Cumulative distribution function and Brownian motion

Question

I am having troubles with one exercise. Your help will be great!

Let be $\Phi$ the C.D.F of a standard gaussian random variable (i.e. mean = 0 and variance = 1) and $(B_t)_{t\geq0}$ a Brownian motion from 0. We denote the following process

$$\begin{equation} \forall 0 \leq t < T, \quad M_t = \Phi(B_t/\sqrt{T-t}) \end{equation}$$

What is the value of $\lim_{t\to T} M_t$ a.s. ?

Thanks!

Answer 1

Note that $\frac {B_t} {\sqrt {T-t}} \to \infty$ on the set $B_T >0$ and $\frac {B_t} {\sqrt {T-t}} \to -\infty$ on the set $B_T <0$. Hence $\lim_{t\to T} M_t =I_{\{B_T >0\}}$ almost surely.