Introduction to Brownian Motion

Brownian motion is the basic building block of stochastic calculus (Itô integrals, stochastic differential equations), which underlies modern mathematical finance: the Black–Scholes option pricing model treats log-stock-prices as drifted, scaled Brownian motion. In physics, it remains the canonical model of diffusion — heat conduction, particle transport, and the Fokker–Planck equation are its PDE counterparts. In statistics, Brownian motion (and its bridge and sheet variants) arises as the limiting process in the theory of empirical processes, underlying Kolmogorov–Smirnov goodness-of-fit tests. In engineering, white noise is formally the (distributional) derivative of Brownian motion, central to signal processing and Kalman filtering. In pure mathematics, Brownian motion connects to potential theory (harmonic functions and hitting probabilities), fractal geometry (its graph has Hausdorff dimension $3/2$), and representation theory of Gaussian processes; it is also the prototypical example motivating the general theory of Markov processes and martingales built on the foundations of Random Walks.

Ask the tutor: How does adding drift and volatility parameters turn standard Brownian motion into the geometric Brownian motion used to model stock prices?

Picture a random walk $S_n = X_1 + X_2 + \cdots + X_n$ where each $X_i = \pm 1$ with probability $1/2$, independently. This is the discrete skeleton. To get a continuous-time, continuous-space limit, imagine speeding up the clock: take $n$ steps per unit time, and shrink each step so the total variance stays finite. Concretely, define for $t \ge 0$

At integer multiples of $1/n$, $\mathrm{Var}(W_n(t)) = \frac{\lfloor nt \rfloor}{n} \to t$. The Central Limit Theorem tells us that $S_{\lfloor nt\rfloor}/\sqrt{n}$ converges in distribution to $N(0,t)$: this is exactly the scaling $\Delta x \sim \sqrt{\Delta t}$ mentioned above. Independence of the walk's increments is inherited by $W_n$'s increments over disjoint time blocks, and this independence survives in the limit. What is not obvious — and is the deep content of Donsker's theorem, which we approach here at the level of finite-dimensional convergence — is that the entire path, not just single-time marginals, converges, and that the limit has continuous paths. The picture to hold in mind: at any fixed resolution the path looks like a smooth wiggly curve, but zoom into any interval and the same jaggedness reappears at every scale — a fractal-like self-similarity coming from the identity $B_{ct} \stackrel{d}{=} \sqrt{c}\, B_t$ for $c>0$. This self-similarity is why no derivative exists: the local increment over $[t, t+h]$ has typical size $\sqrt{h}$, so the difference quotient $\big(B_{t+h}-B_t\big)/h$ has typical size $1/\sqrt{h} \to \infty$.

(Stated here as the signature qualitative feature of the process; the full proof is a substantial result of Paley–Wiener–Zygmund type and is beyond the scope of this introductory treatment, but the heuristic is immediate from scaling: since $B_t - B_s \sim N(0,|t-s|)$, we have $\mathbb E[(B_t-B_s)^2] = |t-s|$, so a typical increment over a tiny interval of length $h$ has size of order $\sqrt h$, not order $h$. A genuine derivative at $s$ would require $(B_{s+h}-B_s)/h \to$ some finite limit, but the typical size of this ratio is $\sqrt h / h = 1/\sqrt h \to \infty$ as $h\to 0$. This scaling obstruction, made rigorous via a Borel–Cantelli argument over dyadic increments, is what forces nondifferentiability at every point simultaneously.)

Example 1: A joint-probability computation with asymmetric, non-round times. Let $(B_t)$ be standard Brownian motion. Compute $\mathbb{P}(B_2 > 1, B_5 > 4)$. Solution. Use the independent increments structure. Write $B_5 = B_2 + (B_5 - B_2)$, where $B_2 \sim N(0,2)$ and $B_5 - B_2 \sim N(0,3)$ are independent (by property (2)–(3) of the definition, using the partition $0<2<5$). Let $X = B_2$ and $Y = B_5 - B_2$, so $X \sim N(0,2)$, $Y\sim N(0,3)$ independent, and we want

Condition on $X = x$: given $X=x$, the event becomes $Y > 4-x$, and since $Y \sim N(0,3)$ independent of $X$,

where $\phi_2$ is the $N(0,2)$ density and $\Phi$ the standard normal CDF. This integral does not have an elementary closed form, but it can be evaluated numerically or expressed via the bivariate normal CDF, since $(X,X+Y)$ is a bivariate Gaussian vector with

matching part (c) of the theorem directly — a useful cross-check. The correlation is

Standardizing, $Z_1 = X/\sqrt2$, $Z_2 = (X+Y)/\sqrt5$ are standard normals with correlation $\rho \approx 0.6325$, and we want

Using the standard bivariate normal survival function $L(h,k;\rho) = \mathbb{P}(Z_1>h, Z_2>k)$ with $h=0.7071$, $k=1.7889$, $\rho = 0.6325$, numerical evaluation (e.g. via the bivariate normal integral tables or software) gives

The key structural point — independent of the numeric integration — is that this problem reduces entirely to Gaussian bivariate machinery because Brownian motion's finite-dimensional distributions are jointly Gaussian with the explicit covariance $\min(s,t)$: no path-integral or process-level reasoning beyond that is needed for finite-time-point questions.

Example 2: Verifying the scaling/self-similarity property with concrete numbers. Show that if $(B_t)_{t\ge0}$ is standard Brownian motion and $c = 4$, then $\widetilde B_t := \tfrac{1}{2} B_{4t}$ is again a standard Brownian motion, and use this to compute $\mathbb{P}(B_{12} - B_4 > 6)$ two ways: directly, and via the rescaled process. Solution. First, the general scaling claim: for $c>0$, define $\widetilde B_t = c^{-1/2} B_{ct}$. Check the four defining properties. 1. $\widetilde B_0 = c^{-1/2}B_0 = 0$ a.s. ✓. 2. For $0=t_0<t_1<\cdots<t_n$, the increments $\widetilde B_{t_j} - \widetilde B_{t_{j-1}} = c^{-1/2}\big(B_{ct_j} - B_{ct_{j-1}}\big)$ are scalar multiples of the increments $B_{ct_j}-B_{ct_{j-1}}$ of the original process over the disjoint, ordered time points $0 \le ct_0 < ct_1 < \cdots < ct_n$; independence of the latter (property (2) for $B$) gives independence of the former. ✓ 3. $B_{ct_j}-B_{ct_{j-1}} \sim N(0, c(t_j-t_{j-1}))$, so $\widetilde B_{t_j}-\widetilde B_{t_{j-1}} = c^{-1/2}(B_{ct_j}-B_{ct_{j-1}}) \sim N\big(0, c^{-1}\cdot c(t_j-t_{j-1})\big) = N(0,t_j - t_{j-1})$. ✓ 4. $t \mapsto \widetilde B_t = c^{-1/2}B_{ct}$ is a continuous function of $t$ whenever $t\mapsto B_t$ is, being a composition of a continuous linear rescaling of the time axis with a continuous path. ✓ So $\widetilde B$ satisfies all four defining properties and is itself a standard Brownian motion. With $c=4$: $\widetilde B_t = \tfrac12 B_{4t}$ is standard Brownian motion. Now compute $\mathbb{P}(B_{12}-B_4 > 6)$ directly: $B_{12}-B_4 \sim N(0, 12-4) = N(0,8)$, so

Via the rescaled process with $c=4$: $B_{12} = 2\widetilde B_3$ and $B_4 = 2\widetilde B_1$, so $B_{12}-B_4 = 2(\widetilde B_3 - \widetilde B_1)$, and $\widetilde B_3 - \widetilde B_1 \sim N(0,2)$ since $\widetilde B$ is standard Brownian motion. Then

matching exactly, since $3/\sqrt2 = 6/\sqrt8$. This confirms the self-similarity computation is internally consistent and gives a genuine alternate route (useful, e.g., for reducing problems at "inconvenient" time scales to unit-scale computations).