  We develop precise bounds on the growth rates and fluctuation sizes of
unbounded solutions of deterministic and stochastic nonlinear Volterra
equations perturbed by external forces. The equation is sublinear for large
values of the state, in the sense that the state--dependence is negligible
relative to linear functions. If an appropriate functional of the forcing term
has a limit $L$ at infinity, the solution of the differential equation behaves
asymptotically like the underlying unforced equation when $L=0$, like the
forcing term when $L=+\infty$, and inherits properties of both the forcing term
and underlying differential equation for values of $L\in (0,\infty)$. Our
approach carries over in a natural way to stochastic equations with additive
noise and we treat the illustrative cases of Brownian and L\'evy noise.
