9.1 Definition and properties of Brownian motion

Definition of Brownian motion

Brownian motion is a continuous-time stochastic process that models random, erratic movement. Named after botanist Robert Brown, who observed pollen grains jittering unpredictably in water, it has become a cornerstone for modeling randomness in physics, finance, and many other fields.

The physical picture is simple: a tiny particle gets bombarded from all sides by molecules in a fluid, producing a jagged, unpredictable trajectory. The mathematical formalization of this idea turns out to be remarkably rich.

Mathematical formulation

A stochastic process $\{B(t), t \geq 0\}$ is a (standard) Brownian motion if it satisfies three defining properties:

1. Starts at zero: $B(0) = 0$.

2. Gaussian increments: For any $0 \leq s < t$, the increment $B(t) - B(s) \sim \mathcal{N}(0,\, t - s)$. That is, it's normally distributed with mean 0 and variance $t - s$.

3. Independent increments: For any collection of non-overlapping time intervals, the corresponding increments are independent random variables.

From these three properties, you can derive the probability density of $B(t)$ at position $x$:

$p(x, t) = \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{x^2}{2t}\right)$

This is just the pdf of a $\mathcal{N}(0, t)$ random variable, which follows directly from property 2 with $s = 0$.

Physical interpretation

At the microscopic level, Brownian motion captures the rapid, irregular changes in a particle's velocity and direction caused by molecular collisions. The trajectory looks completely random.

At the macroscopic level, though, the collective behavior of many such particles becomes predictable. The concentration of particles evolves according to the diffusion equation $\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}$, connecting the probabilistic description to a deterministic PDE. This link between microscopic randomness and macroscopic determinism is one of the deep insights Brownian motion provides.

Properties of Brownian motion

Beyond the three defining properties above, Brownian motion has several additional structural properties that make it both mathematically interesting and practically useful.

Gaussian increments

Already part of the definition, but worth emphasizing: because every increment $B(t) - B(s)$ is Gaussian, the entire finite-dimensional distribution of Brownian motion is determined. Any vector $(B(t_1), B(t_2), \ldots, B(t_n))$ is jointly Gaussian with covariance $\text{Cov}(B(s), B(t)) = \min(s, t)$. This means you can apply the full machinery of Gaussian analysis to study Brownian motion.

Independent increments

For any times $s_1 < t_1 \leq s_2 < t_2$, the increments $B(t_1) - B(s_1)$ and $B(t_2) - B(s_2)$ are independent. This is what makes many computations tractable: knowing what happened in one time window tells you nothing about what happens in a later, non-overlapping window.

Continuous sample paths

The sample paths $t \mapsto B(t)$ are continuous functions of time with probability 1. There are no jumps or discontinuities. This continuity is not obvious from the definition (it's sometimes included as an explicit axiom, sometimes proved as a consequence depending on the construction used), and it's essential for the connection to diffusion processes.

Non-differentiable paths

Here's the striking counterpart to continuity: Brownian paths are almost surely nowhere differentiable. The paths are so jagged that at no point does a well-defined slope exist. Intuitively, the increments over a tiny interval $[t, t+h]$ have standard deviation $\sqrt{h}$, which grows much faster than $h$ as $h \to 0$. So the difference quotient $\frac{B(t+h) - B(t)}{h}$ blows up.

This is exactly why ordinary calculus fails for Brownian motion and why stochastic calculus (Itô calculus) is needed.

Markov property

The future of Brownian motion depends only on where it is now, not on how it got there. Formally, for $s < t$:

$\mathbb{P}(B(t) \in A \mid \mathcal{F}_s) = \mathbb{P}(B(t) \in A \mid B(s))$

where $\mathcal{F}_s$ is the filtration (the accumulated information) up to time $s$. This follows directly from independent increments: the future increment $B(t) - B(s)$ is independent of $\mathcal{F}_s$.

Martingale property

Brownian motion is a martingale with respect to its natural filtration. This means:

$\mathbb{E}[B(t) \mid \mathcal{F}_s] = B(s) \quad \text{for } s < t$

The proof is short: $\mathbb{E}[B(t) \mid \mathcal{F}_s] = \mathbb{E}[B(s) + (B(t) - B(s)) \mid \mathcal{F}_s] = B(s) + 0 = B(s)$, since the increment has mean zero and is independent of $\mathcal{F}_s$.

The martingale property is central to financial modeling, where it underpins the idea of "fair game" pricing of derivatives.

Self-similarity and scaling

Brownian motion is self-similar with exponent $1/2$. Concretely, for any constant $c > 0$, the rescaled process

$\{c^{-1/2} B(ct),\, t \geq 0\}$

is again a standard Brownian motion. You can verify this by checking that the rescaled process satisfies all three defining properties.

This means zooming into a Brownian path (scaling time by $c$ and space by $\sqrt{c}$) produces something statistically identical to the original. The path looks equally jagged at every scale.

Wiener process

The terms "Brownian motion" and "Wiener process" are used interchangeably in most modern texts. The name honors Norbert Wiener, who gave the first rigorous mathematical construction of the process in 1923.

Standard Brownian motion

The standard Wiener process $W(t)$ is defined by the same three axioms listed above: $W(0) = 0$, Gaussian increments with variance $t - s$, and independent increments. It has zero drift and unit diffusion coefficient. This serves as the building block from which more general processes are constructed.

Drift and diffusion coefficients

A Brownian motion with drift is obtained by adding a deterministic linear trend:

$X(t) = \mu t + \sigma W(t)$

• The drift $\mu$ controls the average rate of change. If $\mu > 0$, the process tends to increase over time.
• The diffusion coefficient $\sigma$ controls the magnitude of random fluctuations. Larger $\sigma$ means wider spread.

At time $t$, $X(t) \sim \mathcal{N}(\mu t,\, \sigma^2 t)$.

Generalized Wiener process

More generally, an Itô process allows the drift and diffusion to depend on both time and the current state:

$dX(t) = \mu(t, X(t))\, dt + \sigma(t, X(t))\, dW(t)$

This stochastic differential equation (SDE) is the workhorse of applied stochastic modeling. The notation "$dW(t)$" represents an infinitesimal Brownian increment, made rigorous through the Itô integral.

Note: not every Itô process is called a "generalized Wiener process" in all references. Some texts reserve "generalized Wiener process" for the case where $\mu$ and $\sigma$ are constants or functions of time only. Be aware of the terminology your course uses.

Stochastic integrals and calculus

Because Brownian paths are nowhere differentiable, the classical Riemann-Stieltjes integral $\int f\, dB$ doesn't work in the usual sense. Stochastic calculus provides the tools to define and manipulate such integrals rigorously.

Itô integral

The Itô integral of an adapted process $X(t)$ with respect to Brownian motion is defined as a limit of left-endpoint Riemann sums:

$\int_0^T X(t)\, dW(t) = \lim_{n \to \infty} \sum_{i=0}^{n-1} X(t_i)\, \bigl(W(t_{i+1}) - W(t_i)\bigr)$

where $0 = t_0 < t_1 < \cdots < t_n = T$ is a partition of $[0, T]$.

The key word is left-endpoint: the integrand is evaluated at $t_i$, not at some midpoint. This choice ensures the integral is a martingale and gives the Itô isometry:

$\mathbb{E}\left[\left(\int_0^T X(t)\, dW(t)\right)^2\right] = \mathbb{E}\left[\int_0^T X(t)^2\, dt\right]$

The tradeoff is that the ordinary chain rule breaks. Instead, you must use Itô's lemma, which includes an extra second-order correction term.

Stratonovich integral

The Stratonovich integral uses midpoint evaluation instead:

$\int_0^T X(t) \circ dW(t) = \lim_{n \to \infty} \sum_{i=0}^{n-1} X\!\left(\frac{t_i + t_{i+1}}{2}\right) \bigl(W(t_{i+1}) - W(t_i)\bigr)$

The advantage: the classical chain rule holds, so formulas from ordinary calculus carry over directly. The disadvantage: the Stratonovich integral is not a martingale, which complicates probabilistic analysis.

The two integrals are related by a correction term. For a smooth function $f$:

$\int_0^T f(W(t)) \circ dW(t) = \int_0^T f(W(t))\, dW(t) + \frac{1}{2}\int_0^T f'(W(t))\, dt$

Physics literature tends to prefer Stratonovich (because it respects coordinate changes), while finance and probability theory almost always use Itô.

Stochastic differential equations

An SDE has the general form:

$dX(t) = \mu(t, X(t))\, dt + \sigma(t, X(t))\, dW(t)$

This is shorthand for the integral equation:

$X(t) = X(0) + \int_0^t \mu(s, X(s))\, ds + \int_0^t \sigma(s, X(s))\, dW(s)$

The solution $X(t)$ is a stochastic process driven by both a deterministic component (the drift integral) and a random component (the Itô integral). Under Lipschitz and linear growth conditions on $\mu$ and $\sigma$, a unique strong solution exists.

SDEs appear throughout applied mathematics:

• Finance: Geometric Brownian motion $dS = \mu S\, dt + \sigma S\, dW$ models stock prices.
• Physics: The Langevin equation models particle motion with friction and noise.
• Biology: Population models with environmental randomness.

Applications of Brownian motion

Physics and thermodynamics

Brownian motion originated in physics, and the connection runs deep.

• Einstein's 1905 paper showed that the mean squared displacement of a Brownian particle grows linearly with time: $\mathbb{E}[B(t)^2] = t$. This provided indirect evidence for the existence of atoms.
• The diffusion equation $\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$ governs how particle concentrations spread over time (Fick's second law). The fundamental solution is exactly the Gaussian density of Brownian motion.
• The fluctuation-dissipation theorem connects the intensity of random thermal fluctuations to the system's dissipative (friction) response, linking Brownian motion to thermodynamic equilibrium.

Financial modeling and options pricing

The Black-Scholes model assumes that the log-price of a stock follows Brownian motion with drift, so the stock price itself follows geometric Brownian motion:

$dS(t) = \mu S(t)\, dt + \sigma S(t)\, dW(t)$

Using Itô calculus, Black and Scholes derived a closed-form formula for European option prices. The key insight is that under a risk-neutral measure, the discounted stock price becomes a martingale.

Beyond Black-Scholes, Brownian motion appears in:

• Interest rate models: Vasicek, Cox-Ingersoll-Ross
• Credit risk: Structural models (Merton's model)
• Portfolio theory: Continuous-time mean-variance optimization

Signal processing and filtering

• The Wiener filter estimates a desired signal from noisy observations by minimizing mean squared error, assuming Gaussian noise with known spectral properties.
• The Kalman filter extends this to dynamic state-space models, using Brownian motion to model both process noise and measurement noise. It's the backbone of navigation systems, tracking algorithms, and control engineering.
• Fractional Brownian motion (a generalization with correlated increments and Hurst parameter $H \neq 1/2$) models long-range dependence in signals that standard Brownian motion cannot capture.

Queueing theory and network traffic

In heavy-traffic regimes (where arrival rate is close to service rate), the queue length process can be approximated by a reflected Brownian motion. This diffusion approximation makes it possible to compute performance metrics like mean waiting time and overflow probabilities that would be intractable with the exact discrete model.

For network traffic modeling, empirical data from Ethernet and Internet traffic shows self-similar and long-range dependent behavior. Fractional Brownian motion, with its tunable correlation structure, provides a better fit than standard Brownian motion for these applications.