Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where
$$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$
$B_t$ – is a standard Brownian motion
I would like to show it is a true martingale, so I am looking at these sets:
$$\mathcal{S}_1:=\{ X^{T_n}_t : n\geq 1\} \text{ or }\mathcal{S}_2:=\{X_T : \text{ T is a bounded stopping time} \}$$
And trying to show that either of them is UI. ($T_n$ are the stopping times reducing $X_t$)
I need some help with this step.
Edit
This is a homework exercise, which stipulates the usage of Ito's formula
Best Answer
It seems to me that it is easy to show directly that $X_t$ is a martingale by verifying that $E[X_t \mid \mathcal{F}_s] = X_s$. (Here I assume that $B_t$ is a Brownian motion with respect to the filtration $\mathcal{F}_t$, and that you are trying to show $X_t$ is a martingale with respect to the same filtration.) One just writes $B_t = B_s + (B_t - B_s)$ and uses independence of increments. It helps to check that, for $N \sim N(0, \sigma^2)$ we have $$E[e^{-N}] = e^{\sigma^2/2}, \quad E[N e^{-N}] = -\sigma^2 e^{\sigma^2/2}.$$