14 research outputs found
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
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.
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 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
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.Model uncertainty, duality, utility maximization, hedging
Duality in Robust Utility Maximization with Unbounded Claim via a Robust Extension of Rockafellar's Theorem
We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endowment. To obtain this duality, we prove a robust version of Rockafellar's theorem on convex integral functionals and apply Fenchel's general duality theorem.
On Admissible Strategies in Robust Utility Maximization
The existence of optimal strategy in robust utility maximization is addressed when the utility function is finite on the entire real line. A delicate problem in this case is to find a "good definition" of admissible strategies, so that an optimizer is obtained. Under suitable assumptions, especially a time-consistency property of the set of probabilities which describes the model uncertainty, we show that an optimal strategy is obtained in the class of strategies whose wealths are supermartingales under all local martingale measures having a finite generalized entropy with at least one of candidate models (probabilities).