Levy Process Research Articles

Conditional Least Squares Estimation for Fractional Super Levy Processes in Nonlinear SPDEs

We consider infinite dimensional extension of affine models as super Levy processes satisfying a nonlinear SPDE. We obtain the asymptotics of the conditional least squares estimators. Finally we obtain the Berry-Esseen inequality.

Analisis Penerapan Akuntansi Tarif Retribusi Daerah pada Pelayanan Kesehatan di Puskesmas Gunting Saga Kabupaten Labuhanbatu Utara

The main problem in this research is that the implementation of accounting for regional levies at the Gunting Saga Community Health Center is still not good. The aim of this research is to determine the application of regional levy rate accounting for health services at the Gunting Saga Community Health Center, North Labuhanbatu Regency. The method used in this research is a qualitative research method. Research techniques through interviews and documentation. The results of this research are that the implementation of the Gunting Saga Community Health Center's accounting can be said to be good, but it still needs to be developed and improved so that it complies with applicable accounting standards. The Gunting Saga Community Health Center has carried out the health service levy process well and collects fees in accordance with the health service rates that have been set in accordance with North Labuhanbatu Regency Regional Regulation (PERDA) Number 6 of 2022.

ПОРОГОВІ МОДЕЛІ ДЛЯ ПРОЦЕСІВ ЛЕВІ ТА ОЦІНКА НАБЛИЖЕНОЇ МАКСИМАЛЬНОЇ ПРАВДОПОДІБНОСТІ

Using the Levy process (the solution to the Ito–Skorokhod stochastic differential equation) we propose the construction of the model of the threshold process and the approximate maximum likelihood method based on approximation of the logarithmic function of the likelihood of observations. The estimates for the parameters of the two-mode threshold jump process with discretely sampled data are found. We show that by checking the likelihood ratio, determining the presence of threshold effects is possible. Keywords: threshold jump process, approximate maximum likelihood method, stochastic differential equation.

A Levy process approach coupled to the stochastic Leslie–Gower model

This paper focuses on a two-dimensional Leslie–Grower continuous-time stochastic predator–prey system with Lévy jumps. Firstly, we prove that there exists a unique positive solution of the system with a positive initial value. Then, we establish sufficient conditions for the mean stability and extinction of the considered system. Numerical algorithms of higher order are elaborated. The obtained results show that Lévy jumps significantly change the properties of population systems.

Decay rates of convergence for Fokker-Planck equations with confining drift

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.

Analysis of fractional differential equation and its application to realistic data

This paper aims to propose a new a Fractional differential equation which derived from the classical Lévy model by introducing new parameters in order to fit the realistic data. The model is based on a combination of generalized tempered stable (GTS) process and Lévy stable process (LSP). A numerical method is chosen to solve the fractional partial differential equation associated to Fractional Lévy Stochastic model. Then, using the accurate information collected from the Yahoo Finance, the inverse approach is applied to estimate the unknown parameters for the Fractional Lévy stochastic model. Finally, numerical results are presented and a conclusion is drawn.

Relation between the degree and betweenness centrality distribution in complex networks.

The centrality measures, like betweenness b and degree k in complex networks remain fundamental quantities helping to classify them. It is realized from Barthelemy's paper [Eur. Phys. J. B 38, 163 (2004)10.1140/epjb/e2004-00111-4] that the maximal b-k exponent for the scale-free (SF) networks is η_{max}=2, belonging to SF trees, based on which one concludes δ≥γ+1/2, where γ and δ are the scaling exponents for the distribution functions of the degree and the betweenness centralities, respectively. This conjecture was violated for some special models and systems. Here we present a systematic study on this problem for visibility graphs of correlated time series, and show evidence that this conjecture fails in some correlation strengths. We consider the visibility graph of three models: two-dimensional Bak-Tang-Weisenfeld (BTW) sandpile model, one-dimensional (1D) fractional Brownian motion (FBM), and 1D Levy walks, the two latter cases are controlled by the Hurst exponent H and the step index α, respectively. In particular, for the BTW model and FBM with H≲0.5, η is greater than 2, and also δ<γ+1/2 for the BTW model, while the Barthelemy's conjecture remains valid for the Levy process. We assert that the failure of the Barthelemy's conjecture is due to large fluctuations in the scaling b-k relation resulting in the violation of hyperscaling relation η=γ-1/δ-1 and emergent anomalous behavior for the BTW model and FBM. Universal distribution function of generalized degree is found for these models which have the same scaling behavior as the Barabasi-Albert network.

On inverse-Gamma distribution delayed by Poisson process

The models of stochastic subordination, or random time indexing, has been recently applied to model financial returns series X(t)t≥0 exhibiting sharp, and large variations. These sharp and large variations are linked to information arrivals and/or represent sudden events, and hence we have a model with jumps. For this purpose, by substituting the usual deterministic time t as a subordinator T(t)t≥0 in a stochastic process X(t)t≥0 we obtain a new process X(T(t))t≥0 whose stochastic time is dominated by the subordinator T(t)t≥0. Therefore, we propose in this paper an alternative approach based on a combination of the continuous-time bilinear (COBL) process subordinated by a Poisson process (that it is a Levy process) which permits us to introduce further randomness for the phenomena which exhibit either a speeded up or slowed down behavior. So, the main probabilistic properties of such models are studied and the explicit expression of the higher-order moments properties are given.

Singular Levy processes and dispersive effects of generalized Schrödinger equations

Singular Levy processes and dispersive effects of generalized Schrödinger equations

Asymptotic shape of the concave majorant of a Lévy process

We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at $T$) of any Levy process on $[0,T]$ as $T\to\infty$. The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the Levy measure. The key tool in the proofs is the recent representation of the concave majorant for all Levy processes using a stick-breaking representation.

New approach for multidimensional prognostic of stochastic time series based on sparse-based fractional Levy quaternion extended Kalman filter

Multidimensional data are frequently encountered in numerous industrial systems, standard techniques regarding statistical multichannel prognostic methods naturally cannot fully exploit and capture the coupled nature and spatiotemporal relationship of the available information among channels. To alleviate a bottleneck problem, in this paper, a new approach named sparse-based fractional Levy quaternion extended Kalman filter (SFLQEKF) is formulated for the prognostic of stochastic time series. Initially, the stochastic time series is decomposed into the static component and the dynamic component resorting to the piecewise representation through a new improved sparse low-rank matrix (SLRM) approximation algorithm namely singular value penalty resonance sparse signal decomposition (SVP-RSSD). Afterward, both static component and dynamic component are predicted individually through the proposed fractional Levy quaternion extended Kalman filter (FLQEKF) method, the final time series of interest can be obtained by integrating the predicted static- and dynamic components correspondingly, in which a new predictive derivation of the proposed FLQEKF model is formulated for the first time by considering fractional-order characteristic and Levy flight process damping term, hence the issue of oscillating indeterminacy induced from classical extended Kalman filter (EFK) can be addressed. Eventually, the prognostic behavior of the proposed approach is investigated via the stochastic time series conducted from two experimental cases, the predicted results illustrate the availability and applicability of the proposed approach compared with three state-of-the-art benchmarks.

A Note on α-stable and α-inverse Gaussian Laws

In this article we obtain the first passage time distribution of α-stable Levy processes. We derive the moment estimators of the parameters of α-inverse Gaussian laws and also their asymptotic distribution.

Numerical Solution of a Matrix Integral Equation Arising in Markov-Modulated Lévy Processes

Markov-modulated Levy processes lead to matrix integral equations of the kind $ A_0 + A_1X+A_2 X^2+A_3(X)=0$ where $A_0$, $A_1$, $A_2$ are given matrix coefficients, while $A_3(X)$ is a nonlinear function, expressed in terms of integrals involving the exponential of the matrix $X$ itself. In this paper we propose some numerical methods for the solution of this class of matrix equations, perform a theoretical convergence analysis and show the effectiveness of the new methods by means of a wide numerical experimentation.

Developing New Techniques for Obtaining the Threshold of a Stochastic SIR Epidemic Model with 3-Dimensional Levy Process

This paper considers the classical SIR epidemic model driven by a multidimensional Levy jump process. We consecrate to develop a mathematical method to obtain the asymptotic properties of the perturbed model. Our method differs from previous approaches by the use of the comparison theorem, mutually exclusive possibilities lemma, and some new techniques of the stochastic differential systems. In this framework, we derive the threshold which can determine the existence of a unique ergodic stationary distribution or the extinction of the epidemic. Numerical simulations about different perturbations are realized to confirm the obtained theoretical results.

Levy geometric graphs.

We present a family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's size but only on the considered scale. It follows a gamma distribution and thus presents an exponential decay. Levy flights are particular random walks with some power-law increments of infinite variance. When building the geometric graphs from them, we show from dimensional arguments that the number of connected components (clusters) follows an inverse power of the scale. The distribution of the size of their components, properly normalized, is scale invariant, which reflects the self-similar nature of the underlying process. This allows to test if a graph (including nonspatial ones) could possibly result from an underlying Levy process. When the scale increases, these graphs never tend towards a single cluster, the giant component. In other words, while the autocorrelation of the process scales as a power of the distance, they never undergo a phase transition of percolation type. The Levy graphs may find applications in community detection and in the analysis of collective behaviors as in face-to-face interaction networks.

Approximations for the distribution of perpetuities with small discount rates

AbstractPerpetuities (i.e., random variables of the form play an important role in many application settings. We develop approximations for the distribution of when the “accumulated short rate process”, , is small. We provide: (1) characterizations for the distribution of when and are driven by Markov processes; (2) general sufficient conditions under which weak convergence results can be derived for , and (3) Edgeworth expansions for the distribution of in the iid case and the case in which is a Levy process and the interest rate is a function of an ergodic Markov process.

Multiple change point clustering of count processes with application to California COVID data

• Analysis of 275-day long time series of county level COVID-19 data in California state. • Multiple change point estimation in the framework of mixture modeling and model-based clustering. • Change point given by gap between logit transformed segments in negative binomial nonhomogeneous Levy process. • Three geographically meaningful clusters, each with several change points indicating the spread and decline of infection. In this paper, a model-based clustering algorithm relying on a finite mixture of negative binomial Lévy processes is proposed. The algorithm models heterogeneous stochastic count process data and automatically estimates multiple change points upon fitting the mixture model. Such change point estimation identifies time points when deviation from the standard process has occurred and serves as an important diagnostic tool for analyzing temporal data. The proposed model is applied to the COVID-positive ICU cases in the state of California with very interesting results.

Quasi-likelihood Estimation in Fractional Levy SPDEs from Poisson Sampling

We study the quasi-likelihood estimator of the drift parameter in the stochastic partial differential equations driven by a cylindrical fractional Levy process when the process is observed at the arrival times of a Poisson process. We use a two stage estimation procedure. We first estimate the intensity of the Poisson process. Then we plug-in this estimate in the quasi-likelihood to estimate the drift parameter. We obtain the strong consistency and the asymptotic normality of the estimators.

Modeling Pulsed Evolution and Time-Independent Variation Improves the Confidence Level of Ancestral and Hidden State Predictions.

Ancestral state reconstruction is not only a fundamental tool for studying trait evolution, but also very useful for predicting the unknown trait values (hidden states) of extant species. A well-known problem in ancestral and hidden state predictions is that the uncertainty associated with predictions can be so large that predictions themselves are of little use. Therefore, for meaningful interpretation of predicted traits and hypothesis testing, it is prudent to accurately assess the uncertainty of the predictions. Commonly used constant-rate Brownian motion (BM) model fails to capture the complexity of tempo and mode of trait evolution in nature, making predictions under the BM model vulnerable to lack-of-fit errors from model misspecification. Using empirical data (mammalian body size and bacterial genome size), we show that the distribution of residual Z-scores under the BM model is neither homoscedastic nor normal as expected. Consequently, the 95% confidence intervals of predicted traits are so unreliable that the actual coverage probability ranges from 33% (strongly permissive) to 100% (strongly conservative). Alternative methods such as BayesTraits and StableTraits that allow variable rates in evolution improve the predictions but are computationally expensive. Here, we develop Reconstructing Ancestral State under Pulsed Evolution in R by Gaussian Decomposition (RasperGade), a method of ancestral and hidden state prediction that uses the Levy process to explicitly model gradual evolution, pulsed evolution, and time-independent variation. Using the same empirical data, we show that RasperGade outperforms both BayesTraits and StableTraits in providing reliable confidence estimates and is orders-of-magnitude faster. Our results suggest that, when predicting the ancestral and hidden states of continuous traits, the rate variation should always be assessed and the quality of confidence estimates should always be examined. [Bacterial genomic traits; model misspecification; trait evolution.].

Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-informed Deep Generative Models

We propose a new method for inferring the governing stochastic ordinary differential equations (SODEs) by observing particle ensembles at discrete and sparse time instants, i.e., multiple "snapshots". Particle coordinates at a single time instant, possibly noisy or truncated, are recorded in each snapshot but are unpaired across the snapshots. By training a physics-informed generative model that generates "fake" sample paths, we aim to fit the observed particle ensemble distributions with a curve in the probability measure space, which is induced from the inferred particle dynamics. We employ different metrics to quantify the differences between distributions, e.g., the sliced Wasserstein distances and the adversarial losses in generative adversarial networks (GANs). We refer to this method as generative "ensemble-regression" (GER), in analogy to the classic "point-regression", where we infer the dynamics by performing regression in the Euclidean space. We illustrate the GER by learning the drift and diffusion terms of particle ensembles governed by SODEs with Brownian motions and Levy processes up to 100 dimensions. We also discuss how to treat cases with noisy or truncated observations. Apart from systems consisting of independent particles, we also tackle nonlocal interacting particle systems with unknown interaction potential parameters by constructing a physics-informed loss function. Finally, we investigate scenarios of paired observations and discuss how to reduce the dimensionality in such cases by proving a convergence theorem that provides theoretical support.