Theorem Hold Research Articles

Is (Independent) Subordination Relevant in Equity Derivatives?

ABSTRACTMonroe (1978) demonstrates that any local semimartingale can be represented as a time‐changed Brownian Motion (BM). A natural question arises: does this representation theorem hold when the BM and the time‐change are independent? We prove that a local semimartingale is not equivalent to a BM with a time‐change that is independent from the BM. Our result is obtained utilizing a class of additive processes: the additive normal tempered stable (ATS). This class of processes exhibits an exceptional ability to calibrate the equity volatility surface accurately. We notice that the sub‐class of additive processes that can be obtained with an independent additive subordination is incompatible with market data and shows significantly worse calibration performances than the ATS, especially on short time maturities. These results have been observed every business day in a semester on a dataset of S&P 500 and EURO STOXX 50 options.

Locating the spectrum of a relation matrix through those of its inputs

Let [Formula: see text] and [Formula: see text] be Banach spaces. When [Formula: see text] and [Formula: see text] are linear relations in [Formula: see text] and [Formula: see text], respectively, we denote by [Formula: see text] the linear relation matrix acting on [Formula: see text] of the form [Formula: see text], where [Formula: see text] is the zero operator and [Formula: see text] is a bounded operator from [Formula: see text] to [Formula: see text]. In this paper, we prove that if [Formula: see text] denotes the Weyl spectrum, the Browder spectrum or the Drazin spectrum of a linear relation, then for every [Formula: see text] we have the equality [Formula: see text] where [Formula: see text] a subset of [Formula: see text]. Moreover, we explore how Weyl’s theorem and Browder’s theorem hold for linear relation matrices.

Kernel-based linear system identification: When does the representer theorem hold?

It holds for any physical (bounded) input if and only if the kernel is stable.

Stationary modes for vector nonlinear Schrödinger-type equations: A numerical procedure for complete search and its mathematical background

In this paper, we address the problem of complete description of coexisting stationary localized modes for the system of n nonlinearly coupled Schrödinger-type equations. Our approach is based on the observation that if the nonlinearity is defocusing (repulsive), then generic solutions of the corresponding stationary system have singularities at finite points of the real axis and, consequently, cannot describe profiles of stationary modes. Then the stationary localized modes can be found by a procedure of “filtering out” solutions with singularities. We have formulated and proved three rigorous statements that form the mathematical background of the approach. The first of them, Theorem on parametrization, states that there exists a homeomorphism between S and Rn, where S is the set of solutions of the stationary system that decay to zero componentwise at +∞. This allows to code the solutions from S by n-component vectors. The second statement, Theorem on singularity, states that, under certain conditions, the singular solutions are ubiquitous in the system and gives the conditions for detection of singularities. This statement allows to seek for localized modes by properly arranged numerical scanning in S operating with vectors from Rn. The third theorem, Theorem on collapsing solutions, allows to reduce the scanning procedure to a finite domain in Rn. As a result, if the conditions of the three theorems hold, the scanning procedure yields the complete set of localized modes for the system. We illustrate this method by an examples of two and three nonlinearly coupled equations that describe the steady states of two and three- component Bose-Einstein Condensate. As the main outcome, in the chosen domain of chemical potentials, we have found all coexisting nonlinear modes.

Perron-Frobenius type theorem for nonnegative tubal matrices in the sense of t-product

This paper is focused on the study of Perron-Frobenius (PF) type theorem for nonnegative tubal matrices, where a tubal matrix is a matrix with every element being a tubal scalar (a vector in the usual sense). Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce irreducible tubal matrices and provide two equivalent characterizations. Then, the celebrated PF theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices, such as the existence of a positive eigenvector, can be generalized to the tubal matrix setting, while some are not. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.

Strong Shannon–McMillan–Breiman’s theorem for locally compact groups

AbstractWe prove that for a vast class of random walks on a compactly generated group, the exponential growth of convolutions of a probability density function along almost every sample path is bounded by the growth of the group. As an application, we show that the almost sure and $L^1$ convergences of the Shannon–McMillan–Breiman theorem hold for compactly supported random walks on compactly generated groups with subexponential growth.

Functional Central Limit Theorems for Stick-Breaking Priors

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stick-breaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its marginal distributions to obtain the asymptotics although the computations are highly technical.

The existence and uniqueness of solutions for kernel-based system identification

The notion of reproducing kernel Hilbert space (RKHS) has emerged in system identification during the past decade. In the resulting framework, the impulse response estimation problem is formulated as a regularized optimization defined on an infinite-dimensional RKHS consisting of stable impulse responses. The consequent estimation problem is well-defined under the central assumption that the convolution operators restricted to the RKHS are continuous linear functionals. Moreover, according to this assumption, the representer theorem hold, and therefore, the impulse response can be estimated by solving a finite-dimensional program. Thus, the continuity feature plays a significant role in kernel-based system identification. We show that this central assumption is guaranteed to be satisfied in considerably general situations, namely when the input signal is bounded, the kernel is an integrable function, and in the case of continuous-time dynamics, continuous. Furthermore, the strong convexity of the optimization problem and the continuity property of the convolution operators imply that the kernel-based system identification admits a unique solution. Consequently, it follows that kernel-based system identification is a well-defined approach.

Finite W-algebras associated to truncated current Lie algebras

Finite W-algebras associated to truncated current Lie algebras are studied in this paper. We show that some properties of finite W-algebras in the semisimple case hold in the truncated current case. In particular, Kostant's theorem and Skryabin equivalence hold in our case. As an application, we give a classification of simple Whittaker modules for truncated current Lie algebras in the \(s\ell_2\) case.

Order integrals

We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space E. The monotone convergence theorem, Fatou’s lemma, and the dominated convergence theorem are established; the analogues of the classical \({\mathscr {L}}^1\)- and \({\mathrm L}^1\)-spaces are investigated. The results extend earlier work by Wright and specialise to those for the Lebesgue integral when E equals the real numbers. The hypothesis on E that is needed for the definition of the integral and for the monotone convergence theorem to hold (\(\sigma \)-monotone completeness) is a rather mild one. It is satisfied, for example, by the space of regular operators between a directed partially ordered vector space and a \(\sigma \)-monotone complete partially ordered vector space, and by every JBW-algebra. Fatou’s lemma and the dominated convergence theorem hold for every \(\sigma \)-Dedekind complete space. When E consists of the regular operators on a Banach lattice with an order continuous norm, or when it consists of the self-adjoint elements of a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space, then the finite measures as in the current paper are precisely the strongly \(\sigma \)-additive positive operator-valued measures. When E is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in our sense, but not conversely. Even when a measure falls into both categories, the domain of the integral as defined in this paper can properly contain that of any reasonably defined integral with respect to the vector measure using Banach space methods.

Laplacian comparison theorem on Riemannian manifolds with modified $m$-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

Corona and Wolff theorems for the multiplier algebra of Dirichlet–Morrey spaces

AbstractFor $0&lt;\lambda ,p&lt;1$ , the Dirichlet–Morrey space $\mathcal {D}_p^{\lambda } $ is the space of all analytic function on the unit disc such that the measure $ |f'(z)|^2(1-|z|^2)^pdA(z)$ is a $p\lambda $ -Carleson measure. In this paper, we show that the corona theorem and the Wolff theorem hold for the multiplier algebra of Dirichlet–Morrey spaces.

NON-CLASSICAL FOUNDATIONS OF SET THEORY

AbstractIn this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Poisson processes in the space of (d-1)-dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature -1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all dge 2, it is shown that in case (ii) the central limit theorem holds for din {2,3} and fails if dge 4 and k=d-1 or if dge 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

Prime number theorem for regular Toeplitz subshifts

AbstractWe prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

On the Weak Lefschetz Principle in Birational Geometry

This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics.

A Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”

Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.

Commuting probabilities of infinite groups

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox.

Horváth-Type Theorems on a Star Graph with Mixed Boundary Conditions

In this paper, we deal with some partial inverse problems for the Sturm–Liouville operator on a star graph with Robin and/or Dirichlet boundary conditions in pendant vertices. It is shown that if all but one of the potentials are known a priori, then Horvath-type theorems hold. Our method is based on the related theory about Weyl function and the growth of entire functions.

The Coase Theorem, the Nonempty Core, and the Legal Neutrality Principle

AbstractThe Coase theorem states that where there are externalities and no transaction costs resource allocation is Pareto-optimal and independent of the stakeholders’ legal position. This result has been challenged many times. In the cooperative game approach to resource allocation, the refutation is made by constructing a three-person game which has an empty core under one set of liability rules—which implies that optimal allocations are coalitionally unstable–and a nonempty core under another set. In this example, however, the probability that the core is non-empty is rather high (5/6). Yet, even if coalitionally stable Pareto-optimal arrangements are likely, to establish the plain validity of the Coase theorem it must be shown that the legal neutrality statement also holds. We show that for the three-person cooperative game example mentioned above, the probability that the two assertions of the Coase theorem hold can be as low as 3/8.