Expansion of a filtration with a stochastic process: The information drift

When expanding a filtration with stochastic process it is easily possible for semimartingale no longer to remain in the enlarged filtration. Y. Kchia and P. Protter indicated way avoid this pitfall 2015, but they were unable give decomposition except special cases. We provide compute such decomposition, moreover we sufficient condition Itô processes (Corollary 3.7). This can be interpreted terms of statistical arbitrage Mathematical Finance.