# Theoretical distribution of (geometric) Brownian motion (with drift)

Discussion in 'Finance' started by Emily, Oct 8, 2018.

1. ### EmilyGuest

I am working on a simulation study which focuses on both the Brownian motion with drift (1) and the geometric Brownian motion (2). I denote them by $X_t$.

What are the theoretical distributions of these processes under the measure P (thus equal probabilities of $\frac{1}{2}$)?

I am getting confused. I know that the processes are of the type:

$(1) X_t = \mu t + \sigma W_t, \\ (2) X_t = \exp (\mu t + \sigma W_t).$

where $W_t \sim N(0,t)$.

However, then what is the theoretical distribution of $X_t$? Are they simply the normal distribution and log-normal distribution? My thought was that it follows a normal distribution with mean $\mu t$ and variance $\sigma^2 t$. The geometric BM would then follow a log-normal distribution with the same parameters. Is that correct?

Furthermore, how do these distributions change under a different (equivalent) measure Q?

Thank you!