Introductory Mathematical Analysis for Quantitative Finance.

The purpose of this book is to be a tool for students, with little mathematical background, who aim to study Mathematical Finance. The only prerequisites assumed are one–dimensional differential calculus, infinite series, Riemann integral and elementary linear algebra. In a sense, it is a sort of intensive course, or crash–course, which allows students, with minimal knowledge in Mathematical Analysis, to reach the level of mathematical expertise necessary in modern Quantitative Finance. These lecture notes concern pure mathematics, but the arguments presented are oriented to Financial applications. The n–dimensional Euclidean space is briefly introduced, in order to deal with multivariable differential calculus. Sequences and series of functions are introduced, in view of theorems concerning the passage to the limit in Measure theory, and their role in the general theory of ordinary differential equations, which is also presented. Due to its importance in Quantitative Finance, the Radon–Nykodim theorem is stated, without proof, since the Von Neumann argument requires notions of Functional Analysis, which would require a dedicated course. Finally, in order to solve the Black–Scholes partial differential equation, basics in ordinary differential equations and in the Fourier transform are provided. We kept our exposition as short as possible, as the lectures are intended to be a preliminary contact with the mathematical concepts used in Quantitative Finance and provided, often, in a one–semester course. This book, therefore, is not intended for a specialized audience, although the material presented here can be used by both experts and non-experts, to have a clear idea of the mathematical tools used in Finance.