Optimal investment in incomplete financial markets

We give a review of classical and recent results on maximization of expected utility for an investor who has the possibility of trading in a financial market. Emphasis will be given to the duality theory related to this convex optimization problem. For expository reasons we first consider the classical case where the underlying probability space Ω is finite. This setting has the advantage that the technical difficulties of the proofs are reduced to a minimum, which allows for a clearer insight into the basic ideas, in particular the crucial role played by the Legendre-transform. In this setting we state and prove an existence and uniqueness theorem for the optimal invest-ment strategy, and its relation to the dual problem; the latter consists in finding an equivalent martingale measure optimal with respect to the conjugate of the utility function. We also discuss economic interpreta-tions of these theorems. We then pass to the general case of an arbitrage-free financial market modeled by an Rd-valued semi-martingale. In this case some regularity conditions have to be imposed in order to obtain an existence result for the primal problem of finding the optimal investment, as well as for a proper duality theory. It turns out that one may give a necessary and sufficient condition, namely a mild condition on the asymptotic behavior of the utility function, its so-called reasonable asymptotic elas-ticity. This property allows for an economic interpretation motivating the term “reasonable”. The remarkable fact is that this regularity con-dition only pertains to the behavior of the utility function, while we do not have to impose any regularity conditions on the stochastic pro-cess modeling the financial market (to be precise: of course, we have to require the arbitrage-freeness of this process in a proper sense; also we have to assume in one of the cases considered below that this pro-cess is locally bounded; but otherwise it may be an arbitrary Rd-valued semi-martingale)