Show that at any time \(T>0\), the random variable \(S_{T}:=S_{0} \mathrm{e}^{\sigma B_{T}+\left(\mu-\sigma^{2} / 2ight) T}\) has the lognormal distribution with probability density function

\[ x \longmapsto f(x)=\frac{1}{x \sigma \sqrt{2 \pi T}} \mathrm{e}^{-\left(-\left(\mu-\sigma^{2} / 2ight) T+\log \left(x / S_{0}ight)ight)^{2} /\left(2 \sigma^{2} Tight)}, \quad x>0 \]

with \(\log\)-variance \(\sigma^{2}\) and \(\log\)-mean \(\left(\mu-\sigma^{2} / 2ight) T+\log S_{0}\), see Figures 3.9 and 5.6.

Data from Figure 3.9

Data from Figure 3.9

Transcribed Image Text:

Figure 3.9: Multiplicative Galton board simulation. ***** HE ***** ****** ****** 0 0 98 ....... 0 0 0 0 0 0 0 6 G 9 G9 9 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 00000 GOGO 0 0 0 0 3 000 -30 00