Integration with respect to finite variation processes Hey can anyone explain me (or recommend a book) how we construct stochastic integral of (not necessarily continuous) process $H$ with respect to process with finite variation (also not continuous)?

The stochastic integral is usually constructed for adapted càglàd (continuous on the left, limit on the right) processes as integrands and semimartingales $\big($which are càdlàg (continuous on the right, limit on the left) processes and include the FV-processes$\big)$ as integrator by using predictable simple processes and the property that the predictable simple processes are dense under the ucp-topology in the set of adapted càglàd processes. From this you can extend it to a bigger class of integrands. But all this you can read for example in the book of Protter ("Stochastic Integration and Differential Equations").
By the way, for integrators which are FV-processes you can also directly use the Lebesgue-Stieltjes integration path-by-path. A theorem states (also in the book I recomended) that the Lebesgue-Stieltjes integral and the stochastic integral are indistinguishable in that case.
