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…
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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.
This paper characterizes cryptocurrency market behavior using Levy's stable distributions.
problem Modeling price fluctuations in cryptocurrency markets with fat tails and scaling phenomena.
method Characterization using Levy's stable distribution with under certain time intervals, employing Parseval's relation and GCLT.
result Price fluctuations in cryptocurrency markets can be well described by Levy's stable distribution.
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…
Stock market price fluctuations follow Lévy's stable distribution over long term.
problem Understanding the stability of stock market price fluctuations over different time scales.
method Estimated Lévy's stable parameters from four stock markets over long and short term.
result Stable parameters from different stock markets showed a unique value over long term, but fluctuated with correlation in short term.
Non-Lévy Distribution of Commodity Price Fluctuationscond-mat.stat-mech
Price fluctuations of commodities like cotton and wheat are thought to display probability distributions of returns that follow a Lévy stable distribution. Recent analysis of stocks and foreign exchange markets show that the probability distributions are not Lévy stable, a plausible result since commodity markets have …
Characterizes Lévy-driven Ornstein-Uhlenbeck processes linked to tempered stable distributions.
problem Understanding Lévy-driven Ornstein-Uhlenbeck processes and their properties.
method Characterizes the Lévy triplet and deduces transition laws for finite variation Ornstein-Uhlenbeck processes associated with tempered stable distributions.
result Provides algorithms for generating skeleton of Ornstein-Uhlenbeck processes related to exponentially-modulated tempered stable laws.
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.
Formula found for ruin probabilities in divided insurance companies.
problem Ruin probabilities in divided insurance companies with Lévy processes.
method Formula for supremum distribution of Lévy processes with broken drift.
result Formulas for ruin probabilities in specified proportions.
New method combines simulated annealing and Levy distribution for fast matrix factorization.
problem High complexity and difficulty in parallelizing matrix factorization for large matrices.
method Combining simulated annealing with Levy distribution for matrix factorization.
result Achieves good solutions in acceptable time with low computations.
Long-Time Fluctuations in a Dynamical Model of Stock Market Indicescond-mat.stat-mech
Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. In recent empirical studies of stock market indices it was examined whether the distribution P(r) of returns r(tau) after some time tau can be described by a (truncated) Levy-stable distribution L_{alpha}(r)…
Develops Bilateral Gamma processes for financial market modeling.
problem Modeling financial market fluctuations with Lévy processes.
method Exploration of bilateral Gamma distributions and their Lévy processes.
result Validates Bilateral Gamma processes on real financial data.
Study cliquet options in a jump-diffusion model with Lévy processes.
problem Pricing cliquet options in a complex financial model with jumps.
method Developed semi-analytic expressions using Lévy process distribution and Fourier transform.
result Inferred semi-analytic expressions for cliquet option prices and derived Greeks.
The paper evaluates functions of stable Lévy processes and their extrema efficiently.
problem Efficiently evaluating functions of stable Lévy processes and their extrema.
method Integral representations, conformal acceleration technique, simplified trapezoid rule.
result Efficient numerical procedures for cumulative probability distribution functions (cpdfs) are developed.
The generalized correlation approach, which has been successfully used in statistical radio physics to describe non-Gaussian random processes, is proposed to describe stochastic financial processes. The generalized correlation approach has been used to describe a non-Gaussian random walk with independent, identically d…
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…
New GLPs split Lévy bridges into non-overlapping subprocesses.
problem Creating multivariate stochastic processes with specific properties.
method Defining GLPs by splitting Lévy bridges and using time changes.
result GLPs have terminal values and increments with generalised multivariate Liouville distributions.
The paper simulates Lévy processes and their extremum and hitting time.
problem Simulating Lévy processes and their extremum and hitting time accurately and efficiently.
method Using characteristic functions and conditional characteristic functions, with conformal deformations and precalculated values on multi-grids.
result Accurate and fast simulation of Lévy processes and their extremum and hitting time.
Study cliquet options pricing using geometric Meixner model.
problem Pricing cliquet options in a geometric Meixner model.
method Inference of semi-analytic expressions using Meixner distribution and Fourier transform techniques.
result Inferred semi-analytic expressions for cliquet option price.
A new Lévy process kernel model for robust function extrapolation.
problem Kernel uncertainty in Gaussian process predictions for long-range extrapolation.
method Modeling spectral mixture density with a Lévy process to form a distribution over kernels.
result Automatic and data-efficient learning, long-range extrapolation, and state-of-the-art predictive performance.
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.
Lewis and Mordecki have computed the Wiener-Hopf factorization of a Lévy process whose restriction on of their Lévy measure has a rational Laplace transform. That allows to compute the distribution of . For the same class of Lévy processes, we compute the distribution of $ (…
Finite element approximation of the Levi-Civita connection and its curvature in two dimensionsmath.NA
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
Paper calculates the distribution of time spent below zero in risk models.
problem Analyzing time spent below zero in risk models.
method Analytical expressions for the distribution of occupation times.
result Improved understanding of risk processes by providing distribution formulas.
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…
In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally ne…
Heteroskedastic Levy Flightscond-mat.stat-mech
Truncated Lévy flights are random walks in which the arbitrarily large steps of a Lévy flight are eliminated. Since this makes the variance finite, the central limit theorem applies, and as time increases the probability distribution of the increments becomes Gaussian. Here, truncated Lévy flights with correlated fluct…
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.
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…
Bayesian inference for Levy density with Gibbs posterior in discrete sampling.
problem Inference on Levy density for financial models with jumps.
method Gibbs posterior framework using a loss function for intractable likelihood.
result Gibbs posterior achieves nearly optimal rate of convergence under certain conditions.
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…
SINH-acceleration speeds up probability distribution and option pricing calculations.
problem Efficiently calculating probability distributions and option prices.
method Using SINH-acceleration with specific transformations and the simplified trapezoid rule.
result Significantly faster and more accurate evaluation of integrals.
This paper improves conformal prediction for robust interval estimation under distribution shifts.
problem Robustness of conformal prediction under distribution shifts.
method Modeling distribution shifts using Levy-Prokhorov (LP) ambiguity sets, which capture both local and global perturbations.
result Constructs robust conformal prediction intervals that remain valid under distribution shifts.
Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validationcond-mat.stat-mech
Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively …
This a free translation with additional explanations of {\em Processus à Accroissement Independants Chapitre I: La Décomposition de Paul Lévy}, by J.L. Bretagnolle, in {\em Ecole d'Eté de Probabilités}, Lecture Notes in Mathematics 307, Springer 1973. The Lévy-Khintchine representation of infinitely divisible distribut…
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.
Stochastic Multiplicative Processes for Financial Marketscond-mat.stat-mech
We study a stochastic multiplicative system composed of finite asynchronous elements to describe the wealth evolution in financial markets. We find that the wealth fluctuations or returns of this system can be described by a walk with correlated step sizes obeying truncated Levy-like distribution, and the cross-correla…
On fitting the Pareto-Levy distribution to stock market index data: selecting a suitable cutoff valuecond-mat.other
The so-called Pareto-Levy or power-law distribution has been successfully used as a model to describe probabilities associated to extreme variations of worldwide stock markets indexes data and it has the form from empirical d…
An analytic recursive method for optimal multiple stopping: Canadization and phase-type fittingq-fin.MF
We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Levy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Levy process at a constant horizon and hence t…
Order flow in the financial markets from the perspective of the Fractional Lévy stable motionq-fin.ST
The study examines order flow in financial markets using fractional Lévy stable motion.
problem Challenges in selecting the best models for financial time series data.
method Investigates order disbalance time series from the perspective of fractional Lévy stable motion.
result Orders exhibit stable anti-correlation for 18 randomly selected stocks.
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.
Generalized Expected Discounted Penalty Function at General Drawdown for Lévy Risk Processesq-fin.PR
The paper studies drawdown times in Lévy risk processes, generalizing previous results.
problem Analyzing the time of drawdown in spectrally negative Lévy risk processes.
method Using the joint distribution of drawdown times, maximums, and other related quantities.
result Obtained semi-explicit expressions for the joint distribution in terms of scale functions and Lévy measure.
The purpose of this note is to describe, in terms of a power series, the distribution function of the exponential functional, taken at some independent exponential time, of a spectrally negative Lévy process ξwith unbounded variation. We also derive a Geman-Yor type formula for Asian options prices in a financial marke…
New insights into tail behavior of heavy-tailed random vectors and processes.
problem Understanding tail behavior of aggregates of heavy-tailed random vectors.
method Analyzing multivariate regularly varying random vectors and Lévy processes.
result More than one large jump can determine tail behavior of aggregates.
Generative model for Lévy area improves SDE simulation accuracy.
problem Simulating Lévy areas for high-order SDEs is challenging due to non-Gaussian nature and lack of fast sampling algorithms.
method LévyGAN, a deep-learning model with a GNN-inspired architecture, generates approximate samples of Lévy area.
result LévyGAN matches all joint and conditional odd moments exactly and achieves state-of-the-art performance in 4D Brownian motion.
Efficient evaluation of double-barrier options and joint cpdf of a Lévy process and its two extremaq-fin.CP
Fast method developed for pricing barrier options and joint Lévy process distributions.
problem Accurate pricing of barrier options and joint distributions in Lévy models.
method Dual space calculations, Wiener-Hopf factorization, sinh-deformations, Gaver-Wynn Rho acceleration.
result Achieves precision of in seconds and in fractions of a second.
Method extracts stochastic systems with Lévy noise from data.
problem Identifying stochastic dynamical systems with Lévy noise from short data.
method Estimate Lévy jump measure and noise intensity, approximate drift coefficient.
result Accurate and effective method for discovering stochastic laws.
Paper develops methods for estimating and simulating a Student-t Lévy regression model.
problem Estimation and simulation of Student-t Lévy process with arbitrary degrees of freedom.
method Develops a two-step estimation procedure and simulates increments using inverse Fourier transform.
result Efficient estimation and simulation methods for Student-t Lévy process.