Normal and Lognormal Diffusions. Let \(A(t, \omega)\) be a diffusion, and let \(\mu, \sigma\) be two constants.

(a) Let

\[d A(t, \omega)=\mu d t+\sigma d B(t, \omega)\]

and show that

\[A(t, \omega) \sim N\left(A(0)+\mu t, \sigma^{2} tight)\]

(b) Let

\[\frac{d A(t, \omega)}{A(t, \omega)}=\mu d t+\sigma d B(t, \omega)\]and apply Ito's lemma to \(f(t, A(t, \omega))=\ln (A(t, \omega))\) to show that\[A(t, \omega) / A(0) \sim L N\left(\left(\mu-\frac{1}{2} \sigma^{2}ight) t, \sigma^{2} tight)\]

[Think of \(\int d B(u, \omega)\) as a limiting sum of successive increments of a Brownian motion resulting in \(\int_{0}^{t} d B(u, \omega)=B(t, \omega)-B(0)=B(t, \omega) \sim\) \(N(0, t)\) since \(B(0)=0\) ]