A complete explicit solution to the log-optimal portfolio problem

D. Kramkov and W. Schachermayer [Ann. Appl. Probab. 9 (1999) 904-950] proved the existence of log-optimal portfolios under weak assumptions in a very general setting. For many--but not all--cases, T. Goll and J. Kallsen [Stochastic Process. Appl. 89 (2000) 31-48] 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 of Goll and Kallsen are generalized here in two further respects: First, we allow for random convex trading constraints. Second, 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.