I'm having trouble with the following question:
Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal F_t\}_{t\geqslant 0}, P)$. Using It$\hat{\text{o}}$'s formula, find the semimartingale decomposition of the following process
$$X_t = t + e^{W_t}$$
I know this might be a simple problem but I'm having problems as to where to start. Any help would be appreciated, thanks.
Im surprised you didn't know this, Doob, but anyway:
Apply Ito on $X_t = t+e^{W_t}$ to get
$$ dX_t = e^{W_t} dW_t + (1+\frac{1}{2}e^{W_t})dt $$ In integral form this is $$ X_t = X_0 + \int^t_0 e^{W_u} dW_u + \int^t_0 (1+\frac{1}{2}e^{W_u})du $$ The $dW_u$-integral is local Martingale since it is an integral against Brownian Motion, and the $du$-integral is finite variation since it is integral against finite variation. So it is the form $X_t = X_0 + M_t + A_t$ where $M$ is local martingale, $A$ is finite variation.