On pathwise stochastic integration with respect to semimartingales

Abstrakt

For any real-valued stochastic process X with càdlàg 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. Using properties of a functional called truncated variation we compare the obtained result with classical results of Wong–Zakai and Bichteler on pathwise stochastic integration.