55 research outputs found
Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise
We consider the family of stochastic partial differential equations indexed
by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) =
\eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*}
(t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation,
is a second-order partial differential operator with constant coefficients,
and are smooth functions and is a Gaussian noise, white
in time and with a stationary correlation in space. Let p^\eps_{t,x} denote
the density of the law of u^\eps(t,x) at a fixed point
(t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0}
\eps^2\log p^\eps_{t,x}(y) for a fixed . The results apply to a class
of stochastic wave equations with and to a class of stochastic
heat equations with .Comment: 39 pages. Will be published in the book " Stochastic Analysis and
Applications 2014. A volume in honour of Terry Lyons". Springer Verla
Brownian bridges to submanifolds
We introduce and study Brownian bridges to submanifolds. Our method involves
proving a general formula for the integral over a submanifold of the minimal
heat kernel on a complete Riemannian manifold. We use the formula to derive
lower bounds, an asymptotic relation and derivative estimates. We also see a
connection to hypersurface local time. This work is motivated by the desire to
extend the analysis of path and loop spaces to measures on paths which
terminate on a submanifold
Alternative proof for the localization of Sinai's walk
We give an alternative proof of the localization of Sinai's random walk in
random environment under weaker hypothesis than the ones used by Sinai.
Moreover we give estimates that are stronger than the one of Sinai on the
localization neighborhood and on the probability for the random walk to stay
inside this neighborhood
Systemic Risk and Default Clustering for Large Financial Systems
As it is known in the finance risk and macroeconomics literature,
risk-sharing in large portfolios may increase the probability of creation of
default clusters and of systemic risk. We review recent developments on
mathematical and computational tools for the quantification of such phenomena.
Limiting analysis such as law of large numbers and central limit theorems allow
to approximate the distribution in large systems and study quantities such as
the loss distribution in large portfolios. Large deviations analysis allow us
to study the tail of the loss distribution and to identify pathways to default
clustering. Sensitivity analysis allows to understand the most likely ways in
which different effects, such as contagion and systematic risks, combine to
lead to large default rates. Such results could give useful insights into how
to optimally safeguard against such events.Comment: in Large Deviations and Asymptotic Methods in Finance, (Editors: P.
Friz, J. Gatheral, A. Gulisashvili, A. Jacqier, J. Teichmann) , Springer
Proceedings in Mathematics and Statistics, Vol. 110 2015
Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions
NONLINEAR DIFFUSION LIMIT FOR A SYSTEM WITH NEAREST NEIGHBOR INTERACTIONS
Physics, MathematicalSCI(E)0ARTICLE131-5911
CramĂ©râs Theorem in Banach Spaces Revisited
International audienceThe text summarizes the general results of large deviations for empirical means of independent and identically distributed variables in a separable Banach space, without the hypothesis of exponential tightness. The large deviation upper bound for convex sets is proved in a nonasymptotic form; as a result, the closure of the domain of the entropy coincides with the closed convex hull of the support of the common law of the variables. Also a short original proof of the convex duality between negentropy and pressure is provided: it simply relies on the subadditive lemma and Fatou's lemma, and does not resort to the law of large numbers or any other limit theorem. Eventually a Varadhan-like version of the convex upper bound is established and embraces both results
Large Deviations And The Thermodynamic Formalism: A New Proof Of The Equivalence of Ensembles
: equivalence of ensembles holds at the level of measures whenever it holds at the level of thermodynamic functions. The problem of the equivalence of ensembles is not confined to statistical mechanics; it can be found in other areas of applied probability theory -- in information theory, for example. Here the problem is to prove that a sequence of conditioned measures is a Lecture delivered by J.T. Lewis equivalent, in an appropriate sense, to a sequence of "tilted" measures. Our choice of setting is sufficiently general to cover such applications. Probabilistic methods have been used for at least fifty years to prove results about the equivalence of ensembles: Khinchin (1943) used a local central limit theorem to prove it for a classical ideal (non--interacting) gas; Dobrushin and Tirozzi (1977) proved it for lattice gas models for which they were able to establish a local central limit theorem -- a restriction which ruled--out model
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