Stochastic Processes: From Random Walks to Geometric Brownian Motion

A random walk is the simplest stochastic process

A stochastic process is just a sequence of random variables indexed by time — at every instant, the next value is drawn from a probability distribution rather than computed by a fixed rule. The simplest example is the random walk: a particle that, at each discrete tick, steps left or right with equal probability. It sounds trivial, but it is the seed from which the rest of the field grows.

Run many independent walkers from the same start and their spread grows predictably even though each individual path is unpredictable. For a simple symmetric walk of n steps of size Δx, the variance of the position grows linearly with the number of steps:

x(n) = Σ ξᵢ,   ξᵢ = ±Δx with probability 1/2 each
E[x(n)] = 0                     (no net drift)
Var[x(n)] = n · Δx²             (spread grows linearly in n, so σ ∝ √n)

That square-root scaling — the standard deviation of a random walk grows with the square root of time, not with time itself — is one of the most important facts in all of stochastic modelling. It's why diffusion is slow, why it takes a scent four times as long to spread twice as far, and why a random walk without drift always returns arbitrarily close to its start in one or two dimensions but not in three.

Brownian motion: the continuous-time limit

Let the step size shrink and the step rate grow, in just the right combination, and the discrete random walk converges to Brownian motion (also called a Wiener process, W(t)) — the continuous-time process first used by Einstein in 1905 to explain the jittery motion of pollen grains buffeted by unseen water molecules, and put on rigorous mathematical footing by Norbert Wiener. Its defining properties are that it starts at zero, has independent increments, and each increment over a time span Δt is normally distributed with mean 0 and variance Δt — the same square-root scaling as the discrete walk, now exact rather than approximate.

Geometric Brownian motion and financial prices

Plain Brownian motion can go negative, which makes it a poor model for something like a stock price that must stay positive. Geometric Brownian motion fixes this by modelling the logarithm of the price as Brownian motion with drift, which is equivalent to saying the price itself evolves by the stochastic differential equation

dS = μ·S·dt + σ·S·dW

S = asset price    μ = drift rate (expected return)
σ = volatility      dW = increment of a Wiener process

solution:  S(t) = S(0)·exp[(μ − σ²/2)·t + σ·W(t)]

This is the process at the heart of the Black-Scholes option pricing model: prices can never hit zero, percentage changes rather than absolute changes are what's normally distributed, and the model captures the empirical fact that returns compound multiplicatively rather than additively.

Poisson processes: counting rare, independent events

Not every random process is about position — sometimes you're counting discrete events arriving at random times: customers at a queue, radioactive decays, server requests. A Poisson process models exactly this: events occur independently at a constant average rate λ, and the number of events in any interval of length t follows a Poisson distribution with mean λt. Its defining, slightly counterintuitive property is memorylessness — the waiting time until the next event doesn't depend at all on how long you've already waited, which is why the time between consecutive events is exponentially distributed.

Why everything converges to a bell curve

The reason Brownian motion, Poisson counts and countless other stochastic processes all end up looking approximately Gaussian for large samples is the central limit theorem: the sum of many independent random contributions, regardless of their individual distribution, converges to a normal distribution as the number of terms grows — exactly what happens inside a random walk's sum of ±Δx steps. It's the deep reason a bell curve keeps showing up everywhere from measurement noise to option returns to the diffusion of ink in water.

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