Skip to content
Stochastic Calculus / Discrete Martingales and the Gambling Blueprint

Conditional Expectation and σ-Algebras

The pricing formula Vt=e−r(T−t)EQ[VT∣Ft]V_t = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[V_T \mid \mathcal{F}_t] looks deceptively familiar, but the object EQ[VT∣Ft]\mathbb{E}^{\mathbb{Q}}[V_T \mid \mathcal{F}_t] is not the elementary conditional expectation E[X∣Y=y]\mathbb{E}[X \mid Y = y] from the earlier probability course. It is a random variable — one that depends on which path the market has taken up to time tt — and the σ-algebra Ft\mathcal{F}_t is what makes "all information available at time tt" into a precise mathematical object. Without this precision, the tower property is just a hand-wave, and Girsanov's change of measure cannot be stated.

This note develops the measure-theoretic definition of conditional expectation, proves the six properties it must satisfy, and presents the L2L^2 projection interpretation that connects it to Hilbert-space geometry. These foundations underlie every result in the stochastic calculus unit: martingales (which are just CE constraints on a process), the Feynman-Kac formula (which converts a PDE into a CE), and Girsanov's theorem (which re-weights CE by a density).

Learning objectives. By the end of this note you should be able to:
  • Define a σ-algebra and explain why the trivial and full σ-algebras represent "no information" and "complete information."
  • State the measure-theoretic definition of E[Y∣G]\mathbb{E}[Y \mid \mathcal{G}] and describe its two defining properties.
  • Prove linearity, the tower property, and "taking out what is known" from the definition.
  • Interpret conditional expectation as the L2L^2 orthogonal projection onto the subspace of G\mathcal{G}-measurable random variables.
  • Apply the properties to simplify conditional expectations in finance examples (discounted stock price, option payoff conditioning).

In this lesson

  1. σ-Algebras and information
  2. The measure-theoretic definition
  3. Core properties
  4. The L2L^2 projection interpretation
  5. Key example: a time series

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.