Stochastic integration with respect to a sequence of semimartingales

We propose a theory of stochastic integration with respect to a sequence of semimartingales, starting from the theory of cylindrical integration with respect to a sequence of square-integrable martingales and with respect to a sequence of processes with finite variation. Indeed, by making use of an appropriate change in probability, we replace the integral with respect to a sequence of semimartingales with the sum of an integral with respect to a sequence of square integrable martingales and an integral with respect to a sequence of predictable processes with integrable variation. We show that, with our definition, the stochastic integral keeps some good properties of the integral with respect to a finite-dimensional semimartingale, such as invariance with respect to a change in probability and the so-called ``Memin's theorem". There are however some differences with the finite-dimensional case, and some ``bad properties'', which are pointed out by some examples.