Remarks on the stochastic integral

In Karandikar-Rao 11], the quadratic variation M, M] of a (local) martingale was obtained directly using only Doob’s maximal inequality and it was remarked that the stochastic integral can be defined using M, M], avoiding using the predictable quadratic variation 〈M, M〉 (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in 11], we construct ∫ f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L(X) of predictable processes f such that |f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new.We then discuss the vector stochastic integral ∫ 〈f, dY〉 where f is ? ^d valued predictable process, Y is ? ^d valued semimartingale. This was defined by Jacod 6] starting from vector valued simple functions. Memin 13] proved that for (local) martingales M¹, … M ^d : If N ⁿ are martingales such that N _t ⁿ → N _t for every t and if ?f ⁿ such that N _t ⁿ = ∫ 〈f ⁿ , dM〉, then ?f such that N = ∫ 〈f, dM〉.Taking a cue from our characterization of L(X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above.This completeness result is an important step in the proof of the Jacod-Yor 4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing.