A new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Loéve-Young inequality

Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Lo\'{e}ve-Young inequality for the Riemann-Stieltjes integrals driven by irregular signals. Using this result we strenghten some results of Terry Lyons on the existence of solutions of integral equations driven by moderately irregular signals.