Remarks on F\"ollmer's pathwise It\^o calculus

We extend some results about F"ollmer’s pathwise It^o calculus that have only been derived for continuous paths to c`adl`ag paths with quadratic variation. We study some fundamental properties of pathwise It^o integrals with respect to c`adl`ag integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise It^o integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of F"ollmer’s calculus.