Brownian Motion

Brownian Motion

Brownian motion (equivalently, the Wiener process) is a continuous-time stochastic process $\{B_t{\}}_{t\ge 0}$ taking values in $\mathbb{R}$ with the following defining properties:

1. $B_0=0$ almost surely.
2. Independent increments. For any times $0\le t_0<t_1<\cdots <t_n$, the increments $B_{t_1}-B_{t_0},B_{t_2}-B_{t_1},\dots ,B_{t_n}-B_{t_{n-1}}$ are mutually independent.
3. Gaussian increments. For $0\le s<t$, $B_t-B_s\sim \mathcal{N}(0,t-s)$.
4. Continuous paths. The map $t\mapsto B_t$ is continuous almost surely.

These four axioms determine the process up to indistinguishability. The mean function is $\mathbb{E}[B_t]=0$ and the covariance function is
$K(s,t)=\text{Cov}(B_s,B_t)=\min (s,t).$
The joint law of any finite collection $(B_{t_1},\dots ,B_{t_n})$ is multivariate Gaussian with mean zero and covariance matrix ${\Sigma }_{ij}=\min (t_i,t_j)$. Brownian motion is therefore a Gaussian process with kernel $K(s,t)=\min (s,t)$, the canonical non-stationary kernel.

Despite having continuous sample paths, Brownian motion is nowhere differentiable almost surely. Its paths are Hölder continuous of every order $\alpha <\frac{1}{2}$ and have infinite total variation on every interval, but finite quadratic variation: on $[0,t]$ the sum ${\sum }_{k=0}^{n-1}(B_{t_{k+1}}-B_{t_k}{)}^2$ converges to $t$ in probability as the mesh of the partition goes to zero.

Brownian motion is the continuous-time limit of a symmetric random walk: if $S_n=X_1+\cdots +X_n$ with $X_i\stackrel{\text{iid}}{\sim }\mathcal{N}(0,1)$, then $B_t^{(n)}:=n^{-1/2}{S}_{\lfloor nt\rfloor }$ converges in distribution to $B_t$ as $n\to \infty$ (Donsker’s invariance principle).

Prerequisites

• 6.2 Multivariate Gaussian Distribution: the joint law of $(B_{t_1},\dots ,B_{t_n})$ is Gaussian with ${\Sigma }_{ij}=\min (t_i,t_j)$.
• 2.6c Continuous Distributions: Gaussian density is the building block of the increment distribution.
• 4.4 Random Walks: discrete-time predecessor; Donsker’s theorem gives BM as its limit.

Intuition

Brownian motion is the mathematical idealization of a particle jostled by infinitely many infinitesimally small independent shocks per unit time. At the macroscopic scale the particle’s position is continuous, but zooming in reveals that the trajectory is rougher than any smooth curve: it fluctuates on every time scale. The defining tension is that the increments are independent and centered (so the process has no drift and no memory), yet their accumulated variance grows linearly with time ($\mathrm{Var}(B_t)=t$), producing paths that wander arbitrarily far from the origin.

The covariance $K(s,t)=\min (s,t)$ is the signature that makes BM a tractable Gaussian process. Intuitively, $B_s$ and $B_t$ share exactly the increments accumulated in their common past $[0,\min (s,t)]$; the increments after $\min (s,t)$ are independent and contribute nothing to the covariance. This gives the process its Markov property: the future after time $t$ depends on the past only through $B_t$, because $B_{t+h}-B_t$ is independent of $\{B_s:s\le t\}$ by the independent-increments axiom.

Properties

(Independent Increments) Disjoint-time increments are mutually independent.

For any $0\le t_0<t_1<\cdots <t_n$, the random variables $B_{t_1}-B_{t_0},\dots ,B_{t_n}-B_{t_{n-1}}$ are independent. This is the single most important structural property: it reduces joint probabilities over times $t_1,\dots ,t_n$ to products of univariate Gaussians over the gaps $t_i-t_{i-1}$, and it gives the process its Markov property and its martingale property simultaneously.

(Gaussian Increments) $B_t-B_s\sim \mathcal{N}(0,t-s)$ for $s<t$.

The distribution of an increment depends only on the length of the time interval, not on its position. This makes the process stationary in increments (sometimes called “homogeneous increments”) even though the process itself is non-stationary: $\mathrm{Var}(B_t)=t$ grows with $t$. The standard deviation of the increment is $\sqrt{t-s}$, which is why BM is Hölder-$\alpha$ continuous for every $\alpha <\frac{1}{2}$ but not $\frac{1}{2}$.

(Covariance Function) $K(s,t)=\text{Cov}(B_s,B_t)=\min (s,t)$.

Write $s\le t$ and decompose $B_t=B_s+(B_t-B_s)$. Since $B_s=B_s-B_0$ is independent of $B_t-B_s$ by the independent-increments axiom, $\text{Cov}(B_s,B_t)=\text{Var}(B_s)=s=\min (s,t)$. The symmetric case $t\le s$ gives $\min (s,t)$ by the same argument. This kernel is positive semi-definite but non-stationary: it does not depend only on $|t-s|$. The Gram matrix $[\min (t_i,t_j){]}_{i,j}$ for any finite collection of times is the covariance matrix of the joint law of $(B_{t_1},\dots ,B_{t_n})$.

(Markov Property) $B_{t+h}$ given $\{B_s:s\le t\}$ has the same distribution as $B_{t+h}$ given $B_t$ alone.

The independent-increments axiom gives $B_{t+h}-B_t\perp \{B_s:s\le t\}$, so conditioning on the whole past is no more informative than conditioning on the present. This makes BM a continuous-time Markov process on $\mathbb{R}$, with transition density
$p(t,x;t+h,y)=\frac{1}{\sqrt{2\pi h}}\exp \left(-\frac{(y-x{)}^2}{2h}\right),$
the Gaussian kernel of variance $h$. The density satisfies the heat equation ${\partial }_{t}p=\frac{1}{2}{\partial }_{xx}p$, so BM is the stochastic process associated with Brownian diffusion.

(Martingale Property) $B_t$ is a martingale with respect to its own filtration.

$\mathbb{E}[B_{t+h}\mid {\mathcal{F}}_t]=B_t+\mathbb{E}[B_{t+h}-B_t\mid {\mathcal{F}}_t]=B_t+0=B_t$, using independence of the increment from ${\mathcal{F}}_t=\sigma (B_s:s\le t)$ and the zero-mean property. Together with $B_t^2-t$ being a martingale and $\exp (\lambda B_t-\frac{1}{2}{\lambda }^{2}t)$ being a martingale for every $\lambda \in \mathbb{R}$, these are the standard martingales used in stochastic calculus.

(Non-Differentiability) Sample paths are continuous but nowhere differentiable almost surely.

Formally: with probability one, there is no $t\ge 0$ at which the limit ${\lim }_{h\to 0}(B_{t+h}-B_t)/h$ exists. The heuristic is that the increment $B_{t+h}-B_t$ has standard deviation $\sqrt{h}$, so $(B_{t+h}-B_t)/h$ has standard deviation $h^{-1/2}$, which blows up as $h\to 0$. Paths are Hölder continuous of every order $\alpha <\frac{1}{2}$ but not $\alpha =\frac{1}{2}$. This is why classical calculus fails on BM paths and Itô calculus is needed: integrals $\int f(B_t)d{B}_t$ cannot be defined pathwise as Riemann-Stieltjes integrals.

(Quadratic Variation) ${\sum }_k(B_{t_{k+1}}-B_{t_k}{)}^2\to t$ in probability as the partition mesh $\to 0$.

Although the total variation ${\sum }_k|B_{t_{k+1}}-B_{t_k}|$ is infinite on every interval, the sum of squared increments converges to the length of the interval. Informally, ”$(d{B}_t{)}^2=dt$.” This replaces the usual calculus rule $(df{)}^2=0$ at second order and is the reason the Itô chain rule for $f(B_t)$ carries the extra term $\frac{1}{2}{f}^{\prime \prime }(B_t)dt$:
$df(B_t)=f^{\prime }(B_t)d{B}_t+\frac{1}{2}{f}^{\prime \prime }(B_t)dt.$

(Time Inversion and Self-Similarity) $\{c{B}_{t/c^2}{\}}_{t\ge 0}$ is itself a Brownian motion for every $c>0$; $\{t{B}_{1/t}{\}}_{t>0}$ extended to $0$ at $t=0$ is another BM.

Scaling time by $c^2$ and space by $c$ leaves the law invariant, and this self-similarity is what makes BM a fractal. Time inversion $t\mapsto 1/t$ exchanges the behavior near $0$ with the behavior near $\infty$, which is useful for deducing long-time asymptotics from short-time ones and vice versa.

(Reflection Principle) For $a>0$, $\mathbb{P}[{\max }_{s\le t}{B}_s\ge a]=2\mathbb{P}[B_t\ge a]$.

The running maximum of BM on $[0,t]$ has the distribution of $|B_t|$. The argument: every path that exceeds $a$ at some point before $t$ is equally likely (by reflection about level $a$ after the hitting time) to end up above or below $a$ at time $t$. This gives the explicit formula
$\mathbb{P}\left[{\max }_{s\le t}{B}_s\ge a\right]=2\mathbb{P}[B_t\ge a]=2\Phi (-a/\sqrt{t}),$
and related closed forms for the distributions of first hitting times, last exit times, and the joint density of $(B_t,{\max }_{s\le t}{B}_s)$.

(Construction via Donsker’s Theorem) BM is the scaling limit of simple symmetric random walks.

Let $X_1,X_2,\dots$ be i.i.d. with mean $0$ and variance $1$, and set $S_n=X_1+\cdots +X_n$. The rescaled process
$B_t^{(n)}:=\frac{1}{\sqrt{n}}(S_{\lfloor nt\rfloor }+(nt-\lfloor nt\rfloor )X_{\lfloor nt\rfloor +1})$
converges in distribution (as a process on $C([0,1])$) to standard Brownian motion as $n\to \infty$. This gives a constructive proof that BM exists and justifies the heuristic that BM is “continuous random walk.”

Examples

Joint distribution of $(B_1,B_2,B_3)$

The covariance matrix for $(B_1,B_2,B_3)$ is
$\Sigma =\left(\begin{array}{ccc}\min (1,1)& \min (1,2)& \min (1,3)\\ \min (2,1)& \min (2,2)& \min (2,3)\\ \min (3,1)& \min (3,2)& \min (3,3)\end{array}\right)=\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 2\\ 1& 2& 3\end{array}\right).$
So $(B_1,B_2,B_3)\sim \mathcal{N}(0,\Sigma )$. To compute $\mathbb{P}[B_3>2\mid B_1=1]$, use the standard Gaussian conditioning formula:
$B_3\mid B_1=1\sim \mathcal{N}\left(\frac{\min (1,3)}{\min (1,1)}\cdot 1,\min (3,3)-\frac{\min (1,3{)}^2}{\min (1,1)}\right)=\mathcal{N}(1,2).$
Therefore $\mathbb{P}[B_3>2\mid B_1=1]=1-\Phi ((2-1)/\sqrt{2})=1-\Phi (1/\sqrt{2})\approx 0.24$. This is exactly the computation that underlies Gaussian process regression with the Brownian kernel: the conditional expectation is a kernel-weighted function of the observed values, and the conditional variance is the kernel’s value at the query point minus a kernel-weighted correction.

Hitting probability via the reflection principle

What is the probability that Brownian motion hits level $a=2$ at some point during $[0,4]$?

By the reflection principle,
$\mathbb{P}\left[{\max }_{s\le 4}{B}_s\ge 2\right]=2\mathbb{P}[B_4\ge 2]=2\Phi (-2/\sqrt{4})=2\Phi (-1)\approx \mathrm{0.317.}$
Without the reflection principle, this probability would require integrating a conditional density over all paths that reach level $2$ before time $4$, an essentially impossible calculation by direct methods. Reflection collapses the problem to a single-time Gaussian tail probability.

Quadratic variation of BM on a dyadic partition

Partition $[0,1]$ into $n=2^k$ equal pieces and compute
$Q_n:={\sum }_{j=0}^{n-1}(B_{(j+1)/n}-B_{j/n}{)}^2.$
Each summand has expectation $\mathbb{E}[(B_{(j+1)/n}-B_{j/n}{)}^2]=1/n$, so $\mathbb{E}[Q_n]=n\cdot (1/n)=1$.

The variance is $\mathrm{Var}(Q_n)=n\cdot \mathrm{Var}((B_{1/n}{)}^2)=n\cdot 2/n^2=2/n$ (using $\mathrm{Var}(Z^2)=2$ for standard Gaussian $Z$, then scaling), which goes to $0$. By Chebyshev, $Q_n\to 1$ in probability. This is the quadratic-variation identity ${\int }_0^1(d{B}_s{)}^2=1$ in action, and is the starting point for Itô’s formula.

Derivation

Goal. Derive the covariance function $K(s,t)=\min (s,t)$ directly from the four defining axioms.

For $s\le t$, decompose
$B_t=B_s+(B_t-B_s).$
Both terms have mean zero, so $\text{Cov}(B_s,B_t)=\mathbb{E}[B_s{B}_t]=\mathbb{E}[B_s(B_s+(B_t-B_s))]=\mathbb{E}[B_s^2]+\mathbb{E}[B_s(B_t-B_s)]$.

By the independent-increments axiom, $B_s=B_s-B_0$ is independent of $B_t-B_s$, so the cross term factors: $\mathbb{E}[B_s(B_t-B_s)]=\mathbb{E}[B_s]\mathbb{E}[B_t-B_s]=0\cdot 0=0$.

By the Gaussian-increments axiom, $B_s\sim \mathcal{N}(0,s)$, so $\mathbb{E}[B_s^2]=s$. Combining,
$\text{Cov}(B_s,B_t)=s=\min (s,t)\text{for }s\le t.$
The case $t\le s$ is symmetric, so $K(s,t)=\min (s,t)$ for all $s,t\ge 0$.

For the joint law of $(B_{t_1},\dots ,B_{t_n})$, apply the same decomposition recursively. Writing $t_0=0$ and $\Delta B_k=B_{t_k}-B_{t_{k-1}}$, the independence and Gaussianity of increments give $\Delta B_k\stackrel{\text{ind}}{\sim }\mathcal{N}(0,t_k-t_{k-1})$. Since $(B_{t_1},\dots ,B_{t_n})$ is a linear function of $(\Delta B_1,\dots ,\Delta B_n)$, it is jointly Gaussian with mean zero, and its covariance matrix is ${\Sigma }_{ij}=\min (t_i,t_j)$ by the pairwise calculation above. This jointly-Gaussian property is exactly what makes BM a Gaussian process.

See Also

• 3.10 Gaussian Processes: BM as the canonical GP with kernel $\min (s,t)$.
• 3.10.1 Kernels: the Brownian-motion kernel and its algebra with other kernels.
• 3.10.3 Integrated Brownian Motion: ${\int }_0^t{B}_{s}ds$; cubic-spline posteriors.
• 4.4 Random Walks: discrete-time analog; Donsker’s invariance principle.
• 4.5 Martingales: $B_t$, $B_t^2-t$, and $\exp (\lambda B_t-\frac{1}{2}{\lambda }^{2}t)$ are all martingales.
• 4.3 Markov Chains: BM as a continuous-time continuous-space Markov process.
• 7. Stochastic Calculus: Itô integrals, stochastic differential equations, and BM-driven SDEs.
• 6.2 Multivariate Gaussian Distribution: underlying distribution theory for finite-dimensional marginals.