Young-Stieltjes integrals with respect to Volterra covariance functions

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.