  Complementary regularity between the integrand and integrator is a well known
condition for the integral $\int_0^T f(r) \, \mathrm{d} g(r)$ to exist in the
Riemann-Stieltjes sense. This condition also applies to the multi-dimensional
case, in particular the 2D integral $\int_{[0, T]^2} f(s,t) \, \mathrm{d}
g(s,t)$. In the paper, we give a new condition for the existence of the
integral under the assumption that the integrator $g$ is a Volterra covariance
function. We introduce the notion of strong H\"{o}lder bi-continuity, and show
that if the integrand possess this property, the assumption on complementary
regularity can be relaxed for the Riemann-Stieltjes sums of the integral to
converge.
