A Dynamic Programming Approach for Pricing Options Embedded in Bonds

The aim of this paper is to price options embedded in bonds in a Dynamic Programming (DP) framework, the focus being on call and put options with advance notice. The pricing of interest rate derivatives was usually done via trees or finite differences. Trees are not really very efficient as they deform crudely the dynamic of the underlying asset(s), here the short term risk-free interest rate. They can be interpreted as elementary DP procedures with fixed grid sizes. For a long time, finite differences presented poor accuracy because of the discontinuities of the bond's value that may arise at decision dates. Recently, remedies were given by d'Halluin et al (2001) via techniques related to flux limiters. DP does not suffer from discontinuities that may arise at decision dates and does not require a time discretization. It may also be implemented in discrete-time models. Results show efficiency and robustness. Suggestions to combine DP and finite differences are also formulatedDynamic Programming, Stochastic Processes, Options Embedded in Bonds, American Options

Image Segmentation with Multidimensional Refinement Indicators

We transpose an optimal control technique to the image segmentation problem. The idea is to consider image segmentation as a parameter estimation problem. The parameter to estimate is the color of the pixels of the image. We use the adaptive parameterization technique which builds iteratively an optimal representation of the parameter into uniform regions that form a partition of the domain, hence corresponding to a segmentation of the image. We minimize an error function during the iterations, and the partition of the image into regions is optimally driven by the gradient of this error. The resulting segmentation algorithm inherits desirable properties from its optimal control origin: soundness, robustness, and flexibility

Optimal Multiphase Investment Strategies for Influencing Opinions in a Social Network

We study the problem of optimally investing in nodes of a social network in a competitive setting, where two camps aim to maximize adoption of their opinions by the population. In particular, we consider the possibility of campaigning in multiple phases, where the final opinion of a node in a phase acts as its initial biased opinion for the following phase. Using an extension of the popular DeGroot-Friedkin model, we formulate the utility functions of the camps, and show that they involve what can be interpreted as multiphase Katz centrality. Focusing on two phases, we analytically derive Nash equilibrium investment strategies, and the extent of loss that a camp would incur if it acted myopically. Our simulation study affirms that nodes attributing higher weightage to initial biases necessitate higher investment in the first phase, so as to influence these biases for the terminal phase. We then study the setting in which a camp's influence on a node depends on its initial bias. For single camp, we present a polynomial time algorithm for determining an optimal way to split the budget between the two phases. For competing camps, we show the existence of Nash equilibria under reasonable assumptions, and that they can be computed in polynomial time

Multi-item Auctions for Automatic Negotiation

Available resources can often be limited with regard to the number of demands. In this paper we propose an approach for solving this problem which consists of using the mechanisms of multi-item auctions for allocating the resources to a set of software agents. We consider the resource problem as a market in which there are vendor agents and buyer agents trading on items representing the resources. These agents use multi-item auctions which are viewed here as a process of automatic negotiation, and implemented as a network of intelligent software agents. In this negotiation, agents exhibit different acquisition capabilities which let them act differently depending on the current context or situation of the market. For example, the "richer" an agent is, the more items it can buy, i.e. the more resources it can acquire. We present a model for this approach based on the English auction, then we discuss experimental evidence of such a model. Dans un environnement multiagent, les ressources peuvent toujours s'avérer insuffisantes relativement à un nombre élevé de demandes. Dans ce cahier, nous proposons une approche mixant les enchères et les agents logiciels en vue de contribuer à résoudre ce problème. Cette approche consiste en fait à utiliser le mécanisme d'enchères multi-articles en vue d'allouer les ressources à un ensemble d'agents. À cet effet, nous considérons le problème de ressources comme un marché dans lequel évoluent des agents acheteurs et des agents vendeurs négociant des articles représentant des ressources. Ces agents utilisent des enchères multi-articles et par conséquent ils constituent un processus de négociation automatisé et programmé comme un réseau d'agents logiciels. Dans ce type de négociation, chaque agent exhibe différentes capacités d'acquisition lui permettant ainsi d'agir différemment selon le contexte ou la situation de marché. Par exemple, plus on est riche, plus on peut acheter d'articles. Nous présentons pour ce modèle une enchère anglaise et nous discuterons ses résultats expérimentaux.Multi-agent systems, Negotiations, Multi-item auctions, Systèmes multiagents, négociations, enchères multi items

Approximability of Robust Network Design: The Directed Case

We consider robust network design problems where an uncertain traffic vector belonging to a polytope has to be dynamically routed to minimize either the network congestion or some linear reservation cost. We focus on the variant in which the underlying graph is directed. We prove that an O(?k) = O(n)-approximation can be obtained by solving the problem under static routing, where k is the number of commodities and n is the number of nodes. This improves previous results of Hajiaghayi et al. [SODA\u272005] and matches the ?(n) lower bound of Ene et al. [STOC\u272016] and the ?(?k) lower bound of Azar et al. [STOC\u272003]. Finally, we introduce a slightly more general problem version where some flow restrictions can be added. We show that it cannot be approximated within a ratio of k^{c/(log log k)} (resp. n^{c/(log log n)}) for some constant c. Making use of a weaker complexity assumption, we prove that there is no approximation within a factor of 2^{log^{1- ?} k} (resp. 2^{log^{1- ?} n}) for any ? > 0