  For any real-valued stochastic process $X$ with c\'rdl\'rg paths we define
non-empty family of processes which have locally finite total variation, have
jumps of the same order as the process $X$ and uniformly approximate its paths
on compacts. The application of the defined class is the definition of
stochastic integral with semimartingale integrand and integrator as a limit of
pathwise Lebesgue-Stieltjes integrals. This construction leads to the
stochastic integral with some correction term (different from the Stratonovich
integral). We compare the obtained result with classical results of Wong-Zakai
and Bichteler on pathwise stochastic integration. As a "byproduct" we obtain an
example of a series of double Skorohod maps of a standard Brownian motion,
which is not a semimartingale.
