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 ℝ d valued predictable process, Y is ℝ d valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M 1 , … 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.