Brownian Motion

Tomorrow's stock price is a random variable. A molecule diffuses in a liquid along a random path. Noise in an electric circuit is unpredictable. All of these phenomena are described by Brownian motion-the most fundamental continuous-time random process and the foundation of all stochastic analysis.

• **Financial markets**-the Black-Scholes model assumes the log of the stock price is Brownian motion with drift
• **Diffusion models (DDPM)**-the noise addition and removal process in modern generative AI is based on Brownian motion SDE
• **Physics**-Brownian diffusion, the Fokker-Planck equation, Feynman's path integrals in quantum mechanics all use Brownian paths

Предварительные знания

• Martingale Inequalities and Convergence
• Poisson Process

Brownian Motion

In 1827, botanist Robert Brown observed the chaotic motion of pollen grains in water. In 1905, Einstein explained it as the result of random molecular collisions. The mathematical model-**Brownian motion** (the Wiener process)-was constructed by Norbert Wiener in 1923. It is a continuous random process with independent Gaussian increments: the limit of a normalized random walk.

**Standard Brownian motion**-a process {B(t), t ≥ 0} such that: 1. B(0) = 0 2. Independent increments: B(t)-B(s) ⊥ B(u), u ≤ s 3. Gaussian increments: B(t)-B(s) ~ N(0, t-s) 4. Continuous trajectories: t ↦ B(t) is continuous a.s.

Brownian motion is simultaneously a **martingale** and a **Gaussian process**. The covariance function Cov(B(s), B(t)) = min(s, t) is its complete characterization as a Gaussian process. Brownian motion is the central object of stochastic analysis: Ito integrals, the Black-Scholes equation, and diffusion models in ML are all built on it.

Variants of the Wiener Process

Standard Brownian motion is just one member of a whole family of processes. Geometric Brownian motion (stock prices), the Brownian bridge (connecting two fixed values), Brownian motion with drift (directed diffusion)-all of these are built from B(t) by simple transformations.

**Important variants:** - BM with drift: X(t) = μt + σB(t), E[X(t)] = μt - Geometric BM: S(t) = S(0)·exp((μ-σ²/2)t + σB(t))-stock price model - Brownian bridge: X(t) = B(t) - t·B(1), t ∈ [0,1], X(0)=X(1)=0 - Ornstein-Uhlenbeck: dX = -θX dt + σ dB-'elastic' Brownian motion

**Reflected Brownian motion** |B(t)| and the **Bessel process** are used in problems with boundaries (queues with reflection). **Fractional Brownian motion** with Hurst parameter H ≠ 1/2 models long-range dependence (H > 1/2) or anti-persistence (H < 1/2)-important in financial markets.

Quadratic Variation and Continuity

Classical functions of bounded variation have finite 'length.' Brownian motion is different: its **quadratic variation** over [0,T] equals T (finite), but its **total variation** is infinite. This is not just a technical detail-it is precisely the quadratic variation that explains why Ito's formula differs from the classical Newton-Leibniz rule.

**Quadratic variation of BM:** For a partition π = {0=t_0 < t_1 < ... < t_n = T}: sum_i (B(t_{i+1}) - B(t_i))² → T (in L² and a.s.) Written as [B,B]_T = T, or in differential form: (dB)² = dt. **Total variation:** sum_i |B(t_{i+1}) - B(t_i)| → ∞ for fine partitions-infinite. **Consequence for Ito's formula:** f(B(t)) ≠ ∫f'(B)dB; a correction (1/2)f'' dt is needed.

Hölder continuity: BM trajectories satisfy |B(t)-B(s)| = O(|t-s|^{1/2-ε}) for any ε > 0, but not for ε = 0. The Hausdorff dimension of the trajectory is 3/2-an 'intermediate' object between a curve (dimension 1) and a square (dimension 2): a fractal.

Construction of Brownian Motion

Proving the existence of a process with such properties is non-trivial. Norbert Wiener constructed BM in 1923 explicitly via Fourier series with random coefficients. Later, simpler constructions appeared: via random walks (Donsker's theorem) and via the Ito integral.

**Lévy-Ciesielski construction (Haar wavelets):** B(t) = Z_0·t + sum_{n=0}^{∞} sum_{k=0}^{2^n - 1} Z_{n,k} · H_{n,k}(t) where Z_{n,k} ~ N(0,1) are independent and H_{n,k} are Haar functions. **Donsker's theorem (functional CLT):** The normalized random walk S^n(t) = S_{floor(nt)} / sqrt(n) → B(t) weakly in C[0,1]

Donsker's theorem is a **functional** analogue of the CLT: convergence not just for one time point t, but for the entire trajectory as an element of C[0,1]. This allows proving theorems about extrema, time above zero, and other functionals of the random walk by 'passing to the limit'-Brownian motion.

Key Ideas

• **Brownian motion**-a Gaussian martingale with independent increments B(t)-B(s) ~ N(0,t-s) and continuous trajectories
• **Quadratic variation** [B,B]_t = t: the key property; (dB)² = dt distinguishes BM from differentiable functions
• **Variants**-geometric BM (prices), Brownian bridge (statistics), Ornstein-Uhlenbeck (interest rates)
• **Donsker's theorem**-normalized random walk converges to BM; the functional analogue of the CLT

Related Topics

Brownian motion is the foundation of continuous stochastic analysis:

• Ito Stochastic Integrals — The Ito integral is built over BM trajectories; Ito's formula follows from quadratic variation
• Financial Mathematics — The stock price in Black-Scholes is geometric Brownian motion
• Diffusion Models (ML) — DDPM and score-based models use forward diffusion (adding BM noise) and the reverse SDE

Вопросы для размышления

• If BM is nowhere differentiable, how can we speak of a 'diffusion velocity'? What is the diffusion coefficient σ in dX = μdt + σdB?
• BM trajectories are fractals with Hausdorff dimension 3/2. What does this imply for practical simulations?
• Geometric BM can only be positive. Why is this physically reasonable for a stock price model?

Связанные уроки

• prob-19-brownian