Jan Kallsen, Thomas Goll
A classical problem in mathematical finance is the computation of optimal portfolios, where optimal here refers to maximization of expected utility from terminal wealth or consumption. In this talk, we consider logarithmic utility because it is intuitive and allows to obtain explicit solutions even in inhomogeneous, incomplete markets.
We want to provide an in some sense definitive answer to the log-optimal portfolio problem: For the most general semimartingale model, the optimal investment is determined explicitly in terms of the semimartingale characteristics of the price process. Similarly as transition densities and stochastic differential equations, this notion describes the local behaviour of a stochastic process. Previously obtained explicit solutions in the literature are easily recovered as special cases.
Moreover, we consider neutral derivative pricing in incomplete markets.