Girsanov compensation with square-integrable input ($X=\int YdW,|Y|\le C,M_T\in L^2(P)⟹M-⟨M,X⟩\text{ is a true Q-martingale}$)

Let $Z=\exp (X-⟨X⟩/2)$ and $dQ=Z_{T}dP$. The Girsanov theorem first makes $N=M-⟨M,X⟩$ a continuous local Q-martingale. Boundedness of $Y$ gives $⟨X{⟩}_T\le C^{2}T$ and ${\mathbb{E}}_P{Z}_T^2\le e^{C^{2}T}$: write $Z_T^2=\mathcal{E}(2X{)}_T{e}^{⟨X{⟩}_T}$ and apply Novikov's condition to $2X$.

The quadratic covariation bound gives ${\sup }_{s\le T}|⟨M,X{⟩}_s|\le C\sqrt{T}⟨M{⟩}_T^{1/2}$. For a square-integrable martingale $M$, both its maximal square moment and its expected terminal bracket are finite. Cauchy-Schwarz inequality with the density $Z_T$ therefore yields ${\mathbb{E}}_Q{\sup }_{s\le T}|N_s|<\infty$. The dominated limit of its stopped conditional identities proves the true martingale property. Mere P-integrability of $M_T$ does not suffice.