Chapter 2: Stochastic Processes¶

The central idea of this chapter is that randomness, when combined with information, produces structure. We build stochastic processes in three stages:

Roadmap¶

Key Structure¶

1. Simple Random Walk → Discrete Foundation¶

The simple random walk

provides the fundamental model of accumulated randomness. This section develops:

• algebraic structure (moments, MGF)
• martingale structure
• long-run behavior (recurrence)
• scaling limits

The key result is Donsker's theorem:

which explains the emergence of Brownian motion from discrete randomness.

2. Brownian Motion → Continuous Model¶

Brownian motion is the continuous-time limit of random walks, characterized by:

• Gaussian independent increments
• continuous paths

Its structure is developed through:

• scaling and self-similarity
• path roughness (non-differentiability)
• quadratic variation
• symmetry and hitting behavior

This section provides the geometric and probabilistic structure of continuous randomness.

3. Filtration and Martingale → Information and Dynamics¶

A filtration \((\mathcal{F}_t)\) describes the flow of information. Martingales are processes that evolve without predictable drift:

This section develops:

• conditional expectation (prediction)
• adapted processes (information consistency)
• stopping times (random horizons)
• inequalities and convergence
• structural decomposition (Doob–Meyer)

These tools form the foundation for pricing and stochastic calculus.

Conceptual Flow¶

flowchart LR
A[Random Walk]
--> B[Scaling Limit]
--> C[Brownian Motion]
--> D[Filtration]
--> E[Martingales]