9.2 Wiener process

The Wiener process (also called Brownian motion) is the foundational continuous-time stochastic process with Gaussian increments, independent increments, and continuous sample paths. It underpins stochastic calculus, stochastic differential equations, and a wide range of models in physics, finance, and biology. This guide covers its formal definition, key properties, connections to random walks, quadratic variation, stochastic integration, SDEs, simulation, and applications.

Definition of Wiener process

The Wiener process is a continuous-time stochastic process that serves as the building block for much of stochastic calculus. Informally, it describes a particle that drifts randomly with no memory of its past trajectory, accumulating Gaussian noise over time.

Standard Wiener process

A standard Wiener process $W_t$ is a stochastic process indexed by $t \geq 0$ satisfying four conditions:

1. Initial value: $W_0 = 0$

2. Gaussian increments: For $0 \leq s < t$, the increment $W_t - W_s \sim N(0,\, t - s)$

3. Independent increments: For non-overlapping intervals $[s, t]$ and $[u, v]$ (with $t \leq u$), the increments $W_t - W_s$ and $W_v - W_u$ are independent

4. Continuous paths: The map $t \mapsto W_t$ is continuous almost surely

From these conditions you can immediately read off that $\mathbb{E}[W_t] = 0$ and $\text{Var}(W_t) = t$. The covariance structure follows too: $\text{Cov}(W_s, W_t) = \min(s, t)$.

Brownian motion vs. Wiener process

These two terms are used almost interchangeably, but there's a useful distinction:

• Brownian motion originally refers to the physical phenomenon: the erratic movement of a particle (e.g., pollen grain) bombarded by surrounding molecules.
• Wiener process refers to the mathematical model Norbert Wiener rigorously constructed to describe that motion.

In most stochastic processes courses, the two terms mean the same mathematical object.

Properties of Wiener process

The four defining conditions above give rise to several deeper properties that shape how the Wiener process behaves and how we do calculus with it.

Gaussian increments

Over any interval $[s, t]$, the increment $W_t - W_s$ is normally distributed with mean 0 and variance $t - s$. This means you can write down the exact probability density for any increment:

$W_t - W_s \sim N(0,\, t-s) \quad \Longrightarrow \quad f(x) = \frac{1}{\sqrt{2\pi(t-s)}} \exp\!\left(-\frac{x^2}{2(t-s)}\right)$

Because the distribution of every increment is fully specified, you can compute probabilities and expectations for the process at any collection of times using multivariate normal distributions.

Independent increments

Increments over non-overlapping intervals carry no information about each other. Formally, for $0 \leq s < t \leq u < v$:

$W_t - W_s \perp W_v - W_u$

This is what makes the Wiener process Markovian: the future evolution depends only on the current value, not on how the process got there. It also means the joint distribution of the process at times $t_1 < t_2 < \cdots < t_n$ factors neatly into a product of Gaussian densities for successive increments.

Continuous sample paths

Almost every realization $t \mapsto W_t(\omega)$ is a continuous function. There are no jumps. This distinguishes the Wiener process from processes like the Poisson process or Lévy jump processes, and it's essential for the construction of stochastic integrals.

Non-differentiable paths

Despite being continuous everywhere, the sample paths are nowhere differentiable almost surely. Intuitively, the increments over tiny intervals $\Delta t$ are of order $\sqrt{\Delta t}$, so the difference quotient $\frac{W_{t+\Delta t} - W_t}{\Delta t} \sim \frac{1}{\sqrt{\Delta t}}$ blows up as $\Delta t \to 0$.

This roughness is not a pathology; it's a fundamental feature. It's precisely why ordinary calculus fails for the Wiener process and why we need stochastic calculus (Itô or Stratonovich) instead.

Wiener process as limit of random walk

One of the most illuminating ways to understand the Wiener process is as the continuous limit of a discrete random walk. This connection is made rigorous by Donsker's theorem.

Donsker's theorem

Donsker's theorem (the functional central limit theorem for random walks) says the following:

Let $X_1, X_2, \ldots$ be i.i.d. random variables with $\mathbb{E}[X_i] = 0$ and $\text{Var}(X_i) = \sigma^2 < \infty$. Define the partial sum $S_n = \sum_{i=1}^n X_i$ and the rescaled process

$W_n(t) = \frac{S_{\lfloor nt \rfloor}}{\sigma\sqrt{n}}, \qquad t \in [0,1].$

Then as $n \to \infty$, $W_n(t)$ converges in distribution (in the space of continuous functions $C[0,1]$) to a standard Wiener process $W_t$.

The key steps in the construction:

1. Take a simple random walk with i.i.d. mean-zero, finite-variance steps.
2. Speed up time by a factor of $n$ (take $n$ steps in unit time).
3. Rescale the spatial axis by $\sigma\sqrt{n}$ so the variance per unit time stays 1.
4. In the limit, the piecewise-constant (or linearly interpolated) path becomes a continuous Wiener path.

This theorem justifies using the Wiener process as a continuous-time model whenever the underlying randomness comes from many small, independent contributions.

Functional central limit theorem

The functional CLT generalizes the classical CLT from convergence of single random variables to convergence of entire random functions (stochastic processes). Where the ordinary CLT says $S_n / (\sigma\sqrt{n}) \xrightarrow{d} N(0,1)$, the functional CLT says the process $W_n(\cdot)$ converges to $W(\cdot)$ in the topology of $C[0,1]$. Donsker's theorem is the most important special case of this result.

Quadratic variation of Wiener process

Quadratic variation captures how much a process "wiggles" over time. For smooth deterministic functions, quadratic variation is zero. For the Wiener process, it's not, and this fact drives much of stochastic calculus.

Definition of quadratic variation

For a stochastic process $X_t$ on $[0, T]$, take a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$ with mesh size $|\Pi| = \max_i (t_i - t_{i-1})$. The quadratic variation is

$[X]_T = \lim_{|\Pi| \to 0} \sum_{i=1}^n (X_{t_i} - X_{t_{i-1}})^2$

where the limit is taken in probability (or $L^2$).

Quadratic variation over partitions

For the Wiener process, the quadratic variation has a remarkably clean result:

$[W]_T = T$

To see why this is plausible, note that each squared increment $(W_{t_i} - W_{t_{i-1}})^2$ has expected value $t_i - t_{i-1}$, so the expected value of the sum is $\sum_i (t_i - t_{i-1}) = T$. As the partition refines, the variance of the sum shrinks to zero, and the sum converges to $T$ in probability.

For comparison, any deterministic function of bounded variation has quadratic variation equal to zero.

Stochastic integration with Wiener process

Because Wiener paths are nowhere differentiable, you can't define integrals like $\int f \, dW$ using ordinary Riemann-Stieltjes theory. Two rigorous frameworks exist: the Itô integral and the Stratonovich integral.

Itô integral

The Itô integral evaluates the integrand at the left endpoint of each subinterval. For an adapted process $f(t, \omega)$ (meaning $f$ at time $t$ depends only on information up to time $t$):

$\int_0^T f(t, \omega)\, dW_t = \lim_{|\Pi| \to 0} \sum_{i=1}^n f(t_{i-1}, \omega)\,(W_{t_i} - W_{t_{i-1}})$

Key properties of the Itô integral:

• Linearity: $\int (af + bg)\,dW = a\int f\,dW + b\int g\,dW$
• Itô isometry: $\mathbb{E}\!\left[\left(\int_0^T f\,dW\right)^2\right] = \mathbb{E}\!\left[\int_0^T f^2\,dt\right]$
• Martingale property: The Itô integral $M_t = \int_0^t f\,dW$ is a (local) martingale, so $\mathbb{E}[M_t] = 0$

The martingale property is extremely useful for pricing in finance and for deriving moment formulas, but it comes at a cost: the ordinary chain rule breaks down.

Stratonovich integral

The Stratonovich integral evaluates the integrand at the midpoint:

$\int_0^T f(t, \omega) \circ dW_t = \lim_{|\Pi| \to 0} \sum_{i=1}^n f\!\left(\tfrac{t_{i-1}+t_i}{2},\, \omega\right)(W_{t_i} - W_{t_{i-1}})$

The Stratonovich integral obeys the standard chain rule of calculus, which makes it natural in physics (e.g., when modeling systems with physical symmetries). However, the Stratonovich integral is not a martingale in general.

Stochastic differential equations (SDEs)

SDEs extend ordinary differential equations by adding a noise term driven by a Wiener process. They're the primary tool for modeling systems with continuous random fluctuations.

SDEs driven by Wiener process

The general form of an SDE driven by a Wiener process is:

$dX_t = \mu(t, X_t)\,dt + \sigma(t, X_t)\,dW_t$

• $\mu(t, X_t)$ is the drift coefficient: it describes the deterministic trend.
• $\sigma(t, X_t)$ is the diffusion coefficient: it controls the intensity of random fluctuations.
• $W_t$ is a standard Wiener process.

The integral form makes the meaning more precise:

$X_t = X_0 + \int_0^t \mu(s, X_s)\,ds + \int_0^t \sigma(s, X_s)\,dW_s$

Classical examples include:

• Geometric Brownian motion (Black-Scholes model): $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$, used for stock price modeling
• Ornstein-Uhlenbeck process (Langevin equation): $dX_t = -\theta X_t\,dt + \sigma\,dW_t$, used for mean-reverting phenomena
• Stochastic Lotka-Volterra models in population biology

Itô's lemma

Itô's lemma is the stochastic chain rule. If $X_t$ satisfies $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and $f(t,x)$ is $C^{1,2}$ (once differentiable in $t$, twice in $x$), then:

$df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2}\sigma_t^2 \frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t \frac{\partial f}{\partial x}\,dW_t$

The term $\frac{1}{2}\sigma_t^2 \frac{\partial^2 f}{\partial x^2}$ is the Itô correction. It arises directly from the fact that $[W]_T = T$ (quadratic variation is non-zero). In ordinary calculus this term doesn't appear because smooth functions have zero quadratic variation.

A quick example: applying Itô's lemma to $f(x) = x^2$ with $X_t = W_t$ (so $\mu = 0$, $\sigma = 1$) gives

$d(W_t^2) = 2W_t\,dW_t + dt$

which rearranges to the useful identity $\int_0^T W_t\,dW_t = \frac{1}{2}W_T^2 - \frac{1}{2}T$.

Existence and uniqueness of solutions

Under standard regularity conditions on the coefficients, an SDE has a unique strong solution. The typical sufficient conditions are:

1. Lipschitz continuity: There exists $K > 0$ such that for all $t, x, y$: $|\mu(t,x) - \mu(t,y)| + |\sigma(t,x) - \sigma(t,y)| \leq K|x - y|$

2. Linear growth: There exists $C > 0$ such that for all $t, x$: $|\mu(t,x)| + |\sigma(t,x)| \leq C(1 + |x|)$

A strong solution is a process $X_t$ adapted to the filtration of the given Wiener process that satisfies the SDE almost surely. When strong solutions don't exist (e.g., coefficients are only measurable, not Lipschitz), one can sometimes find weak solutions, where the Wiener process itself is part of the construction, or reformulate the problem as a martingale problem.

Simulation of Wiener process

Simulating Wiener paths is essential for Monte Carlo methods, numerical solution of SDEs, and computational experiments in stochastic modeling.

Discrete approximation methods

The most straightforward approach: discretize time and generate Gaussian increments.

Euler-Maruyama method (for simulating a Wiener path):

1. Choose time points $0 = t_0 < t_1 < \cdots < t_n = T$.
2. Set $W_{t_0} = 0$.
3. For each $i = 0, 1, \ldots, n-1$, draw $Z_i \sim N(0,1)$ independently and compute: $W_{t_{i+1}} = W_{t_i} + \sqrt{t_{i+1} - t_i}\; Z_i$

This method is exact for the Wiener process itself (each increment has exactly the right distribution). The term "Euler-Maruyama" is more commonly used when applying this discretization to solve SDEs, where it introduces a discretization error with strong convergence order 0.5.

For better accuracy when solving SDEs, higher-order schemes exist:

• Milstein scheme: Adds a correction term involving $\sigma \sigma'$, achieving strong order 1.0
• Stochastic Runge-Kutta methods: Provide higher-order convergence at greater computational cost

Exact simulation methods

When you need Wiener process values at specific times without any discretization error, the Brownian bridge technique is the standard tool.

Given $W_0$ and $W_T$, the conditional distribution of $W_t$ for $t \in (0, T)$ is:

$W_t \mid W_0, W_T \;\sim\; N\!\left(W_0 + \frac{t}{T}(W_T - W_0),\; \frac{t(T-t)}{T}\right)$

To generate a full path at times $t_1 < t_2 < \cdots < t_n$:

1. Generate $W_T \sim N(0, T)$.
2. Use the Brownian bridge formula to fill in the midpoint of $[0, T]$.
3. Recursively subdivide each interval, conditioning on the already-generated endpoints.

This produces exact samples (no discretization bias) and is especially valuable for path-dependent quantities like barrier option prices or first-passage time distributions.

Applications of Wiener process

The Wiener process appears across many disciplines wherever continuous random fluctuations need to be modeled.

• Mathematical finance: Geometric Brownian motion (driven by $W_t$) is the foundation of the Black-Scholes option pricing framework. Interest rate models (Vasicek, CIR) and stochastic volatility models (Heston) also rely on Wiener-driven SDEs.
• Physics and engineering: The Langevin equation uses the Wiener process to model thermal noise acting on a particle. Diffusion processes in materials science, signal processing with white noise, and control theory all build on the Wiener process.
• Biology and neuroscience: Stochastic models of gene expression, neural spike trains, and population dynamics use Wiener-driven SDEs to capture intrinsic randomness in biological systems.

In each case, the Wiener process provides the mathematical engine for quantifying how randomness propagates through a system over time.