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)?
2026-04-06 20:30:59.1775507459
Integration with respect to finite variation processes
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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.