Young-Stieltjes integrals with respect to Volterra covariance functions

2018/06/06 by Lim, Nengli
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1806.02214

Abstract

Complementary regularity between the integrand and integrator is a well known condition for the integral ∫₀^T f(r) d g(r) to exist in the Riemann-Stieltjes sense. This condition also applies to the multi-dimensional case, in particular the 2D integral ∫_{[0, T]²} f(s,t) 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 Hölder 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.

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