Universal approximation for rough paths and Lévy processes.
problem Approximating continuous functionals of càdlàg paths.
method Linear functionals of time-extended signatures.
result Universal approximation theorem for continuous functionals of càdlàg paths.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
Holomorphic immersions blocked in 9D real hypersurface with specific signature.
problem Existence of holomorphic immersions of bi-disks into a specific 9D real hypersurface.
method Application of Cartan's method to analyze the existence of immersions.
result Holomorphic immersions are obstructed by specific conditions on the hypersurface.
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…
We construct examples of nondegenerate CR manifolds with Levi form of signature (p,q), 2≤p≤q, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature (1,n−1) which is not locally CR flat and admits …
In this paper we generalize special geometry to arbitrary signatures in target space. We formulate the definitions in a precise mathematical setting and give a translation to the coordinate formalism used in physics. For the projective case, we first discuss in detail projective Kaehler manifolds, appearing in N=1 supe…
For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR struct…
Study on null-projectability of Levi-Civita connections in neutral metrics.
problem Characterizing projectability of Levi-Civita connections along null parallel distributions.
method Analyzing projectability of torsion-free connections along foliations on manifolds, focusing on neutral metric signatures and mid-dimensional distributions.
result Extension of Patterson and Walker's Riemann extension metrics to null parallel distributions of any dimension.
Solves Lie's 3D metric problem for projective vector fields.
problem Describing 3D Levi-Civita metrics with non-trivial projective vector fields.
method Analyzes Riemannian and Levi-Civita metrics of arbitrary signature.
result Solves the analog of Lie's problem in 3D.
We apply E. Cartan's method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds M up to local CR equivalence in the case that the cubic form of M satisfies a certain symmetry property with respect to the Levi form of M. The solution to the equivalence problem is given by a parallelism on a prin…
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).
Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connecti…
Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature l and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature l. We shall assume that there is a holomorphic mapping $H_0\colon U\to \bC\bP^{N_0+1}$, where U is some neighborhood of M in …
Protocol diagnoses neural HJB-PIDE solvers for Lévy jumps, revealing a missing factor in their importance-proposal density.
problem Neural PDE solvers can match scalar diagnostics but miscompute operators, leading to systematic errors.
method Five-step diagnostic protocol decomposes neural solve into components, compares them with independent reference solutions.
result Corrected a missing 1/2-mixture factor in the neural method's importance-proposal density, improving control accuracy.
By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connectio…
Study on Einstein metrics on SU(3) Lie group, including new Lorentzian example.
problem Existence and properties of left-invariant pseudo-Riemannian Einstein metrics.
method Investigation of left-invariant metrics, use of isometry groups, analysis of curvatures.
result First Lorentzian homogeneous Einstein metric on SU(3).
Study on metrizability and Ricci-flatness of Finsler spaces with Kropina metrics.
problem Investigating metrizability and Ricci-flatness of Finsler spaces with specific metrics.
method Analyzing the Levi-Civita connection, covariant derivative of 1-forms, and using cohomology groups.
result Explicitly obtained all Ricci-flat, locally metrizable m-Kropina metrics in (3+1)D.
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.
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…
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 α-divergences from Lévy triplets, identifying Fisher information matrix and α-connection. result Identifies statistical implications and differential-geometric structures of Lévy processes.
The paper revisits expected signatures in semimartingale models, providing new formulae and simplifying complexity.
problem Computing expected signatures in semimartingale models.
method Revisits and provides new formulae for computing expected signatures in a general semimartingale setting.
result Log-transform of expected signatures simplifies complexity, leading to signature cumulants.
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.
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.
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.
Defines knot signature invariant using G-signature theorem.
problem No specific problem stated; focuses on knot theory.
method Uses G-signature theorem to define knot invariant.
result Defines an invariant for strongly invertible knots.
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 L2-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…
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.
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…
Deep signature/log-signature FBSDE algorithm improves accuracy and training time.
problem Solving FBSDEs with state and path dependent features.
method Incorporates deep signature/log-signature transformation into RNN model.
result Improves accuracy and training time compared to existing methods.
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.
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.
A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifol…
Paper calculates perpetual American put option pricing with drawdown event in Lévy market.
problem Pricing perpetual American put options with a drawdown event in a Lévy market.
method Derives explicit price using geometric Lévy process with downward jumps, optimal stopping rule, and martingale arguments.
result Optimal stopping rule is the first time asset price falls below a specific value.
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for inst…
Study determines Lévy exponent from derivative prices.
problem Determine Lévy exponent in asset pricing models.
method Analyzes power-payoff derivatives to infer Lévy exponent structure.
result Lévy exponent can be determined from derivative prices.
Analyzes Lévy flights on manifolds for finding small targets.
problem Finding small targets using Lévy flights on various manifolds.
method Analytic description of Lévy flights on closed Riemannian manifolds, including asymptotics of expected stopping time.
result Computes the expected time for finding a small target by Lévy flight on surfaces.
Maximum Levine-Tristram signature of torus knots follows a reduction formula.
problem Determining the maximum Levine-Tristram signature for torus knots.
method Proved a reduction formula analogous to Gordon-Litherland-Murasugi's classical signature result.
result Maximum Levine-Tristram signature of torus knots satisfies a reduction formula.
Optimizes liquidation strategies for assets with Levy process price dynamics.
problem Maximizing cash received from asset sale with price impact.
method Almgren-Chriss framework, constant absolute risk aversion, Levy process approximation.
result Explicit expression for optimal liquidation trajectories.
New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
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.
Introduces flat discrete signatures for financial data analysis.
problem Representing financial data for machine learning without continuous transformation.
method Introduced flat discrete signatures and discrete signatures, generalizing flat discrete signatures.
result Flat discrete signatures can represent quadratic variation relevant in finance.
Paper extends Lévy models with memory to better price FX double barrier options.
problem Efficiently pricing double barrier options in complex FX models.
method Introduces regime-switching Lévy models with memory and a modified numerical method.
result New models and method improve accuracy of option pricing.