A selected survey of portfolio optimization problems

The aim of this manuscript is to review some results related to portfolio optimization. The models we consider ranges from general semimartingale models to models driven by Itô processes, as well as model driven by Lévy processes. Two optimization problems are studied: the maximization of the utility coming from the final wealth, and the minimization of “shortfall risk” in replicating a contingent claim. In addition to establish some of the fundamental results, we also consider the mathematical tools underneath them such as duality methods and representation theorems. 1 Optimum hedging on semimartingale models In this section, we present some results of Föllmer and Leukert [3]. 1.1 Model: • The discounted price process {Xt}0≤t≤T of the underlying is described as a semimartingale on a probability space (Ω,F ,P) with filtration (Ft)0≤t≤T . • We assume the absence of arbitrage in the sense that there is at least one equivalent probability measure P∗ such that X is a local martingale under P∗. The set of all equivalent martingale measures is denoted by P . • A self-financing strategy is given by an initial capital V0 ≥ 0 and by a predictable process ξ such that the resulting (discounted) value process