On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper

This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price process is locally bounded and admits an equivalent local martingale measure with finite entropy, we show, without further assumption, that in the case of exponential utility the optimal portfolio process is a martingale with respect to each local martingale measure with finite entropy. Moreover, the optimal value always can be attained on a sequence of uniformly bounded portfolios.     

Convex duality for Epstein–Zin stochastic differential utility

This paper introduces a dual problem to study a continuous‐time consumption and investment problem with incomplete markets and Epstein–Zin stochastic differential utilities. Duality between the primal and dual problems is established. Consequently, the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market models, utility specifications, and agent's admissible strategies. Meanwhile, the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the “least favorable” completion of the market.     

On utility maximization under model uncertainty in discrete‐time markets

We study the problem of maximizing terminal utility for an agent facing model uncertainty, in a frictionless discrete‐time market with one safe asset and finitely many risky assets. We show that an optimal investment strategy exists if the utility function, defined either on the positive real line or on the whole real line, is bounded from above. We further find that the boundedness assumption can be dropped, provided that we impose suitable integrability conditions, related to some strengthened form of no‐arbitrage. These results are obtained in an alternative framework for model uncertainty, where all possible dynamics of the stock prices are represented by a collection of stochastic processes on the same filtered probability space, rather than by a family of probability measures.     

INDIFFERENCE PRICE WITH GENERAL SEMIMARTINGALES

Using duality methods, we prove several key properties of the indifference price π for contingent claims. The underlying market model is very general and the mathematical formulation is based on a duality naturally induced by the problem. In particular, the indifference price π turns out to be a convex risk measure on the Orlicz space induced by the utility function.