Let $X_t$be the solution to the SDE:
$dX_t=-X_tdt+dB_t$, $X_0=0$
Then $X_t$ is the Ornstein–Uhlenbeck process $X_t=e^{-t}\int_0^te^sdB_s$.
I want to calculate $\mathbb{E}[e^\tau X_\tau]$ when $\tau=\inf\{t\geq0:|X_t|=1-t^2\}$.
How can we treat the integral $\mathbb{E}[\int_0^\tau e^sdB_s]$?
My attempt for a solution:
We need to calculate $\mathbb{E}[e^{-\tau}X_\tau]=\mathbb{E}[\int_0^\tau e^sdB_s]$. If we can show that $M_t=\int_0^t e^sdB_s$ is a martingale, then we may be able to use the Optional Sampling Theorem.
Observe that: $\mathbb{E}\left
Hence, $M$ is a martingale and so $\mathbb{E}[M_t]=\mathbb{E}[M_0]=0$
If we can show that $\tau$ is bounded, then from the BDG inequality, we can show that $\tau$ is optional for $M$. So the question is, is $\tau$ bounded?
If we set $g(t)=|X_t|-1+t^2$, then $g(0)=-1<0$ and $g(1)=|X_1|>0$. From the Intermediate value Theorem, there exists $t_0>0$ such that $g(t_0)=0$.
So $\tau$ must be bounded. Is that right?