Barriers

Let $D$ be a standard Wiener process with drift $m$ starting from $D(0) = d > 0$, and suppose that there is a reflecting barrier at the origin. Show that the density function $f^{\mathrm{r}}(t, y)$ of $D(t)$ satisfies $f^{\mathrm{r}}(t, y) \rightarrow 0$ as $t \rightarrow \infty$ if $m \geq 0$, whereas $f^{\mathrm{r}}(t, y) \rightarrow 2\left|m\right| e^{-2\left|m\right| y}$ for $y > 0$, as $t \rightarrow \infty$ if $m < 0$.

Let $W$ be a standard Wiener process started at $W(0) = x \in (0, a)$, with absorbing barriers at 0 and $a$. Let $C$ be the process $W$ conditioned on the event that $W$ is absorbed at $a$. Show that the instantaneous mean and variance of $C$, when $C(t) = c \in (0, a)$, are $1 / c$ and $1$, respectively.

Let $T$ be the first passage time by $C$ to the point $a$. Show that $\mathbb {E}_{x}\left[T\right] = \frac{1}{3}\left(a^{2}-x^{2}\right)$ for $x \in (0, a)$.