Weak Regularity Research Articles

Weak regularity of degenerate elliptic equations

Let φ: Ω → D be a conformal mapping of a bounded simply connected planar domain Ω onto the unit disc D ⊂ ℝ2. We prove existence and uniqueness in Ω of weak solutions of a degenerate Poisson equation for a hyperbolic weight h(z) = |φ ′ |2 in a corresponding two weighted Sobolev space W 2 1 (Ω, h, 1).Here φ ′ is a complex derivative. We also study weak regularity of the solutions in conformal regular domains. The domain Ω is a conformal regular domain [4] if (φ −1) ′ ∈ L α (D) for some α > 2.

Finite-Sample Generalized Confidence Distributions and Sign-Based Robust Estimators In Median Regressions with Heterogeneous Dependent Errors

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution -- including no condition on the existence of moments -- allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free testing theory [Coudin and Dufour (2009, The Econometrics Journal)], from which confidence sets and point estimators are subsequently generated. The estimation problem is tackled in two complementary ways. First, we show how confidence distributions for model parameters can be applied in such a context. Such distributions -- which can be interpreted as a form of fiducial inference -- provide a frequency-based method for associating probabilities with subsets of the parameter space (like posterior distributions do in a Bayesian setup) without the introduction of prior distributions. We consider generalized confidence distributions applicable to multidimensional parameters, and we suggest the use of a projection technique for confidence inference on individual model parameters. Second, we propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show they are median unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors, allowing for noncontinuous distributions, heterogeneity and serial dependence of unknown form. These conditions are considerably weaker than those used to show corresponding results for LAD estimators. In a Monte Carlo study of bias and RMSE, we show sign-based estimators perform better than LAD-type estimators in heteroskedastic settings. We present two empirical applications, which involve financial and macroeconomic data, both affected by heavy tails (non-normality) and heteroskedasticity: a trend model for the S&P index, and an equation used to study β-convergence of output levels across U.S. States.

The Bernstein Mechanism: Function Release under Differential Privacy

We address the problem of general function release under differential privacy, by developing a functional mechanism that applies under the weak assumptions of oracle access to target function evaluation and sensitivity. These conditions permit treatment of functions described explicitly or implicitly as algorithmic black boxes. We achieve this result by leveraging the iterated Bernstein operator for polynomial approximation of the target function, and polynomial coefficient perturbation. Under weak regularity conditions, we establish fast rates on utility measured by high-probability uniform approximation. We provide a lower bound on the utility achievable for any functional mechanism that is epsilon-differentially private. The generality of our mechanism is demonstrated by the analysis of a number of example learners, including naive Bayes, non-parametric estimators and regularized empirical risk minimization. Competitive rates are demonstrated for kernel density estimation; and epsilon-differential privacy is achieved for a broader class of support vector machines than known previously.

Dealing with Monotonicity Constraints in Screening Problems: The Principal-Agent Model

This paper contains two parts clearly differentiated: a survey study about the use of monotonicity constraints in screening problems, and a research work extending the space of solutions in agency theory to non-smooth or even discontinuous action profiles. Specifically, we first systematize and compare, under a wide variety of assumptions on the endogenous variables of the model, existing approaches used for the derivation of necessary conditions in screening problems containing monotonicity constraints. Then, after identifying the most robust methods to generalize previous results, the paper derives a complete characterization of optimum contracts in the model of Jullien (2000), without resort to the usual continuity and smoothness assumptions on allocation functions. In this way, we conclude that most results in Jullien (2000) continue holding if, instead of the mentioned assumptions, the endogenous variables of the problem satisfy some weak regularity condition. Moreover, when the reservation utility fails to be type-dependent, as happens for instance in Guesnerie and Laffont (1978), the characterization of optimal contracts still holds without the need of imposing the referred regularity condition.

Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients

We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of $\mathbb{R} ^d$ with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.

Infinite-Dimensional Calculus Under Weak Spatial Regularity of the Processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces having a product structure with the noise in a Hilbert component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus.

Very weak solutions of the stationary Navier–Stokes equations for an incompressible fluid past obstacles

We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb−ke1 on the boundary of each body is sufficiently small with respect to the viscosity ν. The uniqueness of very weak solutions is proved by assuming that F and vb−ke1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇v∈L3/2(Ω).

On the Cauchy problem for lower semicontinuous differential inclusions

We provide a new proof of a classical result by Bressan on the Cauchy problem for first-order differential inclusions with null initial condition. Our approach allows us to prove the result directly for kth order differential inclusions, under weaker regularity assumptions on the involved multifunction. Our result is the following: let a, b, M be positive real numbers, with $M\cdot\max\{a,a^{k}\}\le b$ , and let B and X be the closed balls in $\mathbf{R}^{n}$ , centered at the origin with radius b and M, respectively. Let $F:[0,a]\times B^{k}\to2^{X}$ be a multifunction with nonempty closed values, such that F is $\mathcal{L}([0,a])\otimes\mathcal{B}(B^{k})$ -measurable, and for all $t\in[0,a]$ the multifunction $F(t,\cdot)$ is lower semicontinuous. Then there exists $u\in W^{k,\infty}([0,a],\mathbf{R}^{n})$ such that $u^{(k)}(t)\in F(t,u(t), u^{\prime}(t),\ldots,u^{(k-1)}(t))$ a.e. in $[0,a]$ , and $u^{(i)}(0)=0_{\mathbf{R}^{n}}$ for all $i=0,\ldots, k-1$ .

Many-body effects on graphene conductivity: Quantum Monte Carlo calculations

Optical conductivity of graphene is studied using Quantum Monte Carlo calculations. We start from Euclidean current-current correlator and extract $\sigma (\omega)$ from Green-Kubo relations using Backus-Gilbert method. Calculations were performed both for long-range interactions and taking into account only contact term. In both cases we vary interaction strength and study its influence on optical conductivity. We compare our results with previous theoretical calculations choosing $\omega \approx \kappa$ thus working in the region of the plateau in $\sigma(\omega)$ which corresponds to optical conductivity of Dirac quasiparticles. No dependence of optical conductivity on interaction strength is observed unless we approach antiferromagnetic phase transition in case of artificially enhanced contact term. Our results strongly support previous theoretical studies claimed very weak regularization of graphene conductivity.

Bounds on Order of Indeterminate Moment Sequences

We investigate the order \(\rho \) of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for \(\rho \) which is explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrence. Under a weak regularity assumption, this estimate coincides with a lower estimate, and hence \(\rho \) becomes computable. Dropping the regularity assumption leads to examples where upper and lower bounds do not coincide and differ from the order. In particular, we provide examples for which the order is different from its lower estimate due to M.S. Livsic.

Modeling and Performance Analysis of a Tracking-Area-List-Based Location Management Scheme in LTE Networks

In a cellular network, the network coverage is partitioned into nonoverlapping areas called registration areas to facilitate location management (LM) that keeps track of the location of a user equipment (UE) unit. In the second-generation Global System for Mobile Communications (GSM) and the third-generation Universal Mobile Telecommunications System (UMTS), a registration area is called a location area (LA) in the circuit-switched domain or a routing area (RA) in the packet-switched domain. In the fourth-generation Long-Term Evolution (LTE), a registration area is called a tracking area (TA). To tackle the drawbacks of the LA/RA-based LM scheme used in GSM and UMTS, which is intrinsically a static scheme, LTE adopts a dynamic LM scheme called a TA list (TAL)-based scheme. Under the TAL-based scheme, the network allocates to a UE unit a group of TAs referred to as a TAL instead of only a single TA. The UE can move freely within the allocated TAL without performing any location updates that are called TA updates (TAUs) in LTE. Only when moving into a TA that is not included in the allocated TAL does the UE need to perform a TAU, and afterward, the network will allocate, to the UE, a new TAL. The performance of the TAL-based scheme relies on the allocated TAL. If the allocated TAL is inappropriate with respect to the UE's mobility and traffic characteristics, the TAL-based scheme may produce adverse effects. To find an optimal TAL allocation method for global UE units that exhibit weak regularity in their mobility, this paper develops an embedded Markov chain approach to analyze the signaling cost of the TAL-based scheme. This paper distinguishes itself from existing studies in the following aspects. First, this paper follows LTE technical specifications, that is, the number of TAs in a TAL can vary and, furthermore, is upper bounded, whereas existing studies partially or completely ignore this stipulation. Second, this paper emphasizes the impact of a call handling model that dictates whether a TAU occurs after the completion of a call by considering two call handling models, i.e., a call plus location update and a call without location update. Third, as for the dependence among the cell residence time, the TA residence time, and the TAL residence time, this paper proposes the use of a fluid flow model to describe this dependence, which is simple but does not compromise accuracy. Analytical formulas for the total signaling cost of the TAL-based scheme due to TAU and paging operations, as well as formulas that are useful in designing an optimal paging scheme, are derived, and their accuracy is validated through Mont Carlo simulation. Then, numerical studies are carried out to investigate the impact of diverse parameters on the signaling cost, revealing that the network can allocate, to a UE unit, an optimal TAL that is consistent with the UE's mobility and traffic characteristics to minimize the signaling cost of the TAL-based scheme.

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS IN LÉVY MODELS WITH STOCHASTIC INTEREST RATE

AbstractRecently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.

On Spectral Properties in Vector-Valued Beurling Algebra

Let $G$ be an LCA group, $\om$ be a weight function on $G$, and $\A$ be a semisimple, commutative Banach algebra. We characterize some Banach algebra properties of vector-valued Beurling algebra $\Lgwa$ in terms of $\Lgw$ and $\A$ using the Bochner integration theory. These properties are unique uniform norm property (UUNP), unique $C^{\ast}$-norm property (UC$^{\ast}$NP), quasi divisor of zero property (QDZP), weak regularity (WR), regularity, and complete regularity (CR).

Higher-order regularization and morphological techniques for image segmentation

Image segmentation is an important field in computer vision and one of its most active research areas, with applications in image understanding, object detection, face recognition, video surveillance or medical image processing. Image segmentation is a challenging problem in general, but especially in the biological and medical image fields, where the imaging techniques usually produce cluttered and noisy images and near-perfect accuracy is required in many cases.In this thesis we first review and compare some standard techniques widely used for medical image segmentation. These techniques use pixel-wise classifiers and introduce weak pair-wise regularization which is insufficient in many cases. We study their difficulties to capture higher-level structural information about the objects to segment. This deficiency leads to many erroneous detections, ragged boundaries, incorrect topological configurations and wrong shapes. To deal with these problems, we propose a new regularization method that learns shape and topological information from training data in a non-parametric way using higher-order potentials.

Critical non-Sobolev regularity for continuity equations with rough velocity fields

We present a divergence free vector field in the Sobolev space H1 such that the flow associated to the field does not belong to any Sobolev space. The vector field is deterministic but constructed as the realization of a random field combining simple elements. This construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well-posed flow which has nevertheless only very weak regularity.

Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions

The paper deals with first order necessary optimality conditions for a class of infinite-horizon optimal control problems that arise in economic applications. Neither convergence of the integral utility functional nor local boundedness of the optimal control is assumed. Using the classical needle variations technique we develop a normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable under weak regularity assumptions. The result generalizes some previous results in this direction. An illustrative economical example is presented.

Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups, with Applications to Earth Mover’s Distance

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Holder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Holder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.

Weak regularity and finitely forcible graph limits

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e., a graphon determined by finitely many subgraph densities, is simple structured. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak ε-regular partition with the number of parts bounded by a polynomial in ε−1. We construct a finitely forcible graphon W such that the number of parts in any weak ε-regular partition of W is at least exponential in ε−2/25log⁎⁡ε−2. This bound almost matches the known upper bound and, in a certain sense, is the best possible.

Weakly regular [formula omitted]-symmetric spacetimes. The future causal geometry of Gowdy spacetimes

We investigate the future asymptotic behavior of Gowdy spacetimes on T3, when the metric satisfies weak regularity conditions, so that the metric coefficients (in suitable coordinates) are only in the Sobolev space H1 or have even weaker regularity. The authors recently introduced this class of spacetimes in the broader context of T2-symmetric spacetimes and established the existence of a global foliation by spacelike hypersurfaces when the time function is chosen to be the area of the surfaces of symmetry. In the present paper, we identify the global causal geometry of these spacetimes and, in particular, establish that weakly regular Gowdy spacetimes are future timelike geodesically complete. This result extends a theorem by Ringström for metrics with sufficiently high regularity. We emphasize that our proof of the energy decay is based on an energy functional inspired by the Gowdy-to-Ernst transformation. In order to establish the geodesic completeness property, we prove a higher regularity property concerning the metric coefficients along timelike curves and we provide a novel analysis of the geodesic equation for Gowdy spacetimes, which does not require high-order regularity estimates. Even when sufficient regularity is assumed, our proof provides an alternative and shorter proof of the energy decay and of the geodesic completeness property for Gowdy spacetimes.

The fused Kolmogorov filter: A nonparametric model-free screening method

A new model-free screening method called the fused Kolmogorov filter is proposed for high-dimensional data analysis. This new method is fully nonparametric and can work with many types of covariates and response variables, including continuous, discrete and categorical variables. We apply the fused Kolmogorov filter to deal with variable screening problems emerging from a wide range of applications, such as multiclass classification, nonparametric regression and Poisson regression, among others. It is shown that the fused Kolmogorov filter enjoys the sure screening property under weak regularity conditions that are much milder than those required for many existing nonparametric screening methods. In particular, the fused Kolmogorov filter can still be powerful when covariates are strongly dependent on each other. We further demonstrate the superior performance of the fused Kolmogorov filter over existing screening methods by simulations and real data examples.