Pathwise stochastic integration with finite variation processes uniformly approximating c\`{a}dl\`{a}g processes

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.