Utility Maximisation in Incomplete Markets

Abstract

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 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 [S03] and [S01a] and, in particular, on the notes taken by P. Guasoni during my Cattedra Galileiana lectures at Scuola Normale Superiore in Pisa [S04b]. We also refer to the original papers [KS99] and [S01] for more detailed information on the topics of the present lectures.

Walter Schachermayer: Support by the Austrian Science Foundation (FWF) under the Wittgenstein-Preis program Z36 and grant P15889 and by the Austrian National Bank undergrant ‘Jubiläumsfondprojekt Number 9486’ is gratefully acknowledged.