Martingale transfer under a density process

Let $Z$ be a positive uniformly integrable martingale defining $d\tilde{\mathbb{P}}=Z_{\infty }d\mathbb{P}$. If $Y$ is bounded and $ZY$ is a true $\mathbb{P}$-martingale, then $Y$ is a $\tilde{\mathbb{P}}$-martingale. This follows from the conditional-expectation change-of-measure identity.

The density-product proof of the Girsanov theorem makes drift-corrected zero-starting increments a local martingale under the new measure. Adding the initial random variable requires its integrability under that measure. This is automatic if the density is one at time zero, because the measures agree on the initial sigma-algebra. Without this normalization it can fail: take $P(n)=2^{-n}$, $Q(n)=c/n^2$, constant $M_t=\log (Q(n)/P(n))$ and constant $X_t=n$. These are P-martingales and the prescribed exponential density is valid, but $X_0$ is not Q-integrable.

Let $Z$ be a positive uniformly integrable density martingale. If $Y$ is bounded and $ZY$ is a local martingale under the original measure, the product is a true martingale: its stopped absolute values are dominated by a fixed multiple of the uniformly integrable stopped density. Bayes then makes $Y$ a martingale under the new measure. Stop a locally bounded continuous $Y$ at absolute-value levels to obtain the local version.