34,944 research outputs found
G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE
We introduce a new notion of G-expectation-weighted Sobolev spaces, or in
short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian
motion are in fact path dependent PDEs in the corresponding Sobolev spaces
under G-norms. For the linear case of G corresponding the classical Wiener
probability space with Wiener measure P, we have established a 1-1
correspondence between BSDE and such new type of quasilinear PDE in the
corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1
correspondence between a fully nonlinear PDE in the corresponding G-Sobolev
space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness
of such type of fully nonlinear path-dependence PDE in G-Sobolev space have
been obtained via a recent results of BSDE driven by G-Brownian motion.Comment: 22 page
On finiteness of Type IIB compactifications: Magnetized branes on elliptic Calabi-Yau threefolds
The string landscape satisfies interesting finiteness properties imposed by
supersymmetry and string-theoretical consistency conditions. We study N=1
supersymmetric compactifications of Type IIB string theory on smooth
elliptically fibered Calabi-Yau threefolds at large volume with magnetized
D9-branes and D5-branes. We prove that supersymmetry and tadpole cancellation
conditions imply that there is a finite number of such configurations. In
particular, we derive an explicitly computable bound on the number of magnetic
flux quanta, as well as the number of D5-branes, which is independent of the
continuous moduli of the setup. The proof applies if a number of easy to check
geometric conditions of the twofold base are met. We show that these geometric
conditions are satisfied for the almost Fano twofold bases given by each toric
variety associated to a reflexive two-dimensional polytope as well as by the
generic del Pezzo surfaces dP_n with n=0,...,8. Physically, this finiteness
proof shows that there exist a finite collection of four-dimensional gauge
groups and chiral matter spectra in the 4D supergravity theories realized by
these compactifications. As a by-product we explicitly construct all generators
of the Kaehler cones of dP_n and work out their relation to representation
theory.Comment: 49 pages, 1 figure, 5 tables. Minor corrections and improvements, a
version to appear in JHE
Parameter estimation and model testing for Markov processes via conditional characteristic functions
Markov processes are used in a wide range of disciplines, including finance.
The transition densities of these processes are often unknown. However, the
conditional characteristic functions are more likely to be available,
especially for L\'{e}vy-driven processes. We propose an empirical likelihood
approach, for both parameter estimation and model specification testing, based
on the conditional characteristic function for processes with either continuous
or discontinuous sample paths. Theoretical properties of the empirical
likelihood estimator for parameters and a smoothed empirical likelihood ratio
test for a parametric specification of the process are provided. Simulations
and empirical case studies are carried out to confirm the effectiveness of the
proposed estimator and test.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ400 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Representation Theorems for Quadratic -Consistent Nonlinear Expectations
In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "-consistent nonlinear expectation") to the case
when it is allowed to be dominated by a -expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense, including the
Doob-Meyer type decomposition theorem and the optional sampling theorem. More
importantly, we show that any quadratic -consistent nonlinear
expectation with a certain domination property must be a quadratic
-expectation. The main contribution of this paper is the finding of the
domination condition to replace the one used in all the previous works, which
is no longer valid in the quadratic case. We also show that the representation
generator must be deterministic, continuous, and actually must be of the simple
form
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