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

Abstract

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.