Stochastic Processes Random Walks

The unifying principle

Start with a random walk: a quantity $X$ takes steps $\xi_i$ that are independent, identically distributed, mean zero, variance $\sigma^2$:

$X_n = \sum_{i=1}^{n} \xi_i, \qquad \mathbb{E}[X_n]=0, \quad \operatorname{Var}(X_n)=n\sigma^2.$

The Central Limit Theorem says that for large $n$, $X_n/\sqrt{n}$ is Gaussian regardless of the step distribution. Take the continuum limit — steps of size $\sqrt{\Delta t}$ every $\Delta t$ — and the walk converges (Donsker's theorem) to the Wiener process $W_t$:

$W_0 = 0,\quad W_t - W_s \sim \mathcal{N}(0,\,t-s),\quad \text{independent increments}.$

Its density $p(x,t)$ satisfies the diffusion (heat) equation

$\frac{\partial p}{\partial t} = \frac{D}{2}\frac{\partial^2 p}{\partial x^2}, \qquad p(x,t)=\frac{1}{\sqrt{2\pi D t}}\,e^{-x^2/(2Dt)}.$

The signature of every instance below is the same: spread $\propto \sqrt{t}$, i.e. $\operatorname{Var}\propto t$.

How it shows up in each field

Maths — the Wiener process

Here Brownian motion is the object of study itself: the canonical continuous martingale, nowhere differentiable, with quadratic variation $\langle W\rangle_t = t$. It underpins Itô calculus:

$dX_t = \mu\,dt + \sigma\,dW_t, \qquad (dW_t)^2 = dt.$

Example: For $f(W_t)=W_t^2$, Itô's lemma gives $d(W_t^2)=2W_t\,dW_t + dt$, so $\mathbb{E}[W_t^2]=t$ — the $\sqrt{t}$ spread made rigorous.

Physics — diffusing particles

Einstein (1905) explained the jitter of a pollen grain: countless molecular collisions are the tiny random steps. The mean-squared displacement grows linearly in time,

$\langle x^2 \rangle = 2Dt, \qquad D = \frac{k_B T}{6\pi\eta r},$

the Einstein relation linking diffusion $D$ to temperature $T$ and drag. This let Perrin measure Avogadro's number — proof that atoms are real.

Example: A $1\,\mu\text{m}$ bead in water diffuses $\sim 1\,\mu\text{m}$ in a second, then $\sim\!\sqrt{60}\approx 8\,\mu\text{m}$ in a minute — not $60\times$ further.

Stock-Market — geometric Brownian motion

Prices can't go negative, so it's the log-price that random-walks. Bachelier (1900) and later Black–Scholes model:

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \;\Rightarrow\; S_t = S_0\,e^{(\mu - \tfrac12\sigma^2)t + \sigma W_t}.$

Returns are Gaussian; volatility scales as $\sigma\sqrt{t}$. This gives the Black–Scholes option price, where the $\tfrac12\sigma^2$ correction is pure Itô calculus.

Example: Annualizing daily volatility: $\sigma_{\text{yr}} = \sigma_{\text{day}}\sqrt{252}$ — the same $\sqrt{t}$ rule as the pollen grain.

Biology — genetic drift

In a finite population of $N$ diploid individuals, an allele's frequency $p$ drifts randomly each generation because gametes are sampled with binomial noise:

$\operatorname{Var}(\Delta p) = \frac{p(1-p)}{2N}.$

For large $N$ this becomes the Wright–Fisher diffusion, with generator

$\frac{\partial \phi}{\partial t} = \frac{p(1-p)}{4N}\frac{\partial^2 \phi}{\partial p^2},$

a diffusion equation with state-dependent variance. Alleles wander until fixed ($p=1$) or lost ($p=0$).

Example: A neutral mutation in $N=10^4$ fixes with probability $1/(2N)=5\times10^{-5}$; the expected time to fixation, $\sim 4N$ generations, is a first-passage time of the drift.

Why this bridge matters

The $\sqrt{t}$ scaling is a portable intuition: whenever you see variance growing linearly with time, suspect an underlying random walk and reach for the diffusion equation.

• Itô calculus (maths) hands physicists and quants the same toolkit: the mysterious $-\tfrac12\sigma^2$ drift in Black–Scholes and Langevin dynamics is one identity, $(dW)^2=dt$.
• The Einstein relation (physics) — fluctuation tied to dissipation — reappears in biology as drift ($\propto 1/N$) versus selection, and in finance as volatility versus drift.
• First-passage / absorbing boundaries: fixation of a gene, a stock hitting a barrier option, and a particle reaching a wall are the same boundary-value problem for the heat equation.
• Recognizing the shared skeleton means a result proven once (e.g. Kolmogorov's forward equation) instantly serves all four domains.

Connections

• 01 Central Limit Theorem
• 02 Wiener Process & Itô Calculus
• 03 Diffusion Equation
• 04 Einstein Relation & Langevin Dynamics
• 05 Geometric Brownian Motion
• 06 Black–Scholes Model
• 07 Wright–Fisher & Genetic Drift
• 08 First-Passage Times

#bridge