Let $W(t)$ be a Brownian motion and let $\mathcal{F}(t), t \geq 0$, be an associated filtration.
3.10 Exercises
119
(i) For $\mu \in \mathbb{R}$, consider the Brownian motion with drift $\mu$ :
$$
X(t)=\mu t+W(t) .
$$
Show that for any Borel-measurable function $f(y)$, and for any $0 \leq s<t$, the function
$$
g(x)=\frac{1}{\sqrt{2 \pi(t-s)}} \int_{-\infty}^{\infty} f(y) \exp \left\{-\frac{(y-x-\mu(t-s))^2}{2(t-s)}\right\} d y
$$
satisfies $\mathbb{E}[f(X(t)) \mid \mathcal{F}(s)]=g(X(s))$, and hence $X$ has the Markov property. We may rewrite $g(x)$ as $g(x)=\int_{-\infty}^{\infty} f(y) p(\tau, x, y) d y$, where $\tau=t-s$ and
$$
p(\tau, x, y)=\frac{1}{\sqrt{2 \pi \tau}} \exp \left\{-\frac{(y-x-\mu \tau)^2}{2 \tau}\right\}
$$
is the transition density for Brownian motion with drift $\mu$.
(ii) For $\nu \in \mathbb{R}$ and $\sigma>0$, consider the geometric Brownian motion
$$
S(t)=S(0) e^{\sigma W(t)+\nu t} .
$$
Set $\tau=t-s$ and
$$
p(\tau, x, y)=\frac{1}{\sigma y \sqrt{2 \pi \tau}} \exp \left\{-\frac{\left(\log \frac{y}{x}-\nu \tau\right)^2}{2 \sigma^2 \tau}\right\} .
$$