A considerable part of the vast development in Mathematical Finance over the last two decades was determined by the application of convex analysis. Particular attention will be devoted to the investigation of innovative and advanced methods from stochastic analysis, convex analysis and duality theory that play a fundamental role in the mathematical modelling of finance, and in particular that arise in the context of arbitrage asset pricing, optimization problems and risk measurement.

The Lectures will focus on the following three topics:

1) The expected utility maximization problem in continuous time stochastic markets, which can be traced back to the seminal work by Merton, received a renovated impulse in the middle of the eighties, when the so-called convex duality approach to the problem was first developed. During the past twenty years, the theory has constantly improved, and in the last few years the general case of semimartingale stochastic models was tackled with great success.

2) The importance of the analysis of the utility maximization problem is also revealed in the theory of asset pricing in incomplete markets, where the agent's preferences have again to be taken in serious consideration. Indeed, different notion of utility based prices - as the concept of indifference price - have been introduced in the literature, since the middle of the nineties. These concepts determine pricing rules which are often non linear outside the set of marketed claims. Depending on the utility function that is selected, these pricing kernels share many properties with non-linear valuations: we are bordering here the realm of risk measures and capital requirements.

3) Coherent or convex risk measures have been intensively studied in the last ten years with particular emphasis on their dual representation. More recently risk measures have been cons

idered in a dynamic context and the theory of non-linear expectations is very appropriate for dealing with the genuinely dynamic aspects of risk measures. In recent papers, risk measures are defined on Orlicz spaces, in order to allow the evaluation of possibly unbounded risk. The main tools for a detailed study of this topics are found in the theory of convex analysis and Frechet lattices. We will also analyse the more recent concept of quasiconvex risk measure in the static and in the dynamic setting.