Girsanov's Theorem and Equivalent Martingale Measures

1. Two probability measures $\mathbb{P}$ and $\mathbb{Q}$ are equivalent if:

2. The Girsanov kernel $Z_t = \exp\!\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right)$ satisfies which SDE?

3. Under $\mathbb{P}$, a stock follows $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t^{\mathbb{P}}$. After the Girsanov change of measure with market price of risk $\theta = (\mu - r)/\sigma$, what are the dynamics of $S_t$ under $\mathbb{Q}$?

4. In a complete Black-Scholes market, the equivalent martingale measure is unique.

5. The Novikov condition $\mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \theta_t^2 \, dt\right)\right] < \infty$ is required to ensure:

6. Under the forward measure $\mathbb{Q}^T$ (taking the zero-coupon bond $P(t,T)$ as numeraire), which of the following is a $\mathbb{Q}^T$-martingale?