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

Abstract

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 p ≥ 1 $p\ge1$ , we introduce the space of regulated signals f : [ a , b ] → W $f:[a,b]\rightarrow W$ ( a 0 $\delta>0$ by signals whose total variation is of order δ 1 − p $\delta^{1-p}$ as δ → 0 + $\delta\rightarrow0+$ 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.

Keywords

• regulated path
• total variation
• p-variation
• truncated variation
• the Riemann-Stieltjes integral
• the Loéve-Young inequality