  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.
