766 research outputs found
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 rĂłwnowagi 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 governmentâs policy objectives as shown by a theoretical model.equilibrium exchange rate;developing country exchange rate;China economic strategy;kurs rĂłwnowagi;kurs walutowy kraju rozwijajÄ
cego siÄ;chiĆska 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
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