Study on higher order Levi forms on homogeneous CR manifolds, improving previous results.
problem Investigating nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds.
method Improving previous results by proving order constraints for real forms in complex flag manifolds and constructing CR vector bundles with arbitrary orders of nondegeneracy.
result General orbits of real forms in complex flag manifolds have order less or equal 3 and compact ones less or equal 2.
We construct a complete convergent normal form for a real hypersurface in $\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. In particular, we obtain, in the spirit of the work of Chern and Moser \cite{chern}, distinguished curves in the …
The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.
problem Understanding the Levy Laplacian on manifolds and its heat flow behavior.
method Various definitions of the Levy Laplacian are proven equivalent. Heat flows of differential forms are used to construct solutions.
result Solutions of the heat equation with the Levy Laplacian tend to locally constant functionals over time.
The paper constructs CR manifolds with arbitrary Levi nondegeneracy.
problem Creating CR manifolds with specific Levi nondegeneracy properties.
method Using CR algebras from su(2) representations, studying iterated Levi forms, and local model equations. result Explicit construction and analysis of homogeneous CR manifolds with arbitrary Levi nondegeneracy.
We discuss various analytical and geometrical aspects of the Levi form, which is associated with a CR manifold having any CR dimension and any CR codimension.
We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result and which may be helpful in the study of other smooth nor…
Classifies Calabi hypersurfaces with parallel Fubini-Pick form.
problem Classifying Calabi hypersurfaces with specific geometric properties.
method Classification based on parallel Fubini-Pick form and Levi-Civita connection.
result Classification of 2 and 3-dimensional Calabi hypersurfaces.
The paper proves unique Levi-Civita connections on noncommutative forms.
problem Existence and uniqueness of Levi-Civita connections on noncommutative differential forms.
method Combining Hilbert module and algebraic techniques, proving conditions for existence and uniqueness of Hermitian torsion-free connections.
result Existence and uniqueness of Levi-Civita connections on θ-deformations of compact Riemannian manifolds.
Complete normal forms for specific real hypersurfaces in complex space are constructed.
problem Constructing complete normal forms for real hypersurfaces in C3. method Utilizing equivariant moving frames for systematic symbolic manipulation.
result Complete normal forms for 5-dimensional real hypersurfaces in C3 are found. We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.
problem CR hypersurfaces in C^4 with constant rank Levi form and their defining equations.
method Obtained a complete normal form for models of real analytic uniformly 2-nondegenerate CR hypersurfaces in C^4.
result Found explicit formulas for infinitesimal symmetries of homogeneous 2-nondegenerate models.
The paper uses moment matching method for pricing spread options under Lévy models.
problem Pricing spread options under Lévy models with mean-variance mixture.
method Moment matching method applied to Lévy models with mean-variance mixture.
result Obtains semi-closed form formulas for spread option prices.
We classify tube domains in Cn+1 (n≥1) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least 2(n−2)(n−3).
In this article, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of CN of codimension d ≥ 1 under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the n…
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat…
Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.
problem Connection between Yang-Mills fields and modified Lévy Laplacians on 4-manifolds.
method Analysis of modified Lévy Laplacians and their relation to Yang-Mills equations under nontrivial holonomy groups.
result Existence of a modified Lévy Laplacian related to Yang-Mills self-duality equations.
This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the (1,1) curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. …
Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.
Let M2n+1, n≥1, be a smooth manifold with a pseudo-convex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets M=S0⊃S1⊃⋯⊃Sn, where Sq is the set of points where the Levi-form has nullity $\ge q…
Energy functional on Teichmüller space is plurisubharmonic but not strictly so.
problem Characterizing points where energy functional fails to be strictly plurisubharmonic.
method Analyzing the kernel of the Levi form and relating it to Higgs bundles and Hitchin fibration.
result For generic choices, energy functional is strictly plurisubharmonic.
This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.
problem Classifying hypersurfaces with parallel Fubini-Pick form in \(\mathbb{R}^{n+1}\).
method Defining a generalized Calabi product and proving decomposition theorems.
result Complete classification of Calabi hypersurfaces in \(\mathbb{R}^{n+1}\) with parallel Fubini-Pick form.
Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension (2n+1) have maximal symmetry dimension n2+4n+3. We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is n2+4 for Levi-indefinite structures and n2+3 for Levi-definite structures when $n>1…
The paper constructs Levi flat structures using structure sheaves and differential complexes.
problem Global solvability and regularity of Levi flat structures.
method Employing formal integrability and differential complexes, the paper constructs a resolution for the structure sheaf.
result Global exactness and Sobolev regularity of the differential complex for Levi flat structures.
Calibrating a Lévy process usually requires characterizing its jump distribution. Traditionally this problem can be solved with nonparametric estimation using the empirical characteristic functions (ECF), assuming certain regularity, and results to date are mostly in 1D. For multivariate Lévy processes and less smooth …
Formula for European option pricing under jump diffusion model.
problem Option pricing under complex stochastic processes.
method Infinite series of Black-Scholes terms for Levy-driven processes.
result Series solution converges with a radius of convergence.
In this paper we construct a large class of new normal forms for Levi-nondegenerate real hypersurfaces in complex spaces. We adopt a general approach illustrating why these normal forms are natural and which role is played by the celebrated Chern-Moser normal form. The latter appears in our class as the one with the "m…
The paper characterizes potential functions whose level sets are orbits in mechanical systems.
problem Characterizing smooth potential energy functions on the plane with specific level set properties.
method Analyzing inverse curvature flow and properties of level sets.
result Analytic or functions with totally path-disconnected critical sets must be radial, while every compact convex set is a critical set of a Levi potential.
Study of degenerate contrast functions on Lie groupoids and their geometric structures.
problem Understanding geometric structures on Lie groupoids with degenerate metrics.
method Using Lie groupoids and algebroids, analyze contrast functions and degenerate two-forms.
result Reduction of degenerate two-forms to pseudometric structures under regular conditions.
We consider a stochastic volatility model with Lévy jumps for a log-return process Z=(Zt)t≥0 of the form Z=U+X, where U=(Ut)t≥0 is a classical stochastic volatility process and X=(Xt)t≥0 is an independent Lévy process with absolutely continuous Lévy measure ν. Small-time expansio…
The paper studies affine models driven by independent Lévy processes and their calibration.
problem Characterizing and classifying affine models driven by Lévy processes.
method Analyzing the short rate equation with independent Lévy processes and characterizing the generator.
result A precise form of the generator and classification of affine models with canonical representations.
Modeling risk and performance with Levy-stable distributions.
problem Understanding risk and performance in financial markets with non-Gaussian distributions.
method Developed a finite-horizon model using Levy-stable scaling, identified parameters from data, derived formulas for various financial ratios.
result Horizon-correct formulas for risk measures are derived and validated across different horizons.
We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniquen…
Introduces Levi core for CR manifolds, linking it to global invariants.
problem Understanding global invariants of CR manifolds.
method Introduces Levi core, relates to Diederich-Fornæss index and D'Angelo class.
result Levi core is trivial under certain conditions, nontrivial otherwise.
We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-C…
Study extends Lévy models to capture market propagation delays.
problem Capturing sudden events in related markets with stochastic delays.
method Extend multivariate Lévy models using self-decomposability and multivariate subordination.
result Derived closed-form expressions for characteristic function and implemented Monte Carlo scheme.
The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter…
We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface M and its symmetry algebra s one has either: (i) dims=15 and M is spherical (with Levi form of signature either (2,0) or (1,1) everywhere), or (ii) dims≤11 where $\di…
Financial markets based on Lévy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to app…
We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan. In particular, we consider Merton…
Develops Palatini formalism in generalized geometry for string theory.
problem Formulating Palatini variation in generalized geometry.
method Palatini formalism within generalized Riemannian geometry of Courant algebroids.
result Natural emergence of generalized Levi-Civita connection and string effective actions.
We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets; that is, possible instantaneous drift, volatility and jump characteristics of the price process. We show that an optimal investment…
New method for Bayesian inference of Lévy-driven SDEs with jumps.
problem Bayesian inference for Lévy-driven SDEs is challenging due to discontinuities and heavy tails.
method Neural exponential tilting framework for variational inference.
result Accurately captures jump dynamics and reliable posterior inference in heavy-tailed regimes.
We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Le…
New structure for quantum algebra representations.
problem Understanding representations of quantum algebras.
method Constructing a quotient category of annular quantum gln webs. result Equivalent to finite dimensional representations of quantum Levi subalgebras.
We establish an explicit pricing formula for the class of Lévy-stable models with maximal negative asymmetry (Log-Lévy model with finite moments and stability parameter 1<α≤2) in the form of rapidly converging series. The series is obtained with help of Mellin transform and the residue theory in C2. T…
The paper proves extension theorems for complex manifolds with Levi q-concave domains.
problem Holomorphic extension theorems for complex manifolds with Levi q-concave domains. method The proof relies on holomorphic Morse inequalities, the Kohn-Rossi extension theorem, and a general Nakano-Griffiths inequality.
result Holomorphic extension theorems for (0,ℓ)-forms on Levi q-concave domains. Study shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.
problem Existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems of Novikov equation.
method Kato's semi-group approach to establish well-posedness and C1 regularity. result Existence of exactly two families of periodic one-forms defining metrics and Levi-Civita connections.
Characterizes term structure models driven by Lévy processes.
problem Modeling non-negative short rates with Lévy processes.
method Analyzes affine term structure models driven by independent Lévy martingales.
result All possible solutions of the models can be obtained using stable processes.
Asymptotic expansions for call prices and implied volatilities in exponential Lévy models.
problem Developing precise call-price and implied volatility approximations for asset-price models.
method Analyzing the asymptotic behavior of at-the-money call prices and implied volatilities for Lévy-driven asset-price models.
result First-order asymptotic expansions for at-the-money call prices and implied volatilities in exponential Lévy models.