stochastic calculus - Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$

Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb P$-Brownian motion.

Let $dX_t = \theta_tdt +dW_t$ be an Ito process where $(\theta_t)_{t \in [0,T]}$ is $\mathscr{F}_t$-adapated and $E[\int_0^T \theta_s^2 ds] < \infty$ and

$$Y_t := X_tL_t, \ \ L_t = \exp Z_t, \ \ Z_t = -\int_0^t \theta_s dW_s - \frac{1}{2}\int_0^t\theta_s^2ds$$

Suppose Novikov's condition holds.

Prove $Y_t$ is a $(\mathscr{F}_t, \mathbb{P})$-martingale.

---

I was able to show that $dY_t = (L_t - \theta_tY_t)dW_t$ from deriving that $dZ_t = -\theta_tdW_t -\frac 1 2 \theta_t^2 dt$ and $dL_t = e^{Z_t}(-\theta_tdW_t)$.

Assuming that this is right, does the fact that there is no drift term in $dY_t$ already establish that $Y_t$ is a $(\mathscr{F}_t, \mathbb{P})$-martingale and not merely that it is a local martingale or merely that $E[Y_t | \mathscr{F}_u] = Y_u$?

---

Edit: It seems that according to this, a solution of an SDE is a martingale if it is unique.

1. 
   

   $E[Y_0^2] = E[X_0^2]< \infty$, I guess? No initial condition is given for $X_t$

   
   


2. 
   

   Show $\exists K \in \mathbb R$ s.t.

   
   

$|L_t - \theta_tx| \le K(1+|x|)$

$|(L_t - \theta_tx) - (L_t - \theta_ty)| \le K|x-y|$

We have:

$$|L_t - \theta_tx| \le |L_t| + |\theta_t||x| \le |\theta_t|(1+|x|)$$

$$|(L_t - \theta_tx) - (L_t - \theta_ty)| \le |\theta_t||x-y|$$

I don't suppose $E[\int_0^T \theta_s^2 ds] < \infty$ means that $\theta_t$ is bounded, does it?