Stability of stochastic integrals under change of filtration

Let (Ω,,P) be a probability space equipped with two filtrations {_t} and {_t} satisfying the usual conditions. Assume that X is a semimartingale and that h is locally bounded and predictable for each of the two filtrations {_t} and {_t}. New examples of such processes are given. Utilizing and extending partial results of Zheng (1982), this paper extends the available results on the relationship between the stochastic integral processes ∫^th_s dX_s taken respectively in the sense of {_t} and of {_t}. In particular, it is shown that these stochastic integrals differ at most by a continuous process with quadratic variation defined and equal to 0. If both stochastic integrals are {_t∩_t} semimartingales, then it is proved that the stochastic integral ∫^th_s dX _s taken in {_t} sense is indistinguishable from that taken in {_t} sense.