Young-Stieltjes integrals with respect to Volterra covariance functions

TL;DR

This paper introduces a new condition for the existence of Young-Stieltjes integrals with respect to Volterra covariance functions, relaxing traditional regularity requirements by using strong Hölder bi-continuity of the integrand.

Contribution

The paper proposes a novel condition based on strong Hölder bi-continuity that allows for the existence of Young-Stieltjes integrals with Volterra covariance functions, extending classical regularity conditions.

Findings

New existence condition for Young-Stieltjes integrals with Volterra covariance functions.

Relaxation of regularity assumptions on the integrand.

Convergence of Riemann-Stieltjes sums under the new condition.

Abstract

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.