Wiener Process

The Wiener process is the rigorous mathematical model of Brownian motion: a continuous process starting at zero whose increments over disjoint intervals are independent and normally distributed with variance equal to the elapsed time.

Definition

The Wiener process is a stochastic process with continuous paths starting at the origin, having independent increments, and with the increment over any interval normally distributed with mean zero and variance equal to the length of the interval, providing the canonical model of Brownian motion.

Scope

This topic covers the defining properties of the Wiener process, its existence and Wiener's construction, the continuity yet nowhere-differentiability of its paths, its quadratic variation equal to elapsed time, the strong Markov property and reflection principle, scaling and time-inversion invariances, and the law of the iterated logarithm describing its fine fluctuations.

Core questions

• What axioms define the Wiener process and guarantee its existence?
• Why are its paths continuous but nowhere differentiable?
• What is its quadratic variation and why does it equal elapsed time?
• How do the reflection principle and strong Markov property describe its behaviour?

Key theories

Path properties and quadratic variation

Wiener-process paths are almost surely continuous yet nowhere differentiable and of infinite total variation, but their quadratic variation over any interval equals the interval's length, the property that makes stochastic integration possible.

Strong Markov property and reflection principle

The process restarts afresh at stopping times, and reflecting the path after it first reaches a level gives the distribution of the running maximum and of first-passage times, a powerful tool for hitting-time computations.

Clinical relevance

The Wiener process models the thermal motion of microscopic particles, serves as the driving noise in stochastic differential equations and the Black-Scholes model of asset prices, appears as the scaling limit of random walks through Donsker's invariance principle, and underlies signal-plus-noise models in engineering.

History

Bachelier modelled stock prices with the process in 1900 and Einstein gave its physical theory in 1905, but it was Wiener who in 1923 proved that a probability measure with the required properties exists on the space of continuous functions, after which Levy and others mapped out its remarkable path properties.

Key figures

• Norbert Wiener
• Albert Einstein
• Louis Bachelier
• Paul Levy

Seminal works

• morters2010

Frequently asked questions

Is the Wiener process the same as Brownian motion?

Yes; the Wiener process is the mathematically rigorous definition of Brownian motion, named after Norbert Wiener who first constructed it as a measure on continuous paths.

How can a path be continuous but nowhere differentiable?

The path never jumps, so it is continuous, yet it oscillates so violently at every scale that no tangent direction exists at any point, which is why its total variation is infinite.