Mapping Geometric Brownian motion to Brownian motion and back

(Work in progress!) This post is inspired by recent work I've been doing on the evolution of income inequality over time. A commonly used model for the distribution of incomes in a nation is the log-normal distribution. There are a whole host of reasons why this doesn't capture important aspects of income inequality, but this does serve as a useful "null model."

The Log-normal Distribution

We say that $X \sim LogNorm(\mu, \sigma)$ if it has the density $$ f(x \,;\, \mu, \sigma) = \frac{1}{x \sigma \sqrt{2 pi}} e^{-\frac{ln(x)^2 - \mu}{2 \sigma^2}}. $$ The mean and median are often easier to use to parameterize a log-normal distribution, and so we note that $$ E(X) = \exp\left(\mu + \frac{\sigma^2}{2}\right) \text{ and } \text{Med}(X) = \exp(\mu). $$ The log-normal distribution gets its name from the observation that if $Z \sim \text{Norm}(0, 1)$, then $$ e^{\mu + \sigma Z} \sim \text{LogNorm}(\mu, \sigma). $$

Geometric Brownian motion

A standard Brownian motion $\{B(t)\}_{t \geq 0}$ is related to the normal distribution through the relationship that for every $t > 0$, $B(t) \sim \text{Norm}(0,t)$. Naturally, $$ X(t) := x\exp\big(\mu t + \sigma B(t)\big) \sim \text{LogNorm}(ln(x) + \mu t, \sigma) $$ We call $X(t)$ Geometric Brownian motion. Using It\^o calculus, we can readily find that $X(t)$ it satisfies the stochastic differential equation (SDE) $$ \mathrm{d} X(t) = \alpha X(t) \mathrm{d} t + \sqrt{2 \beta} X(t) \mathrm{d}B(t) $$ where $$ \alpha = \mu + \frac{\sigma^2}{2} t, \text{ and } \beta = \frac{\sigma^2}{2} $$