2,948 research outputs found
Entropies, convexity, and functional inequalities
Our aim is to provide a short and self contained synthesis which generalise
and unify various related and unrelated works involving what we call
Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies
can be seen in particular as an inclusive interpolation between Poincare and
Gross logarithmic Sobolev inequalities. In addition to the known material,
extensions are provided and improvements are given for some aspects. Stability
by tensor products, convolution, and bounded perturbations are addressed. We
show that under simple convexity assumptions on Phi, such inequalities hold in
a lot of situations, including hyper-contractive diffusions, uniformly strictly
log-concave measures, Wiener measure (paths space of Brownian Motion on
Riemannian Manifolds) and generic Poisson space (includes paths space of some
pure jumps Levy processes and related infinitely divisible laws). Proofs are
simple and relies essentially on convexity. We end up by a short parallel
inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic
Gases and Information Theories.Comment: Formerly "On Phi-entropies and Phi-Sobolev inequalities". Author's
www homepage: http://www.lsp.ups-tlse.fr/Chafai
Time--consistent investment under model uncertainty: the robust forward criteria
We combine forward investment performance processes and ambiguity averse
portfolio selection. We introduce the notion of robust forward criteria which
addresses the issues of ambiguity in model specification and in preferences and
investment horizon specification. It describes the evolution of time-consistent
ambiguity averse preferences.
We first focus on establishing dual characterizations of the robust forward
criteria. This offers various advantages as the dual problem amounts to a
search for an infimum whereas the primal problem features a saddle-point. Our
approach is based on ideas developed in Schied (2007) and Zitkovic (2009). We
then study in detail non-volatile criteria. In particular, we solve explicitly
the example of an investor who starts with a logarithmic utility and applies a
quadratic penalty function. The investor builds a dynamical estimate of the
market price of risk and updates her stochastic utility in
accordance with the so-perceived elapsed market opportunities. We show that
this leads to a time-consistent optimal investment policy given by a fractional
Kelly strategy associated with . The leverage is proportional to
the investor's confidence in her estimate
Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model
We study a family of mean field games with a state variable evolving as a
multivariate jump diffusion process. The jump component is driven by a Poisson
process with a time-dependent intensity function. All coefficients, i.e. drift,
volatility and jump size, are controlled. Under fairly general conditions, we
establish existence of a solution in a relaxed version of the mean field game
and give conditions under which the optimal strategies are in fact Markovian,
hence extending to a jump-diffusion setting previous results established in
[30]. The proofs rely upon the notions of relaxed controls and martingale
problems. Finally, to complement the abstract existence results, we study a
simple illiquid inter-bank market model, where the banks can change their
reserves only at the jump times of some exogenous Poisson processes with a
common constant intensity, and provide some numerical results.Comment: 37 pages, 6 figure
Convex hull of n planar Brownian paths: an exact formula for the average number of edges
We establish an exact formula for the average number of edges appearing on
the boundary of the global convex hull of n independent Brownian paths in the
plane. This requires the introduction of a counting criterion which amounts to
"cutting off" edges that are, in a specific sense, small. The main argument
consists in a mapping between planar Brownian convex hulls and configurations
of constrained, independent linear Brownian motions. This new formula is
confirmed by retrieving an existing exact result on the average perimeter of
the boundary of Brownian convex hulls in the plane.Comment: 14 pages, 8 figures, submitted to JPA. (Typo corrected in equation
(14).
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