Dual formulation of the utility maximization problem: the case of nonsmooth utility

We study the dual formulation of the utility maximization problem in incomplete markets when the utility function is finitely valued on the whole real line. We extend the existing results in this literature in two directions. First, we allow for nonsmooth utility functions, so as to include the shortfall minimization problems in our framework. Second, we allow for the presence of some given liability or a random endowment. In particular, these results provide a dual formulation of the utility indifference valuation rule

A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod Embedding Problem (SEP). Instead, we provide a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem. We then show that this formulation allows us to recover previously known results about lookback options. In particular, our methodology induces a new proof of the optimality of Az\'{e}ma-Yor solution of the SEP for a certain class of lookback options. Unlike the SEP technique, our approach applies to a large class of exotics and is suitable for numerical approximation techniques.Comment: Published in at http://dx.doi.org/10.1214/13-AAP925 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal consumption and investment with bounded downside risk for power utility functions

We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.Comment: 36 page

A Range of Earth Observation Techniques for Assessing Plant Diversity

AbstractVegetation diversity and health is multidimensional and only partially understood due to its complexity. So far there is no single monitoring approach that can sufficiently assess and predict vegetation health and resilience. To gain a better understanding of the different remote sensing (RS) approaches that are available, this chapter reviews the range of Earth observation (EO) platforms, sensors, and techniques for assessing vegetation diversity. Platforms include close-range EO platforms, spectral laboratories, plant phenomics facilities, ecotrons, wireless sensor networks (WSNs), towers, air- and spaceborne EO platforms, and unmanned aerial systems (UAS). Sensors include spectrometers, optical imaging systems, Light Detection and Ranging (LiDAR), and radar. Applications and approaches to vegetation diversity modeling and mapping with air- and spaceborne EO data are also presented. The chapter concludes with recommendations for the future direction of monitoring vegetation diversity using RS

Direct Characterization of the Value of Super-Replication under Stochastic Volatility and Portfolio Constraints.

We study the problem of minimal initial capital needed in order to hedge a European contengent claim without risk. The financial market presents incompleteness arising from two sources: stochastic volatility and portfolio constraints described by a closed convex set. In contrast with previous literature which uses the dual formulation of the problem, we usean original dynamic programming principle stated directly on the initial problem, as in Soner and Touzi.CAPITAL ; PROGRAMME EVALUATION ; RISK

The fundamental theorem of asset pricing with cone constraints

International audienceIn frictionless securities markets, the characterization of the no arbitrage condition by the existence of equivalent martingale measures in discrete time is known as the fundamental Theorem of asset pricing. In the presence of cone constraints on the trading strategies, we extend the fundamental theorem of asset pricing under a nondegeneracy assumption. We also prove a one-period version of this theorem when there are transaction costs. (C) 1999 Elsevier Science S.A. All rights reserved

Continuous-Time Dynkin Games with Mixed Strategies.

In order to get rid of the condition XGAMES ; MATHEMATICAL ANALYSIS ; STRATEGIC PLANNING

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time–space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions