Standard Brownian Motion and Its Defining Properties

What Defines Standard Brownian Motion?

Brownian motion is specified by several conditions working together. Its value starts at zero, its paths are continuous, and each increment over a fixed time interval has a centered Gaussian distribution whose variance is the interval's length. The increments must also be independent of the information already available at the start of that interval. That last condition makes the definition relative to a filtration, not merely to the process's own past values.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\). A filtration is an increasing family of sigma-fields: \(\mathcal{F}_s\subseteq\mathcal{F}_t\subseteq\mathcal{F}\) whenever \(s\le t\). It describes the information available by each time. A process \(B_t\) is adapted if \(B_t\) is \(\mathcal{F}_t\)-measurable for each \(t\).

Here \(N(0,h)\) denotes the Gaussian distribution with mean zero and variance \(h\), for \(h>0\). The starting value is deterministic almost surely; it need not equal zero on every outcome in a null set. At \(s=t\), the increment is identically zero, so it has the degenerate distribution at zero rather than a nondegenerate Gaussian distribution.

Immediate Consequences of the Definition

For \(0\le s<t\), put \(h=t-s\). The definition directly gives the increment's mean and variance, and hence its second moment:

Taking \(s=0\) and using \(B_0=0\) almost surely shows that \(B_t\sim N(0,t)\) for \(t>0\), so \(\mathbb{E}[B_t]=0\) and \(\mathbb{E}[B_t^2]=t\). In particular, \(B_t\) is integrable: the Cauchy–Schwarz inequality gives \(\mathbb{E}|B_t|\le\sqrt{\mathbb{E}[B_t^2]}=\sqrt{t}\). At time zero, these statements hold with \(B_0=0\).

The Gaussian law also gives an exact scaling for a single increment. If \(Z\sim N(0,1)\), then \(B_t-B_s\) has the same distribution as \(\sqrt{t-s}\,Z\). This describes the law of an increment; it does not assert a pathwise identity between the increment and a rescaled random variable chosen in advance.

Continuity provides a separate, pathwise consequence. On the probability-one event where the path is continuous, for fixed \(s\), \(B_t-B_s\to0\) as \(t\to s\). A useful probability version also follows from the increment variance. For any \(\varepsilon>0\), Chebyshev's inequality gives, when \(t>s\),

Thus increments over shrinking time intervals tend to zero in probability as well. The pathwise continuity condition and the Gaussian increment condition are distinct parts of the definition; neither should be silently substituted for the other.

Why the Increments on Disjoint Intervals Are Independent

The definition requires each new increment to be independent of the information available at its starting time. This implies mutual independence of increments over any finite collection of nonoverlapping time intervals. In particular, the intervals need not be separated by a positive gap: adjacent intervals are allowed.

Proof. First remove any intervals of length zero: their increments are constant, so they do not affect independence. If no intervals remain, the assertion concerns an empty family and holds. Otherwise relabel the positive-length intervals in their time order. For each \(j\), the random variable \(B_{b_j}-B_{a_j}\) is measurable with respect to \(\mathcal{F}_{b_j}\), because \(B\) is adapted. For each \(j>1\), every earlier increment is therefore measurable with respect to \(\mathcal{F}_{a_j}\): the earlier endpoint \(b_i\) satisfies \(b_i\le a_j\), and the filtration is increasing. The defining independence condition says that \(B_{b_j}-B_{a_j}\) is independent of \(\mathcal{F}_{a_j}\), and hence independent of the sigma-field generated by all earlier increments. Inductively, the first increment is independent of the empty collection, and each subsequent increment is independent of the sigma-field generated by its predecessors. This is precisely mutual independence of the finite family. \(\square\)

The filtration-relative condition is stronger and more precise than merely saying that the next increment is independent of the current value \(B_s\). It guarantees independence from all information in \(\mathcal{F}_s\), including information about the process's earlier history.

Brownian Motion Is a Martingale

A martingale is an adapted, integrable process \(M_t\) such that \(\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s\) almost surely whenever \(s\le t\). The increment definition verifies this property for Brownian motion. The proof uses independence from the filtration, not just the increment's unconditional mean.

Proof. Adaptedness is part of the definition. For every \(t\ge0\), the increment law and the initial condition give \(\mathbb{E}[B_t^2]=t\), so \(\mathbb{E}|B_t|\le\sqrt{t}<\infty\); thus \(B_t\) is integrable. Fix \(0\le s<t\). The random variable \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has mean zero. To verify the conditional-expectation claim, for any \(A\in\mathcal{F}_s\), independence and integrability give \(\mathbb{E}[\mathbf{1}_A(B_t-B_s)]=\mathbb{P}(A)\mathbb{E}[B_t-B_s]=0\). These equalities for all \(A\in\mathcal{F}_s\), together with \(\mathcal{F}_s\)-measurability of the zero random variable, are the defining test for \(\mathbb{E}[B_t-B_s\mid\mathcal{F}_s]=0\) almost surely. Since \(B_s\) is \(\mathcal{F}_s\)-measurable, conditional linearity yields \(\mathbb{E}[B_t\mid\mathcal{F}_s]=B_s\). For \(s=t\) the equality is immediate. Hence \(B\) is a martingale. \(\square\)

A Common Filtration Pitfall

Knowing independence from \(B_s\) is not enough. A tempting but invalid argument is: “\(B_t-B_s\) is independent of \(B_s\), so it is independent of \(\mathcal{F}_s\).” The conclusion does not follow, because \(\mathcal{F}_s\) may contain more information than \(B_s\).

For a concrete failure, start with a standard Brownian motion under its natural filtration and define the larger filtration \(\mathcal{G}_t=\sigma(B_u:u\ge0)\) for every \(t\). The process still has its usual Gaussian increments and continuous paths, but it is not Brownian motion with respect to \((\mathcal{G}_t)\). Let \(A=\{B_1-B_0>0\}\). This event belongs to \(\mathcal{G}_0\), and its probability is \(\tfrac12\), since \(B_1-B_0\sim N(0,1)\). If the increment were independent of \(\mathcal{G}_0\), independence applied to \(A\) itself would require \(\mathbb{P}(A\cap A)=\mathbb{P}(A)^2=\tfrac14\). But \(A\cap A=A\), so \(\mathbb{P}(A\cap A)=\tfrac12\), a contradiction. The correction is to check independence from the stated \(\mathcal{F}_s\), not simply from \(B_s\) or the process's natural past.

The filtration condition is what supports the martingale conclusion: it ensures each future increment has conditional mean zero given all current information. Together with the Gaussian law and continuity, it forms the basic structure used in later analysis of Brownian paths and processes built from them.