Lectures on the Foundations of Mathematical Finance

The aim of these lectures is to give an introduction to the mathematical foundations of finance, rather than to mathematical finance per se. The reader is assumed to know the basics of stochastic differential equations and mathematical finance (at the level of Shreve’s textbooks [5], [6]). 1 No-arbitrage for Finite Probability Spaces The notion of arbitrage will be one of the main themes of the course. We will start the course by examining models based on finite probability spaces with discrete time. By studying this toy model we can introduce the necessary ideas and language of func-tional analysis at a relatively non-technical level. Later we will study models based on arbitrary probability spaces with continuous time. In this lecture we shall follow sections 2.1–2 of Delbaen and Schachermayer [2] very closely. 1.1 Finite Model of a Financial Market ([2], §2.1) The financial market model that we shall consider here is based on a finite probability space (Ω,F, P), where Ω = ω1,..., ωN is a finite set, P is a probability measure with P (ω)> 0 for all ω ∈ Ω, and F is the σ-algebra of all subsets of Ω. In addition, we consider a filtration F0 ⊂ · · · ⊂ FT = F of sub-σ-algebras of F with F0 = {∅,Ω}. Definition 1.1.1. Asset prices are given by an Rd+1–valued adapted process S ̂ = (Ŝ0t,..., Ŝ d t