P and Q martingales

Question

Let $\mathbb{P},\mathbb{Q}$ be two equivalent probability measures. Could someone come up with an example of a $\mathbb{P}$-martingale, which is not a $\mathbb{Q}$-martingale and another example of a martingale which is both a $\mathbb{P}$ and $\mathbb{Q}$-martingale?

Answer 1

By the Girsanov theorem: when $W_t$ is a Brownian motion under $\mathbb P$ and $$ \frac{d\mathbb Q}{d\mathbb P}=e^{W_t-t/2} $$ then $W_t-t$ is a BM under $\mathbb Q\,.$ I hope you know which of $W_t$ and $W_t-t$ is a martingale under $\mathbb P$ resp. $\mathbb Q\,.$ A constant is an example that is a martingale under both measures. A more interesting example is a Brownian motion that is independent of $W_t\,.$