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.