A Complete Explicit Solution to the Log-Optimal Portfolio Problem

Kramkov and Schachermayer (1999) proved the existence of log-optimal portfolios under weak assumptions in a very general setting. For many – but not all – cases, Goll and Kallsen (2000) obtained the optimal solution explicitly in terms of the semimartingale characteristics of the price process. By extending this result, this paper provides a complete explicit characterization of log-optimal portfolios without constraints. Moreover, the results in Goll and Kallsen (2000) are generalized in two further respects: Firstly, we allow for random convex trading constraints. Secondly, the remaining consumption time – or more generally the consumption clock – may be random, which corresponds to a life-insurance problem. Finally, we consider neutral derivative pricing in incomplete markets.