Sum of Centered Gaussians
Consider the sum of two independent random variables and , where each variable is normally distributed with zero mean. Denote the variance of as and the variance of as . Recall that the sum of and is distributed with variance .
A Continuous Random Walk
Imagine a continuous-time random walk for which observed differences between and are always normally distributed with zero mean:
Assume also that the random walk is uncorrelated with itself when comparing non-overlapping time intervals.
Consistency between Eq. (2) and Eq. (1), i.e., dictates that Let us choose to define such that the constant of proportionality is 1, identifying this random walk as a Wiener process. For positive, infinitesimal differences in time, we may write
If we consider infinitesimal differences in time and recall that, for any random variable For a Wiener process, this implies We may therefore rewrite Eq. (3) in terms of a quadratic variation This observation has an important consequence: First-order approximations in of a process depending on must account for the quadratic variation in , where, in expectation,
An Itô drift-diffusion process may be represented in terms of differentials in and as where and are deterministic (e.g., as functions of the process up to time ) and is a Wiener process as described above. Note that and are the mean and variance, respectively, in this example for , not for !
Twice-differentiable functions applied to drift-diffusion stochastic processes also define drift-diffusion stochastic processes. For example, given a twice-differentiable function where the second argument is given by a stochastic process , it follows that where, Taylor expanding according to Eq. (4) and applying the chain rule twice yields Itô’s Lemma: Importantly, we note that this transformation differs from the classical chain rule:
Geometric Brownian Motion
Consider a stochastic process for which the proportional growth is an affine transformation of a Wiener process, i.e.,
for constant and .
We provide two means of solving for :
By Itô’s Lemma
First, we consider mapping mapping geometric Brownian motion by the function .
Given We may use Itô’s Lemma to solve for : where Therefore, This differential equation has solution for constant determined by boundary conditions (i.e., ). Substituting , we conclude
By Quadratic Variation
Another approach is to note that both the definition of geometric Brownian motion (Eq. (5)) and a Taylor expansion of (Eq. (4)) must be consistent.
By the independence of and , this provides two equations: From the second equation, we note that Substituting into the first equation, it follows that Recognizing an exponential as the solution class for again, for , we arrive at the unique solution
Noting that describes a Galton distribution, it follows that
Imagine investing in a security , the valuation of which (e.g., relative to USD) grows by a ratio each month, for constants and .
Should we model the security according to geometric Brownian motion, which applies in the continuous-time limit, or is this inappropriate when changes happen in discrete time?
First, what is the statistical behavior of after months when the process evolves discretely? For we have
The statistics of this process agree with those of geometric Brownian motion when we identify