Maximum Lebesgue Extension of Monotone Convex Functions

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a "nice" dual representation of the function.Comment: To Appear in Journal of Functional Analysis, 32 page

A Note on Utility Maximization with Unbounded Random Endowment

This paper addresses the applicability of the convex duality method for utility maximization, in the presence of random endowment. When the price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true, for a wide class of utility functions and unbounded random endowments. We show this duality by exploiting Rockafellar's theorem on integral functionals, to a random utility function.Utility maximization, Convex duality method, Martingale measures

Robust Exponential Hedging and Indifference Valuation

We discuss the problem of exponential hedging in the presence of model uncertainty expressed by a set of probability measures. This is a robust utility maximization problem with a contingent claim. We first consider the dual problem which is the minimization of penalized relative entropy over a product set of probability measures, showing the existence and variational characterizations of the solution. These results are applied to the primal problem. Then we consider the robust version of exponential utility indifference valuation, giving the representation of indifference price using a duality result.

Robust Exponential Hedging in a Brownian Setting

This paper studies the robust exponential hedging in a Brownian factor model, giving a solvable example using a PDE argument. The dual problem is reduced to a standard stochastic control problem, of which the HJB equation admits a classical solution. Then an optimal strategy will be expressed in terms of the solution to the HJB equation.Robust Utility Maximization, Stochastic Control, Duality