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…
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48 results for “Levi decomposition”
We prove the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.
A Levi-Malcev type decomposition for -step solvable Lie algebras with a complex structure
problem Decomposition of -step solvable Lie algebras with a complex structure
method Proving a Levi-Malcev type decomposition
result Fino-Vezzoni conjecture holds for -step solvable unimodular Lie algebras
The Levy-Ito theorem explains how certain random processes can be broken down.
problem Understanding how certain random processes can be decomposed.
method Martingale methods are used to prove the Lévy-Ito decomposition theorem.
result The Lévy-Khintchine representation of infinitely divisible distributions is derived.
Constructs a unique Levi-Civita connection for generalised metrics.
problem Non-uniqueness of generalised Levi-Civita connections.
method Geometrically constructs a canonical generalised Levi-Civita connection.
result Decomposes the generalised Riemann curvature tensor in terms of classical geometric data.
These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…
Study finds closed G2-structures on non-solvable Lie groups.
problem Existence of closed G2-structures on non-solvable Lie groups.
method Investigation of left-invariant closed G2-structures on specific Lie groups.
result First examples of closed G2-structures on non-solvable Lie groups.
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.
The paper studies algebraic properties of bounded Killing vector fields on Riemannian manifolds.
problem Algebraic properties of bounded Killing vector fields on Riemannian manifolds.
method Analysis of Levi decomposition and Jordan decomposition of Killing vector fields.
result Eigenvalues of the adjoint operator are all imaginary.
Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced w…
In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) Lévy processes to model the underlying asset of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scal…
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
The paper develops a valuation framework for GLWB-LTC contracts with Levy dynamics and stochastic interest rates.
problem Valuation of GLWB-LTC contracts with financial guarantees, longevity protection, and health-contingent LTC payments.
method Coupling a recombining Hull-White trinomial tree with an IMEX finite difference scheme, incorporating a seven-state health model.
result Hybrid tree-IMEX method delivers stable long-maturity prices consistent with simulation benchmarks.
Study prolongations of nilpotent Lie algebras with specific structural subalgebras.
problem Understanding prolongations of nilpotent Lie algebras with specific structural subalgebras.
method Analyzing finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, focusing on those with decomposable reductive structural subalgebras.
result Obtained Levi-Malčev and Levi-Chevalley decompositions and precise properties of prolongations.
We establish a nondominated version of the optional decomposition theorem in a setting that includes jump processes with nonvanishing diffusion as well as general continuous processes. This result is used to derive a robust superhedging duality and the existence of an optimal superhedging strategy for general contingen…
Hedging of defaultable claims in a structural model using a locally risk-minimizing approachq-fin.MF
A method for hedging defaultable claims using locally risk-minimizing in a structural model.
problem Hedging defaultable claims in a structural model with jumps and non-risk-neutral probabilities.
method Locally risk-minimizing approach in a structural model with finite variation Levy process.
result Derivation of Follmer-Schweizer decompositions for hedging.
Paper proves isoperimetric inequalities for non-reversible Finsler manifolds.
problem Proving isoperimetric inequalities for non-reversible Finsler manifolds.
method Constructing needle decompositions and using curvature-dimension condition CD(K,N).
result Established isoperimetric inequality for non-reversible Finsler manifolds.
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…
Study on Hodge decompositions for Lie algebroids on manifolds with boundary.
problem When does the Chevalley-Eilenberg differential admit a Hodge decomposition?
method Introduce concepts like Cauchy-Riemann structures, elliptic and non-elliptic boundary points, q-convexity, and use them to prove Hodge decompositions.
result Hodge decompositions for q-convex elliptic Lie algebroids on manifolds with boundary.
We revisit Merton's portfolio optimization problem under boun-ded state-dependent utility functions, in a market driven by a Lévy process extending results by Karatzas et. al. (1991) and Kunita (2003). The problem is solved using a dual variational problem as it is customarily done for non-Markovian models. One of …
Proofs Lie's classification of certain vector field subalgebras.
problem Classifying finite dimensional subalgebras of vector fields on the complex plane.
method Representation theory of sl(2, C) and previous classifications.
result Completes the classification of vector field subalgebras.
Paper improves American option valuation in complex models.
problem Valuation of American options in time-dependent jump-diffusion models.
method Integral equations and characteristic functions for explicit exercise boundary determination.
result Efficient and accurate pricing method for American options in various models.
Modeling dependent defaults with multivariate Cox processes.
problem Capturing dependence in default times.
method Multivariate generalized Cox process with càdlàg, increasing processes.
result Closed-form expressions for joint survival probabilities.
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
New approach to parabolic subalgebras and buildings.
problem Understanding the geometry of parabolic subalgebras and buildings.
method Elementary approach over arbitrary fields, focusing on parabolic subalgebras and their properties.
result Derivation of structure theory from root systems to Bruhat decomposition.
Study reveals finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.
problem Finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.
method Simulations and analysis of Levy-Levy-Solomon model with different random number generators and stopping criteria.
result Low-quality pseudo random number generators significantly impact simulation results.
New result on Levi-flat hypersurfaces' normal bundles without positive curvature.
problem Understanding Levi-flat hypersurfaces' normal bundles and their curvature properties.
method Analyzing the normal bundle of Levi-flat real hypersurfaces in complex manifolds.
result The normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature.
Formulae for curvature of quaternionic Kähler manifolds derived from hyper-Kähler data.
problem Deriving curvature formulae for quaternionic Kähler manifolds.
method Using HK/QK correspondence and Levi-Civita connection, we express curvature in terms of hyper-Kähler data.
result Formulae for curvature of quaternionic Kähler manifolds, including norm of curvature tensor.
The problem of an arbitrary truncated Levy flight description using the method of cumulant approach has been solved. The set of cumulants of the truncated Levy distribution given the assumption of arbitrary truncation has been found. The influence of truncation shape on the truncated Levy flight properties in the Gauss…
Develops information geometry for Lévy processes in finance.
problem Understanding the statistical properties of Lévy processes for financial modeling.
method Deriving -divergences from Lévy triplets, identifying Fisher information matrix and -connection.
result Identifies statistical implications and differential-geometric structures of Lévy processes.
Quantitative isoperimetry on manifolds with curvature bounds.
problem Controlling distances of isoperimetric sets from geodesic balls.
method Analysis of transport-rays decompositions in metric measure spaces.
result Deficit between manifold and sphere diameters quantitatively bounded.
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.
Classifies homogeneous Levi non-degenerate hypersurfaces in complex 3-space.
problem Classifying specific types of complex hypersurfaces.
method Analyzing hypersurfaces with symmetry algebra of dimension at least 6.
result All such hypersurfaces are classified.
Aspects of the Levi formmath.CV
Analyzes the Levi form on CR manifolds of any dimension.
problem Understanding the Levi form on CR manifolds of varying dimensions and codimensions.
method Analytical and geometrical study of the Levi form.
result Comprehensive insights into the Levi form on CR manifolds.
The pricing of options in exponential Levy models amounts to the computation of expectations of functionals of Levy processes. In many situations, Monte-Carlo methods are used. However, the simulation of a Levy process with infinite Levy measure generally requires either to truncate small jumps or to replace them by a …
Efficient methods for Lévy models using SINH-regular processes.
problem Efficient numerical methods for evaluating Lévy models.
method Defining SL-processes and sSL-processes, deriving properties of characteristic exponent, and showing all popular Lévy processes can be subordinated to Brownian motion.
result All crucial properties of characteristic exponent are consequences of a specific representation, and all popular Lévy processes are SL- or sSL-subordinated Brownian motion.
Paper proves existence of Lévy term structure models.
problem Existence proof for Lévy term structure models.
method Proof of existence and uniqueness for Heath-Jarrow-Morton type equation.
result Full proof of existence and uniqueness of Lévy term structure models.
This paper classifies minimal complex surfaces with Levi-Civita Ricci-flat metrics.
problem Classifying minimal complex surfaces with specific geometric properties.
method Study of compact complex manifolds with Levi-Civita Ricci-flat metrics.
result Minimal complex surfaces with Levi-Civita Ricci-flat metrics are Kähler Calabi-Yau surfaces and Hopf surfaces.
Study of Lévy flights on Zoll surfaces, revealing geometric information.
problem Understanding the mean first capture time of Lévy flights on Zoll surfaces.
method Analysis of geodesic Lévy processes on Zoll surfaces, focusing on the first correction term.
result The first correction term encodes geometric information, specifically the degree of the conjugate point.
Study shows convergence rates for BSDEs approximated by compound Poisson processes.
problem Analyzing convergence rates of BSDEs driven by Lévy processes.
method Approximating Lévy processes by compound Poisson processes and studying BSDEs.
result Optimal convergence rates derived for BSDEs in -norm and Wasserstein distance.
New method calibrates multivariate Lévy processes using neural networks.
problem Calibrating multivariate Lévy processes with less smooth densities.
method Approximate Lévy density with parametrized functional form, estimate characteristic function using numerical integration with deep neural networks.
result Deep neural networks robustly capture sharp transitions in Lévy densities.
Levy copulas are the most general concept to capture jump dependence in multivariate Levy processes. They translate the intuition and many features of the copula concept into a time series setting. A challenge faced by both, distributional and Levy copulas, is to find flexible but still applicable models for higher dim…
Study on Parisian ruin for a refracted Lévy process with adaptive premium rate.
problem Investigating the probability of Parisian ruin for a Lévy insurance risk process with a refracted premium rate.
method Generalization of Loeffen et al. (2013) for a refracted Lévy process with a deterministic implementation delay.
result Compact and similar structure result for the probability of Parisian ruin.
We introduce an algorithm for the pricing of finite expiry American options driven by Lévy processes. The idea is to tweak Carr's `Canadisation' method, cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted algorithm is viable for any Lévy process whose law at an independent, exponentially distrib…
Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremum of Lévy processes. In particular, we extend the results of Broadie et al. (1999) to jump-diffusion models. We also derive bounds for general exponential Lévy models.
Paper compares two hedging strategies for Lévy models.
problem Difference between locally risk-minimizing and delta hedging strategies.
method Model-independent upper estimations and numerical examples for two Lévy models.
result Upper estimations for the difference between strategies.
The paper analyzes drawdowns in Lévy processes, focusing on magnitude, asymptotics, and duration.
problem Magnitude, asymptotics, and duration of drawdowns in Lévy processes.
method Approximation and asymptotic analysis of drawdown quantities for spectrally negative Lévy processes.
result The law of duration of drawdowns for a wide class of Lévy processes, including TTR.
Study of bandit problem with Poisson decision times and Lévy processes.
problem Continuous-time multi-armed bandit problem with Poisson decision times.
method Gittins index policy applied to spectrally one-sided Lévy processes.
result Gittins index converges to classical Lévy bandit index.