Let $(\Omega,\mathcal F, \mathbb F,\mathbb P)$ be a filtered probability space under the usual conditions and suppose $\mathbb Q\sim\mathbb P$ is an equivalent probability measure. Let $X$ be a $\mathbb P$-semimartingale and $H$ a predictable process $\mathbb P$-integrable with respect to $X$. 


By Girsanov-Meyer theorem we know that $X$ is also a $\mathbb Q$ semimartingale. So we can ask: 


1) Is $H$ $Q$-integrable with respect to $X$?


2) If so, is the stochastic integral $\int H dX$ the same under any of the two probabilities?



Is the stochastic integral invariant under equivalent change of probability?
The measure-invariance of stochastic integrals with (locally) bounded integrands is shown in Meyer (1976, VI.26) or Dellacherie & Meyer (1978, VIII.12). Analogous statement is available for general integrands, e.g. Protter (2004, Theorem II.14).


*Meyer, P. A.*, [**Un cours sur les integrales stochastiques**](http://www.numdam.org/item?id=SPS_1976__10__245_0), Semin. Probab. X, Univ. Strasbourg 1974/75, Lect. Notes Math. 511, 245-400 (1976). [ZBL0374.60070](https://zbmath.org/?q=an:0374.60070).


*Dellacherie, Claude; Meyer, Paul-Andre*, Probabilities and potential. Transl. from the French, North-Holland Mathematics Studies, 29. Amsterdam - New York -Oxford: North-Holland Publishing Company. VIII, 189 p. (1978). [ZBL0494.60001](https://zbmath.org/?q=an:0494.60001).


*Protter, Philip E.*, Stochastic integration and differential equations, Applications of Mathematics 21. Berlin: Springer (ISBN 3-540-00313-4/hbk; 978-3-642-05560-7/pbk). xiii, 415 p. (2004). [ZBL1041.60005](https://zbmath.org/?q=an:1041.60005).


