  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.
