= Girsanov compensation with square-integrable input
{c}
{title2=$X=\int Y\,dW,\ |Y|\leq C,\ M_T\in L^2(P)\implies M-\langle M,X\rangle\text{ is a true Q-martingale}$}
Let $Z=\exp(X-\langle X\rangle/2)$ and $dQ=Z_TdP$. The <Girsanov theorem> first makes $N=M-\langle M,X\rangle$ a continuous local Q-martingale. Boundedness of $Y$ gives $\langle X\rangle_T\leq C^2T$ and $\mathbb E_PZ_T^2\leq e^{C^2T}$: write $Z_T^2=\mathcal E(2X)_T e^{\langle X\rangle_T}$ and apply <Novikov's condition> to $2X$.
The <quadratic covariation> bound gives $\sup_{s\leq T}|\langle M,X\rangle_s|\leq C\sqrt T\langle M\rangle_T^{1/2}$. 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 $\mathbb E_Q\sup_{s\leq T}|N_s|<\infty$. The dominated limit of its stopped conditional identities proves the true martingale property. Mere P-integrability of $M_T$ does not suffice.
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