Brownian Motion

Author

John Robin Inston

Published

September 25, 2026

1 Intuition

Brownian motion is the continuous-time limit of a random walk. It was first observed as the erratic motion of pollen grains in water, which are kicked about by a huge number of tiny, independent molecular collisions. Each kick is small and unpredictable, and the total displacement over a time interval is a sum of many independent contributions, so by the Central Limit Theorem it should be Gaussian. Two features follow:

  • the displacement over an interval depends only on the length of the interval, not on when it starts (stationarity);
  • displacements over disjoint intervals are unrelated (independence).

Brownian motion is the unique continuous process with these two properties (up to a choice of drift and scale). It is the basic source of randomness in Stochastic Calculus, Stochastic Differential Equations and [[financial-mathematics]].

3 Definition

Consider a continuous-time, continuous-state stochastic process \(B:=(B_{t})_{t\geq 0}\) taking values in \(\mathbb{R}\). For \(s<t\) the increment over \([s,t]\) is \(B_{t}-B_{s}\).

A Brownian motion (BM) or Wiener process is a stochastic process \((B_{t})_{t \geq 0}\) taking values in \(\mathbb{R}\) such that:

  1. \(B_{0}=0\) almost surely (starts at zero);
  2. for all \(0 \leq t_{1}<\dots<t_{n}\) the increments \(B_{t_{2}}-B_{t_{1}},\dots,B_{t_{n}}-B_{t_{n-1}}\) are mutually independent (independent increments);
  3. for all \(s < t\), \(B_{t}-B_{s}\sim\mathcal{N}(0,t-s)\) (normal stationary increments);
  4. \(t\mapsto B_{t}(\omega)\) is continuous for almost every \(\omega\) (continuous paths, no jumps).

Note: independent increments means that non-overlapping intervals carry no information about each other. If \(\sigma^2\neq 1\) in (3) we speak of a BM with variance parameter \(\sigma^2\); this is just \(\sigma B\) for a standard BM \(B\). A BM with drift \(\mu\) is \(\mu t+\sigma B_{t}\).

3.1 Brownian motion with respect to a measure

Conditions (1)-(4) are statements about probabilities, so “being a Brownian motion” is not a property of the paths alone, it depends on the probability measure and, for stochastic calculus, on the filtration.

Let \((\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})\) be a filtered probability space (filtration). A process \(W\) is an \((\mathcal{F}_{t})\)-Brownian motion under \(\mathbb{P}\) (a \(\mathbb{P}\)-BM) if it is adapted, has continuous paths, \(W_{0}=0\), and for all \(s<t\) \[ W_{t}-W_{s}\perp \mathcal{F}_{s}\quad\text{and}\quad W_{t}-W_{s}\sim\mathcal{N}(0,t-s)\ \text{ under }\mathbb{P}. \]

What this means in practice:

  • The probabilities (independence and the \(\mathcal{N}(0,t-s)\) law) are all computed with \(\mathbb{P}\). The same process \(W\) can fail to be a Brownian motion under another measure \(\mathbb{Q}\) that weights paths differently.
  • Independence of \(\mathcal{F}_{s}\) (rather than only of earlier increments) says the future increments are unpredictable given everything known at time \(s\). This is what makes \(\int H\,dW\) well defined for adapted \(H\) (Itô Integral).
  • Changing measure changes the drift but not the volatility. By Girsanov Theorem, if \(\mathrm{d}\mathbb{Q}/\mathrm{d}\mathbb{P}\) is built from a process \(\theta\) (see Radon-Nikodym Derivative), then \[ W^\mathbb{Q}_{t}=W_{t}+\int_{0}^{t}\theta_{s}\,ds \] is a \(\mathbb{Q}\)-BM, while \(W\) itself is a \(\mathbb{Q}\)-BM with drift \(-\theta\).

Under \(\mathbb{P}\) let \(W\) be a BM and fix \(\mu\in\mathbb{R}\). The process \(X_{t}=W_{t}+\mu t\) has \(X_{t}-X_{s}\sim\mathcal{N}(\mu(t-s),t-s)\) so it is not a \(\mathbb{P}\)-BM. Define \(\mathbb{Q}\) by \[ \frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\left(-\mu W_{T}-\tfrac{1}{2}\mu^2T\right) \] on \(\mathcal{F}_{T}\). Then \(X\) is a \(\mathbb{Q}\)-BM on \([0,T]\): the density tilts the probability of paths exactly enough to cancel the drift. This is the mechanism behind the risk-neutral measure in [[financial-mathematics]], where asset prices acquire the risk-free drift under \(\mathbb{Q}\).

4 Results

Throughout \(B\) is a standard BM under \(\mathbb{P}\).

  1. Gaussian process: \(B\) is a Gaussian process with mean \(0\) and covariance \(\mathrm{Cov}(B_{s},B_{t})=\min(s,t)\).
  2. Martingales: \(B_{t}\), \(B_{t}^2-t\) and \(\exp(\lambda B_{t}-\tfrac{1}{2}\lambda^2t)\) are martingales.
  3. Markov property: for any \(s\), \((B_{s+t}-B_{s})_{t\geq 0}\) is a BM independent of \(\mathcal{F}_{s}\) (Markov Property). The same holds at Stopping Times.
  4. Scaling: for \(c>0\), \((c^{-1/2}B_{ct})_{t\geq 0}\) is a BM.
  5. Time inversion: \(tB_{1/t}\) (with value \(0\) at \(t=0\)) is a BM.
  6. Reflection principle: for \(a>0\), \(\mathbb{P}\left(\max_{s\leq t}B_{s}\geq a\right)=2\,\mathbb{P}(B_{t}\geq a)\).

Almost surely, the paths of \(B\)

  • are continuous but nowhere differentiable;
  • have infinite total variation on every interval;
  • have quadratic variation \([B]_{t}=t\), i.e. for partitions \(0=t_{0}<\dots<t_{n}=t\) with mesh going to zero, \(\sum_{i}(B_{t_{i+1}}-B_{t_{i}})^2\to t\) in \(L^2\);
  • oscillate without bound and return to \(0\) infinitely often (recurrence in one dimension).

Proof. Sketch for quadratic variation. Write \(\Delta_{i}=B_{t_{i+1}}-B_{t_{i}}\). Then \(\mathbb{E}[\Delta_{i}^2]=t_{i+1}-t_{i}\) so \(\mathbb{E}\sum_{i}\Delta_{i}^2=t\), and by independence \[ \mathrm{Var}\left(\sum_{i}\Delta_{i}^2\right) =\sum_{i}2(t_{i+1}-t_{i})^2\leq 2t\max_{i}(t_{i+1}-t_{i})\to 0. \] A random variable with variance tending to zero converges in \(L^2\) to its mean, giving \([B]_{t}=t\). Infinite variation follows because finite variation would force the quadratic variation to be zero.

The identity \((dB_{t})^2=dt\) is the source of the extra second-order term in the Itô Formula, and it is why ordinary calculus fails for Brownian paths.

A continuous adapted process \(W\) with \(W_{0}=0\) is an \((\mathcal{F}_{t})\)-BM under \(\mathbb{P}\) if and only if both \(W_{t}\) and \(W_{t}^2-t\) are continuous \(\mathbb{P}\)-martingales (equivalently \(W\) is a continuous martingale with \([W]_{t}=t\)).

Lévy’s characterisation is the standard way to verify that a process is a Brownian motion under a new measure, and it is what drives the proof of the Girsanov Theorem.

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