Brownian Motion - Standard & General Processes, Wiener Process

Physical Background & Mathematical Definition

In 1827, British botanist Robert Brown observed "irregular motion of pollen suspended in water." In 1905, Einstein explained this phenomenon using "random collisions of water molecules." In 1923, Wiener provided the rigorous mathematical definition. Brownian motion, also known as the Wiener process, is the core stochastic process describing "continuous-time, continuous-state irregular motion," widely applied in physical diffusion, financial pricing, and other fields.

Standard Brownian Motion (Wiener Process)

A continuous-time stochastic process {B(t); t ≥ 0} satisfying four conditions:

• Initial condition: B(0) = 0
• Stationary independent increments
• Increment normal distribution: B(t) - B(s) ~ N(0, t-s)
• Sample path continuity

General Brownian Motion

If {B(t)} is standard Brownian motion, introducing drift coefficient μ and diffusion coefficient σ > 0:

X(t) = μt + σB(t)

μ > 0: average rightward motion, μ < 0: average leftward motion

Core Properties of Standard Brownian Motion

Normal Process Characteristics

For any n ≥ 1 and 0 < t₁ < t₂ < ... < tₙ, the n-dimensional random variable (B(t₁), B(t₂), ..., B(tₙ)) follows an n-dimensional normal distribution.

All statistical properties can be completely determined by mean and covariance functions.

Numerical Characteristics

• Mean function: μ_B(t) = E[B(t)] = 0
• Variance function: D_B(t) = Var[B(t)] = t
• Covariance function: Cov[B(s), B(t)] = min(s, t)
• Autocorrelation: R_B(s, t) = min(s, t)

Markov Property

For any 0 ≤ s ≤ t, P{B(t) ≤ y | B(u), u ≤ s} = P{B(t) ≤ y | B(s)}.

Future depends only on current state, not on historical trajectory.

Self-Similarity

For any constant α > 0, {B(αt)/√α; t ≥ 0} is still standard Brownian motion.

Time scaling by α, position scaling by √α, motion pattern unchanged.

Reflection Principle & First Passage Time

The reflection principle is a powerful tool for analyzing Brownian motion behavior, particularly useful for calculating first passage times and maximum distributions.

First Passage Time Distribution

For a > 0, first passage time T_a = inf{t ≥ 0 | B(t) = a}:

P{T_a ≤ t} = 2P{B(t) ≥ a}

Using reflection principle: "T_a ≤ t" equivalent to "B(t) ≥ a or B(t) ≤ -a"

Maximum Distribution

For t > 0, maximum in (0, t]: M(t) = max{0≤s≤t} B(s):

P{M(t) ≥ a} = P{T_a ≤ t}

"Maximum ≥ a" equivalent to "first passage time ≤ t"

Brownian Bridge Process

When we need to "force Brownian motion to return to origin at t = 1" (like fixed-end elastic string vibration), we introduce the Brownian bridge process.

Definition

Let {B(t); t ≥ 0} be standard Brownian motion, define:

X(t) = B(t) - tB(1)

{X(t); 0 ≤ t ≤ 1} is called Brownian bridge process, fixed at both ends at 0.

Core Properties

• Boundary conditions: X(0) = 0, X(1) = 0
• Normal process characteristics
• Mean function: μ_X(t) = 0
• Covariance: Cov[X(s), X(t)] = s(1-t)
• Variance: D_X(t) = t(1-t)

Applications & Real-World Examples

Financial Applications

Geometric Brownian Motion (Stock Price Model):

S(t) = S₀ e^(μt + σB(t))

μ: expected return rate, σ: volatility rate, {B(t)}: standard Brownian motion

Log returns follow normal distribution, widely used in option pricing and risk management.

Physical Applications

Diffusion Processes:

Brownian motion models particle diffusion in liquids/gases. Pollen (or molecules) experience random collisions, position changes follow Brownian motion.

Diffusion equation: probability density satisfies Fourier diffusion equation, diffusion coefficient positively correlated with temperature.