On pathwise uniform approximation of processes with c\`adl\`ag trajectories by processes with finite total variation

Abstract

For any real-valued stochastic process X with c\`adl\`ag paths we define non-empty family of processes, which have finite total variation, have jumps of the same order as the process X and uniformly approximate its paths: This allows to decompose any real-valued stochastic process with c\`adl\`ag paths and infinite total variation into a sum of uniformly close, finite variation process and an adapted process, with arbitrary small amplitude but infinite total variation. Another application of the defined class is the definition of the stochastic integral with respect to the process X as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term.