Show that this process is a local martingale

Question

Let $B$ be a Brownian Motion, $a>0$ and $b>0$. We set $\tau_{a,b}:=inf\{t\geq0:B_t\in\{-a;b\}\}$.

How can I show that: $$ M_t=sh(\theta(B_t+a))exp(-\frac{\theta^2}{2}t) $$ is a local martingale $\forall \theta \in \mathbb{R}$.

thanks!

Answer 1

Just use Ito's formula.

Let $f(t,x) = sh(\theta(x+a))e^{-\frac{\theta^2}{2}t}$.

Then $f_x(t,x) = \theta ch(\theta(x+a))e^{-\frac{\theta^2}{2}t}$, $f_{xx}(t,x) = \theta^2 sh(\theta(x+a))e^{-\frac{\theta^2}{2}t}$ and $f_t(t,x) = -\frac{\theta^2}{2} sh(\theta(x+a))e^{-\frac{\theta^2}{2}t}$.

Then $$df(t,B_t) = f_xdB_t + f_tdt + \frac{1}{2}f_{xx}dt = \theta ch(\theta(B_t+a))e^{-\frac{\theta^2}{2}t}dB_t$$ Thus $f(t,B_t)$ is a local martingale.