A level-set approach for stochastic optimal control problems under controlled-loss constraints

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach. With this approach, the state constraints can be managed through an exact penalization technique

A comparison principle for PDEs arising in approximate hedging problems: application to Bermudan options

In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard, Elie and Touzi in [1] and is known to solve an Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in [2]. [1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et al. Bsdes with weak terminal condition. The Annals of Probability, 43(2):572-604,2015

Chore division on a graph

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

Mean-field games of optimal stopping: a relaxed solution approach

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence

A backward dual representation for the quantile hedging of Bermudan options

Within a Markovian complete financial market, we consider the problem of hedging a Bermudan option with a given probability. Using stochastic target and duality arguments, we derive a backward numerical scheme for the Fenchel transform of the pricing function. This algorithm is similar to the usual American backward induction, except that it requires two additional Fenchel transformations at each exercise date. We provide numerical illustrations

Renminbi Equilibrium Exchange Rate: an Agnostic View. Kurs równowagi dla waluty chinskiej: zdanie odrebne.

The alleged undervaluation of the renminbi has been the subject of intensive academic research over the past few years. Using equilibrium exchange rate models many authors have concluded that the renminbi is undervalued by 15 to 30% against the US dollar. Yet China has been experiencing strong economic growth for a decade and does not seem to suffer from the supposed misalignment of its exchange rate, with low inflation rate and current account surpluses. The estimations assume that the economy is at full-employment, a strong hypothesis for China, where unemployment amounts to 150 million people. This article claims that a low exchange rate is suited for the objectives of Chinese economic policy. The exchange rate can be undervalued according to traditional models and in equilibrium compared to the government’s policy objectives as shown by a theoretical model.equilibrium exchange rate;developing country exchange rate;China economic strategy;kurs równowagi;kurs walutowy kraju rozwijającego się;chińska strategia gospodarcza;

Complexity of Manipulating Sequential Allocation

Sequential allocation is a simple allocation mechanism in which agents are given pre-specified turns and each agents gets the most preferred item that is still available. It has long been known that sequential allocation is not strategyproof. Bouveret and Lang (2014) presented a polynomial-time algorithm to compute a best response of an agent with respect to additively separable utilities and claimed that (1) their algorithm correctly finds a best response, and (2) each best response results in the same allocation for the manipulator. We show that both claims are false via an example. We then show that in fact the problem of computing a best response is NP-complete. On the other hand, the insights and results of Bouveret and Lang (2014) for the case of two agents still hold

Fair Division of a Graph

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents -or, less restrictively, the number of agent types- is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape