Brownian Motion

The continuous-time, continuous-path limit of a random walk, with independent Gaussian increments.

1. The idea

Standard Brownian motion ${B_{t}}_{t \geq 0}$ starts at $B_{0} = 0$ , has continuous paths, and independent increments: $B_{t} - B_{s}$ for $t > s$ is independent of the path up to time $s$ and distributed $N (0, t - s)$ .

It arises as the scaling limit of a symmetric random walk: as step size and time step shrink together appropriately, $S_{⌊ n t ⌋} / n \to B_{t}$ . This is why $Var (B_{t}) = t$ mirrors $Var (S_{n}) = n$ .

Because increments over disjoint intervals are independent Gaussians, covariance between two times is determined purely by overlap: $Cov (B_{s}, B_{t}) = min (s, t)$ .

2. Why it matters

Brownian motion is the driving noise behind the Black-Scholes model and most continuous-time finance, and its covariance structure is a recurring building block for pricing and hedging arguments.

3. Core mathematics

Initial condition

B_{0} = 0

Increment distribution

B_{t} - B_{s} \sim N (0, t - s) for t > s

Independent increments

B_{t_{2}} - B_{t_{1}} ⊥ B_{t_{4}} - B_{t_{3}} for disjoint intervals

Covariance

Cov (B_{s}, B_{t}) = min (s, t)

Variance

Var (B_{t}) = t

4. How to recognize it

Think of this when…

Continuous-time processes described as having independent, normally distributed increments
Questions asking for $Cov (B_{s}, B_{t})$ or the distribution of $B_{t} - B_{s}$
Setup phrases mentioning "standard Brownian motion" or "Wiener process"
Precursor to option pricing or stochastic differential equation questions

5. Worked example

Worked example

For standard Brownian motion, find $Var (B_{5} - B_{2})$ and $Cov (B_{2}, B_{5})$ .

1
Increments over $[2, 5]$ are $N (0, 5 - 2)$ , so the variance is the length of the interval.
$Var (B_{5} - B_{2}) = 5 - 2 = 3$
2
Covariance uses the min-of-times formula directly.
$Cov (B_{2}, B_{5}) = min (2, 5) = 2$
Answer: $Var (B_{5} - B_{2}) = 3$ ; $Cov (B_{2}, B_{5}) = 2$ .

6. Common mistakes

Computing $Cov (B_{s}, B_{t})$ as $0$ by mistakenly assuming all Brownian motion values are independent
Forgetting that $B_{t} - B_{s}$ has variance $t - s$ , not $t$ or $s$ alone
Treating Brownian paths as differentiable when they are continuous but nowhere differentiable
Mixing up increments over overlapping vs. disjoint intervals when applying independence

7. Quick check

What is $Var (B_{9})$ for standard Brownian motion?

What is $Cov (B_{3}, B_{7})$ ?

8. Practice

13 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Recursive Expectation / First-Step Analysis

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