arXiv research

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We con…

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…

New GLPs split Lévy bridges into non-overlapping subprocesses.

The basic class of the non-integrable almost complex manifolds with a pair of Norden metrics are considered. The interconnections between corresponding quantities at the transformation between the two Levi-Civita connections are given. A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is…

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…

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique o…

We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stoc…

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …

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…

New method for Bayesian inference of Lévy-driven SDEs with jumps.

In the "positive interest" models of Flesaker-Hughston, the nominal discount bond system is determined by a one-parameter family of positive martingales. In the present paper we extend this analysis to include a variety of distributions for the martingale family, parameterised by a function that determines the behaviou…

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fernández and Gray. This relies on the construction of some G_2 solvmanifolds, whose Levi-Civita connection is known to give a paralle…

Study flat connections on Courant algebroids using Lie groups.

The important application of semi-static hedging in financial markets naturally leads to the notion of quasi self-dual processes. The focus of our study is to give new characterizations of quasi self-duality for exponential Lévy processes such that the resulting market does not admit arbitrage opportunities. We derive …

New formulas for Riemannian gradient and Hessian on manifold metrics.

Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considere…

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of t…

Investigates stock models using tempered stable processes for option pricing.

We analyze the Levy processes produced by means of two interconnected classes of non stable, infinitely divisible distribution: the Variance Gamma and the Student laws. While the Variance Gamma family is closed under convolution, the Student one is not: this makes its time evolution more complicated. We prove that -- a…

Study shows adversarial attacks are more likely when test instances are concentrated, leading to vulnerabilities in robust classifiers.

In this paper we exhibit a one-parameter family of new Taub-NUT instantons parameterized by a half-line. The endpoint of the half-line will be the reducible Yang-Mills instanton corresponding to the Eguchi-Hanson-Gibbons L^2 harmonic 2-form, while at an inner point we recover the Pope-Yuille instanton constructed as a …

We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex Kähler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite …

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…

Study reveals finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.

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…

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…

Ellipsoids approach Gaussian distribution in high dimensions.

This paper considers options pricing when the assumption of normality is replaced with that of the symmetry of the underlying distribution. Such a market affords many equivalent martingale measures (EMM). However we argue (as in the discrete-time setting of Klebaner and Landsman, 2007) that an EMM that keeps distributi…

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.

Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.

A new shrinkage-based construction is developed for a compressible vector $\boldsymbol{x}\in\mathbb{R}^n$, for cases in which the components of $\xv$ are naturally associated with a tree structure. Important examples are when $\xv$ corresponds to the coefficients of a wavelet or block-DCT representation of data. The me…

We study the geometry of the tangent bundle equipped with a two-parameter family of Riemannian metrics. After deriving the expression of the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. Specializing to the case of space forms, we characterise the metric…

Study on deformations of LC Spin(7) instantons simplifies the problem.

Bi-flat F-structures link to differential bicomplexes and Gauss-Manin connections.

Analyzes the Levi form on CR manifolds of any dimension.

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.

Paper proves existence of Lévy term structure models.

In this paper we perform a statistical analysis of the high-frequency returns of the IBEX35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data us…

The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The sec…

Study of Lévy flights on Zoll surfaces, revealing geometric information.

Study shows convergence rates for BSDEs approximated by compound Poisson processes.

New method calibrates multivariate Lévy processes using neural networks.

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 shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.

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.