336 research outputs found
An overview of Viscosity Solutions of Path-Dependent PDEs
This paper provides an overview of the recently developed notion of viscosity
solutions of path-dependent partial di erential equations. We start by a quick
review of the Crandall- Ishii notion of viscosity solutions, so as to motivate
the relevance of our de nition in the path-dependent case. We focus on the
wellposedness theory of such equations. In partic- ular, we provide a simple
presentation of the current existence and uniqueness arguments in the
semilinear case. We also review the stability property of this notion of
solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme
approximation method. Our results rely crucially on the theory of optimal
stopping under nonlinear expectation. In the dominated case, we provide a
self-contained presentation of all required results. The fully nonlinear case
is more involved and is addressed in [12]
Some flows in shape optimization
Geometric flows related to shape optimization problems of Bernoulli type are
investigated. The evolution law is the sum of a curvature term and a nonlocal
term of Hele-Shaw type. We introduce generalized set solutions, the definition
of which is widely inspired by viscosity solutions. The main result is an
inclusion preservation principle for generalized solutions. As a consequence,
we obtain existence, uniqueness and stability of solutions. Asymptotic behavior
for the flow is discussed: we prove that the solutions converge to a
generalized Bernoulli exterior free boundary problem
Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
The main purpose of this paper is to approximate several non-local evolution
equations by zero-sum repeated games in the spirit of the previous works of
Kohn and the second author (2006 and 2009): general fully non-linear parabolic
integro-differential equations on the one hand, and the integral curvature flow
of an interface (Imbert, 2008) on the other hand. In order to do so, we start
by constructing such a game for eikonal equations whose speed has a
non-constant sign. This provides a (discrete) deterministic control
interpretation of these evolution equations. In all our games, two players
choose positions successively, and their final payoff is determined by their
positions and additional parameters of choice. Because of the non-locality of
the problems approximated, by contrast with local problems, their choices have
to "collect" information far from their current position. For integral
curvature flows, players choose hypersurfaces in the whole space and positions
on these hypersurfaces. For parabolic integro-differential equations, players
choose smooth functions on the whole space
Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems
This paper deals with existence and uniqueness, in viscosity sense, of a
solution for a system of m variational partial differential inequalities with
inter-connected obstacles. A particular case of this system is the
deterministic version of the Verification Theorem of the Markovian optimal
m-states switching problem. The switching cost functions are arbitrary. This
problem is connected with the valuation of a power plant in the energy market.
The main tool is the notion of systems of reflected BSDEs with oblique
reflection.Comment: 36 page
Pequeños eventos deportivos y su impacto en el turismo local: el caso de la prueba cicloturista La Quebrantahuesos
La proliferación de eventos deportivos ha despertado el interés por conocer su uso como herramienta de atracción turística en aquellas localidades que los albergan. Frente a la innegable repercusión de los grandes acontecimientos deportivos, surge la necesidad de evaluar si en el caso de eventos en zonas o localidades pequeñas, se producen efectos positivos desde un punto de vista turístico. En el presente trabajo se analiza el caso particular de la prueba cicloturista la Quebrantahuesos en la localidad de Sabiñánigo, situada al norte de la provincia de Huesca. A partir de una encuesta a los participantes en la edición de 2016, hemos podido conocer su valoración respecto a la experiencia vivida, los servicios y actividades que se ofrecen antes, durante y después de la prueba, así como la capacidad de atracción al destino y una aproximación al gasto medio realizado durante la estancia. A partir de los resultados de este estudio, se puede afirmar que La Quebrantahuesos es un producto consolidado y excelentemente bien valorado, con un impacto positivo sobre la promoción turística de un municipio de pequeño tamaño.
The proliferation of sport events has awakend an interest in learning how to use them as a tool for attracting tourists in the places that host them. In view of the undeniable impact of major sport events, there is a need to evaluate whether in the case of events in small areas or localities, positive effects are produced from a tourist perspective. This paper analyses the specific case of the cycling race La Quebrantahuesos in the town of Sabiñánigo, located in the North of the province of Huesca. Based on a survey conducted among the participants in the 2016 edition, we determined their assessment about their experience, the services and activities offered before, during and after the race, as well as the attraction capacity of the destination and their estimated average expenditure during their stay. From the results obtained in the study, it can be maintained that La Quebrantahuesos is a consolidated and a very highly valued product, with a positive impact on the tourist promotion of a small-sized municipality
Dislocation dynamics: from microscopic models to macroscopic crystal plasticity
In this paper we study the connection between four models describing
dislocation dynamics: a generalized 2D Frenkel-Kontorova model at the atomic
level, the Peierls-Nabarro model, the discrete dislocation dynamics and a
macroscopic model with dislocation densities. We show how each model can be
deduced from the previous one at a smaller scale
Tightness for a stochastic Allen--Cahn equation
We study an Allen-Cahn equation perturbed by a multiplicative stochastic
noise which is white in time and correlated in space. Formally this equation
approximates a stochastically forced mean curvature flow. We derive uniform
energy bounds and prove tightness of of solutions in the sharp interface limit,
and show convergence to phase-indicator functions.Comment: 27 pages, final Version to appear in "Stochastic Partial Differential
Equations: Analysis and Computations". In Version 4, Proposition 6.3 is new.
It replaces and simplifies the old propositions 6.4-6.
Nonlinear Parabolic Equations arising in Mathematical Finance
This survey paper is focused on qualitative and numerical analyses of fully
nonlinear partial differential equations of parabolic type arising in financial
mathematics. The main purpose is to review various non-linear extensions of the
classical Black-Scholes theory for pricing financial instruments, as well as
models of stochastic dynamic portfolio optimization leading to the
Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both
problems can be represented by solutions to nonlinear parabolic equations.
Qualitative analysis will be focused on issues concerning the existence and
uniqueness of solutions. In the numerical part we discuss a stable
finite-volume and finite difference schemes for solving fully nonlinear
parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387
On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples
We show a concise extension of the monotone stability approach to backward
stochastic differential equations (BSDEs) that are jointly driven by a Brownian
motion and a random measure for jumps, which could be of infinite activity with
a non-deterministic and time inhomogeneous compensator. The BSDE generator
function can be non convex and needs not to satisfy global Lipschitz conditions
in the jump integrand. We contribute concrete criteria, that are easy to
verify, for results on existence and uniqueness of bounded solutions to BSDEs
with jumps, and on comparison and a-priori -bounds. Several
examples and counter examples are discussed to shed light on the scope and
applicability of different assumptions, and we provide an overview of major
applications in finance and optimal control.Comment: 28 pages. Added DOI
https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final
publication, corrected typo (missing gamma) in example 4.1
Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term
International audienceIn this paper we consider second order fully nonlinear operators with an additive superlinear gradient term. Like in the pioneering paper of Brezis for the semilinear case, we obtain the existence of entire viscosity solutions, defined in all the space, without assuming global bounds. A uniqueness result is also obtained for special gradient terms, subject to a convexity/concavity type assumption where superlinearity is essential and has to be handled in a different way from the linear case
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