Let [ P_{t}=P_{0} exp left(r_{t}+cdots+r_{1} ight) ] where the (r_{i}) are distributed iid as (mathrm{N}left(mu, sigma^{2} ight)). Show that (P_{t}) is distributed as lognormal with

Let

\[ P_{t}=P_{0} \exp \left(r_{t}+\cdots+r_{1}\right) \]

where the \(r_{i}\) are distributed iid as \(\mathrm{N}\left(\mu, \sigma^{2}\right)\). Show that \(P_{t}\) is distributed as lognormal with parameters \(t \mu+\log \left(P_{0}\right)\) and \(t \sigma\). (If \(\mu=0\), this is a lognormal geometric random walk with parameter \(\sigma\).)