  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.
