Brownian Motion Formulas - Standard & General Processes, Wiener Process

Standard Brownian Motion (Wiener Process)

Definition & Properties

Initial Condition

B(0) = 0

Increment Distribution

B(t) - B(s) ~ N(0, t-s)

Sample Path Continuity

B(t, ω) continuous in t

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)

General Brownian Motion with Drift & Diffusion

Process Definition

General Brownian Motion

X(t) = μt + σB(t)

μ: drift coefficient, σ: diffusion coefficient

Increment Distribution

Increment Properties

X(t) - X(s) ~ N(μ(t-s), σ²(t-s))

Mean = μ × time difference, Variance = σ² × time difference

Numerical Characteristics

Mean Function

μ_X(t) = E[X(t)] = μt

Variance Function

D_X(t) = Var[X(t)] = σ²t

Physical Interpretation

• μ > 0: average rightward motion

• μ < 0: average leftward motion

• σ: controls irregularity (larger σ = more irregular)

• σ²: diffusion coefficient (temperature dependent)

Reflection Principle & First Passage Time

First Passage Time

Definition

T_a = inf{t ≥ 0 | B(t) = a}

Distribution Function

P{T_a ≤ t} = 2P{B(t) ≥ a}

Standard Normal Form

P{T_a ≤ t} = 2(1 - Φ(a/√t))

Maximum Distribution

Definition

M(t) = max{0≤s≤t} B(s)

Distribution Function

P{M(t) ≥ a} = P{T_a ≤ t}

Probability

P{M(t) ≥ a} = 2(1 - Φ(a/√t))

Reflection Principle Key Insight

"T_a ≤ t" ⇔ "B(t) ≥ a or B(t) ≤ -a"

The reflection principle states that the probability of reaching level a by time t equals twice the probability of being above level a at time t.

Brownian Bridge Process

Process Definition

Definition

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

Domain

0 ≤ t ≤ 1

Boundary Conditions

X(0) = 0, X(1) = 0

Statistical Properties

Mean Function

μ_X(t) = 0

Variance Function

D_X(t) = t(1-t)

Covariance Function

Cov[X(s), X(t)] = s(1-t)

Physical Interpretation

Fixed-End Elastic String Vibration

The Brownian bridge models a stochastic process that is constrained to return to the origin at t = 1, similar to an elastic string fixed at both ends. Maximum variance occurs at t = 0.5.

Geometric Brownian Motion & Financial Applications

Stock Price Model

Process Definition

S(t) = S₀e^(μt + σB(t))

Log Returns

ln S(t) - ln S₀ ~ N(μt, σ²t)

Expected Price

E[S(t)] = S₀e^(μt + σ²t/2)

Financial Parameters

S₀

Initial stock price

Expected annual return rate

Annual volatility rate

Applications in Finance

Option Pricing

Black-Scholes model uses geometric Brownian motion to price European options, providing closed-form solutions for call and put options.

Risk Management

Value at Risk (VaR) calculations and portfolio optimization rely on geometric Brownian motion for modeling asset price movements.

Self-Similarity & Scaling Properties

Self-Similarity

Scaling Property

{B(αt)/√α; t ≥ 0}

Distribution Invariance

B(αt)/√α ~ N(0, t)

Time Scaling

t → αt, B(t) → B(αt)/√α

Physical Interpretation

• Time scaling by factor α

• Position scaling by factor √α

• Motion pattern remains unchanged

• Fractal-like behavior at all scales

• Important for modeling phenomena at different time scales