A new inequality for the Riemann-Stieltjes integrals driven by irregular signals in Banach spaces

Rafał M Łochowski

doi:10.1186/s13660-018-1611-4

Abstract

We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces, and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any pge1, we introduce the space of regulated signals f:[a,b]rightarrow W (a< b are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy delta>0 by signals whose total variation is of order delta^{1-p} as deltarightarrow0+ and prove that they satisfy the assumptions of the theorem. Finally, we derive more exact, rate-independent characterisations of the irregularity of the integrals driven by such signals.

Highlights

The first aim of this paper is a generalisation of the results of [1] and [2] to the functions attaining their values in R but in more general spaces
To obtain more precise results, for any p ≥ 1, we introduce the space U p([a, b], W ) of regulated functions/signals f : [a, b] → W (a < b are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy δ > 0 by functions whose total variation is of order δ1–p as δ → 0+
In the modern theory of rough paths developed by Terry Lyons, the existence of integrals of the form t 0 f dxs and y satisfying y(t)

Summary

Introduction

The first aim of this paper is a generalisation of the results of [1] and [2] to the functions attaining their values in R but in more general spaces. The restriction to the Banach spaces stems from the fact that the method of our proof requires multiple application of summation by parts and proceeding to the limit of a Cauchy sequence, which may be done in a straightforward way in any Banach space This way we will obtain a general theorem on the existence of the Riemann-Stieltjes integral along a path in some Banach space (E, · E) (with the integrand being a path in the space L(E, V ) of continuous linear mappings F : E → V , where V is another Banach space) and an improved version of the Loéve-Young inequality for integrals driven by irregular paths in this space. B f dg – f (a) g(b) – g(a) ≤ Cp,q V p f , [a, b] 1/p V q g, [a, b] 1/q

Here n

From the elementary inequality x