Utility Maximisation in Incomplete Markets

Abstract

Preface In these lectures we give a short introduction to the basic concepts of Mathematical Finance, focusing on the notion of "no arbitrage", and subsequently apply these notions to the problem of optimizing dynamically a portfolio in an incomplete financial market with respect to a given utility function U. In the first part we mainly restrict ourselves to the situation where the underlying probability space (Ω, F , P) is finite, in order to reduce the functional-analytic difficulties to simple linear algebra. In my opinion, this allows-at least as a first step-for a clearer picture of the Mathematical Finance issues. We then treat the problem of utility maximisation and, in particluar, its duality theory for a general semi-martingale models of financial market. Here we are rather informal and concentrate mainly on explaining the basic ideas, e.g., the notion of the asymptotic elasticity of a utility function U. These notes are largely based on the surveys [S 03] and [S 01a] and, in particular, on the notes taken by P. Guasoni during my Cattedra Galileiana lectures at Scuola Normale Superiore in Pisa [S 04a]. We also refer to the original papers [KS 99] and [S 01] for more detailed information on the topics of the present lectures. 1 Problem Setting We consider a model of a security market which consists of d + 1 assets. We denote by S = ((S i t) 1≤t≤T) 0≤i≤d the price process of the d stocks and