Formule

This document covers fundamental mathematical and stochastic concepts essential for finance students:

• Topology and Measure Theory: Introduces core ideas like topological spaces, σ-algebras, measures, and Lebesgue integration, which are foundational for probability theory.
• Probability and Random Variables: Defines probability spaces, random variables, their distributions (CDF, PDF), expectation, variance, covariance, and the concept of independence, including conditional expectation and the Tower Law.
• Stochastic Processes: Explores the definition of stochastic processes, filtrations, and adapted processes, setting the stage for dynamic financial models.
• Wiener Process (Brownian Motion): Provides a detailed definition and properties of the Wiener process, a cornerstone of continuous-time financial modeling, including its independent and normally distributed increments.
• Stochastic Integral (Itô Integral): Explains the construction and properties of the Itô integral for simple and general stochastic processes, highlighting the Itô isometry.
• Itô Formula: Presents the crucial Itô lemma in both unidimensional and multidimensional forms, which is indispensable for differentiating functions of stochastic processes.
• Martingales: Defines martingales and notes that Itô integrals of adapted processes are martingales, a concept critical for risk-neutral pricing.
• Ordinary Differential Equations (ODEs): Covers basic definitions of ODEs and Cauchy problems, including existence and uniqueness theorems (Peano's, Cauchy-Lipschitz).
• Stochastic Differential Equations (SDEs): Introduces SDEs, their Lipschitz and sublinear conditions for existence and uniqueness, and provides examples like Geometric Brownian Motion and linear SDEs with explicit solutions.
• Partial Differential Equations (PDEs): Discusses relevant PDEs in financial modeling, particularly those of the Black-Scholes type, and their terminal conditions.
• Connection between SDEs and PDEs: Explains how the solution of certain PDEs can be represented as the expected value of a function of an SDE's terminal state, formalized by the Feynman-Kač representation formula.
• Girsanov's Theorem: Touches upon Girsanov's theorem, a powerful tool for changing probability measures, which is fundamental for risk-neutral pricing in arbitrage-free markets.