Sigma-Algebras

Motivation: why this matters in quant finance

A probability model does not assign probabilities to arbitrary English sentences. It assigns probabilities to events, and in modern probability the events must form a sigma-algebra. This matters whenever the model has infinitely many possible outcomes: stock paths, Brownian paths, default times, and continuously distributed returns.

In a finite model, one can usually take every subset of $\Omega$. In an uncountable model, Bertsekas flags the measure-theoretic issue: some unusual subsets cannot be assigned meaningful probabilities. A sigma-algebra is the rulebook that keeps the event collection closed under the logical operations probability needs while excluding pathological sets.

For finance, sigma-algebras carry two roles. The full $\mathcal{F}$ says which events can be assigned probabilities. Sub-sigma-algebras such as $\mathcal{F}_t$ say which events are observable at a time or under a given information set.

The informal idea

Read a sigma-algebra as a collection of yes/no questions about the outcome. If $A\in\mathcal{F}$, the model can ask "did $A$ happen?" and assign a probability to the answer.

The axioms say the collection is logically stable:

• If you can ask whether $A$ happened, you can ask whether it did not.
• If you can ask about each event in a countable list, you can ask whether at least one occurred.
• The certain event $\Omega$ is always askable.

A smaller sigma-algebra asks fewer questions and therefore represents coarser information. A larger sigma-algebra asks more questions and represents finer information.

Formal definitions

Sigma-algebra

1. $\Omega\in\mathcal{F}$.
2. If $A\in\mathcal{F}$, then $A^c\in\mathcal{F}$.
3. If $A_1,A_2,\ldots\in\mathcal{F}$, then $\bigcup_{n=1}^{\infty}A_n\in\mathcal{F}$.

These imply $\emptyset\in\mathcal{F}$ and closure under countable intersections.

Generated sigma-algebra

For a collection $\mathcal{C}$ of subsets of $\Omega$, the sigma-algebra generated by $\mathcal{C}$ is

It is the smallest sigma-algebra that contains the original collection.

Borel sigma-algebra

The Borel sigma-algebra on $\mathbb{R}$ is

It contains open intervals, closed intervals, countable unions and intersections of those sets, and the ordinary sets used by probability distributions.

Sigma-algebra generated by a random variable

For a random variable $X$,

This is the information revealed by observing $X$.

Key properties

Intersections preserve sigma-algebras

Any intersection of sigma-algebras on the same $\Omega$ is a sigma-algebra. This is why generated sigma-algebras are well-defined.

Unions do not usually preserve sigma-algebras

If $\mathcal{G}$ and $\mathcal{H}$ are sigma-algebras, $\mathcal{G}\cup\mathcal{H}$ may fail to be closed under intersections or unions across the two collections. The combined information is $\sigma(\mathcal{G}\cup\mathcal{H})$.

Coarser and finer encode information

If $\mathcal{G}\subseteq\mathcal{H}$, then $\mathcal{G}$ is coarser and $\mathcal{H}$ is finer. Conditional expectation with respect to $\mathcal{G}$ must average over distinctions that $\mathcal{G}$ cannot see.

Measurability is compatibility with information

A random variable $X$ is $\mathcal{F}$-measurable when $\sigma(X)\subseteq\mathcal{F}$. The event collection knows enough to answer every question of the form "$X\in B$."

Trivial and full sigma-algebras are extremes

The smallest sigma-algebra is $\{\emptyset,\Omega\}$; it distinguishes nothing. The largest is $2^\Omega$; it distinguishes every subset. Most useful information structures sit between these extremes.

Worked examples

Example 1: generated sigma-algebra on four outcomes

Let $\Omega=\{1,2,3,4\}$ and $\mathcal{C}=\{\{1\},\{2\}\}$. Closing under complements and unions gives

Outcomes $3$ and $4$ are never separated. The information can identify $1$, identify $2$, or identify "not 1 or 2", but it cannot distinguish $3$ from $4$.

Example 2: observing a payoff loses information

Let $S_T$ take values $90,100,110$, and let a call payoff be $H=(S_T-100)^+$. Then $H$ takes values $0$ and $10$. Observing $H=0$ tells you $S_T\le100$, but not whether $S_T=90$ or $100$. Thus

and the inclusion is strict. A payoff is often a coarsening of the underlying price.

Example 3: first-toss information

For two coin tosses, $\Omega=\{HH,HT,TH,TT\}$. The sigma-algebra generated by the first toss is

It answers "was the first toss heads?" but cannot answer "was the second toss heads?" This is the finite-state prototype of a time-$1$ market information sigma-algebra.

Common confusions and pitfalls

Where this goes next

• Random Variables: Measurability is defined through preimages of Borel sets.
• Filtrations and Information: A filtration is an increasing family of sigma-algebras.
• Conditional Expectation: Conditions on a sub-sigma-algebra as an information set.
• Radon-Nikodym Theorem: Supplies the measure-theoretic machinery behind densities and conditional expectation.
• Change of Measure: Keeps the sigma-algebra fixed while changing the probability measure.

References

• Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability (2nd ed.). Athena Scientific. Ch. 1 §1.1 (Sets), §1.2 (Probabilistic Models). Bertsekas flags the uncountable-sample-space measurability issue but does not develop sigma-algebras in full; the generated and Borel constructions are the standard measure-theoretic extension.