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Financial Mathematics by Roman N. Makarov, Giuseppe Campolieti

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Chapter 9

Essentials of General Probability Theory

In this chapter we present some measure-theoretic foundations of probability theory and random variables. This then leads us into the main tools and formulas for computing expectations and conditional expectations of random variables (more importantly, continuous random variables) under different probability measures. The main formulas that are provided here are used in further chapters for the understanding and quantitative modelling of continuous-time stochastic financial models.

9.1 Random Variables and Lebesgue Integration

Within the foundation of general probability theory, the mathematical expectation of any real-valued random variable X defined on a probability space (Ω, ℱ, ℙ) is a so-called ...

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