# 2Conditional Expectation

This chapter is dedicated to the concept of conditional expectation. This is a very important concept that generalizes the concept of conditional probability, when conditioning occurs not only on a single event, but on an entire family of events (a σ-algebra). It is also the basic concept to define martingales, as we will see in Chapter 4. The readers who may wish to explore the concepts in this chapter in greater detail can consult [BAR 07, BIL 12, CHU 01, DAC 83, FOA 03, KAL 02, OUV 08, OUV 09].

Let us begin in section 2.1 by reviewing the definition of conditional probability with respect to an event with strictly positive probability. Following this, section 2.2 gives the general definition for conditional expectations and its chief properties. Next, section 2.3 gives a geometric interpretation as an orthogonal projection in the particular cases of square integrable random variables. Section 2.4 explores the relations between conditional expectation and independence. Finally, section 2.5 offers exercises in working with all these concepts.

Throughout this chapter, we will work on a probability space (Ω, , ℙ).

## 2.1. Conditional probability with respect to an event

Conditional probability accounts for the information brought in by the occurrence of one event on the probability of the occurrence of another event.

DEFINITION 2.1.– Let B ∈ such that ℙ( ...

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