniqueness without infinite-dimensional stochastic analysis.

Findings

Explicit exponential-affine formulas for Fourier-Laplace functionals.

Existence and uniqueness of solutions for specific state spaces.

Generalization of rough volatility models in finance.

Abstract

We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.