A Martingale approach to optimal portfolios with jump-diffusions and benchmarks

We consider various portfolio optimization problems when the stock prices follow jump-diusion processes. In the first part the classical optimal consumption-investment problem is considered. The investor's goal is to maximize utility from consumption and terminal wealth over a finite investment horizon. We present results that modify and extend the duality approach that can be found in Kramkov and Schachermayer (1999). The central result is that the optimal trading strategy and optimal equivalent martingale measure can be determined as a solution to a system of non-linear equations. In another problem a benchmark process is introduced, which the investor tries to outperform. The benchmark can either be a generic jump-diusion process or, as a special case, a wealth process of a trading strategy. Similar techniques as in the first part of the thesis can be applied to reach a solution. In the special case that the benchmark is a wealth process, the solution can be deduced from the first part's consumption-investment problem via a transform of the parameters. The benchmark problem presented here gives a dierent approach to benchmarks as in, for instance, Browne (1999b) or Pra et al. (2004). It is also, as far as the author is aware, the first time that martingale methods are employed for this kind of problem. As a side effect of our analysis some interesting relationships to Platen's benchmark approach (cf. Platen (2006)) and change of numeraire techniques (cf. German et al. (1995)) can be observed. In the final part of the thesis the set of trading strategies in the previous two problems are restricted to constraints. These constraints are, for example, a prohibition of shortselling or the restriction on the number of assets. Conditions are provided under which a solution to the two problems can still be found. This extends the work of Cvitanic and Karatzas (1993) to jump diffusions where the initial market set-up is incomplete