From continuous returns to lognormal prices

In short

• Continuously compounded return: $r = \ln(P_1/P_0) = \ln(1 + R)$; the future price is $P_T = P_0 e^{r_{0,T}}$.
• Over several periods, continuous returns simply add: $r_{0,T}$ is the sum of the one-period returns.
• i.i.d. returns are independent (the past does not predict the future) and identically distributed (stationary: same mean and variance every period).
• With i.i.d. returns, the mean and the variance grow in proportion to T; the standard deviation grows with $\sqrt{T}$.
• A sum of normal returns is normal; a sum of non-normal i.i.d. returns is approximately normal by the central limit theorem. Either way, $P_T$ is lognormal.