Generalization of the anticipative Girsanov theorem

We study the Itô formula and Girsanov theorem in the anticipative setting using the stochastic integral of adapted and instantly independent processes. The results of the present paper extend several of the previously known theorems. The generalization presented here can be summarized as a domain extension as we allow for a more general class of processes to be treated by the Itô formula and more general shifts to be used in the change of measure in the Girsanov theorem. Finally, we apply our results to present a toy problem of the Black–Scholes formula for a market that knows the future but not the past.