Conditional Expextation-II

The paragraph discusses the concept of conditional expectation beyond the context of a countable partition of the sample space, omega. It explains that while a countable partition allows for a straightforward proof of conditional expectation, the general case with any sigma-algebra requires deeper probability theory. The speaker introduces the idea of constructing a new probability measure from an existing one, suggesting that this is key to understanding conditional expectation in a more general setting. The paragraph also hints at the Radon-Nikodym theorem, which is central to this discussion but is not fully explored due to the complexity of the topic.

This paragraph delves into the process of constructing a new probability measure from an existing one, using a non-negative integrable random variable X that equals 1. The function Q is introduced as a way to map elements of a sigma-algebra to real numbers, which leads to the creation of a new probability measure. The paragraph also touches on the concept of absolute continuity between two probability measures, P and Q, and how it relates to the zeros of P implying the zeros of Q. The importance of this process in finance and probability theory is emphasized.

The Radon-Nikodym theorem is the focus of this paragraph, which states that if P(A) = 0 implies Q(A) = 0 for events A in a sigma-algebra, then there exists a non-negative, integrable random variable Y, known as the Radon-Nikodym derivative, that allows for the construction of Q from P. The theorem is highlighted as a deep result in measure theory and probability, emphasizing the importance of proving the existence of such a variable, even though it may not be unique.

The paragraph presents the result that for an integrable random variable X on a probability space with a sub-sigma field G, there exists a random variable E(X|G) that satisfies the partial averaging property. It is noted that while this random variable may not be unique, its existence is guaranteed, which is crucial for understanding conditional expectation in a general setup. The paragraph also discusses the implications of finding another random variable Y that satisfies the same property, suggesting that Y must be the conditional expectation of X given G.

This paragraph outlines the properties of conditional expectation that are essential for understanding and applying the concept in finance. The first property, 'taking out what is known,' allows for the simplification of expressions involving known random variables and their conditional expectations. The second property is the expectation of a conditional expectation, which simplifies to the original expectation. The paragraph promises further exploration of these properties through exercises and homework, indicating their foundational role in the study of finance and related topics.

The concept of independence between a random variable X and a sigma-algebra G is introduced, explaining that if X is independent of G, the conditional expectation of X given G is a constant. The paragraph also introduces the 'tower property' of conditional expectation, which describes how the conditional expectation of X given G can be further conditioned on a sub-sigma field H, effectively simplifying to E(X|H). The importance of these properties in the study of finance and Martingales is emphasized, with a teaser for the upcoming discussion on these topics.

In the concluding paragraph, the speaker emphasizes the importance of understanding the properties of conditional expectation, especially in the context of finance and derivative pricing. The speaker encourages students to engage with the material through exercises and to utilize various resources for learning. There is a nod to the practical applications of these concepts in the stock market and an anticipation of future lessons on Martingales and geometric Brownian motion, which are foundational for understanding financial markets.