continuous local martingale brownian motion $B$ is a one-dimensional Brownian motion and $X_t$ is defined as$\\$
$X_t:=f_{1-t}(B_t)$, $0\le t<1$ and $0$, $1\le t<\infty$ where $f_s(x)=\frac{1}{\sqrt{2\pi s}}e^{-\frac{x^2}{2s}}$. I have to show that $X$ is a continuous local martingale.
$f$ is continuous as a combination of continuous functions. I tried to use Ito's Formula, but i cannot see how it helps me. I also tried to show that it is a martingale by using stopping times.
I would be thankful for any help.
 A: Ito's lemma is one way to prove it. Using the definition of $f_{1-t}$, we have
$$X_t = f_{1-t}(B_t) = \frac{1}{\sqrt{2\pi (1-t)}} \exp\left\{\frac{-B_t^2}{2(1-t)}\right \}:=F(t,B_t),$$
and according to Ito's lemma:
$$dX_t = dF(t,B_t) = \left(\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial B_t^2}\right)dt+ \frac{\partial F}{\partial B_t}dW_t.$$
Computing the partial derivatives: 
\begin{align}
\frac{\partial F}{\partial t} &= \frac{\exp\left\{\frac{-B_t^2}{2(1-t)}\right\}}{2\sqrt{2\pi(1-t)}}\left(\frac{1}{1-t}-\frac{B_t^2}{(1-t)^2}\right) \\
\frac{\partial F}{\partial B_t} &= \frac{\exp\left\{\frac{-B_t^2}{2(1-t)}\right\}}{\sqrt{2\pi(1-t)}} \left(\frac{-B_t}{1-t}\right) \\
\frac{\partial^2 F}{\partial B_t^2} &= \frac{\exp\left\{\frac{-B_t^2}{2(1-t)}\right\}}{\sqrt{2\pi(1-t)}} \left(\frac{-1}{1-t}+\frac{B_t^2}{(1-t)^2}\right).
\end{align}
It follows that 
$$\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial B_t} = 0.$$
Hence, we obtain
$$dX_t = -\frac{B_t}{1-t}X_tdW_t,$$
which implies that $X_t$ is a (continuous) local martingale ($X_t$ is a stochastic integral w.r.t Brownian motion). 
