Class Of Stochastic Models Research Articles

Intertwining and Propagation of Mixtures for Generalized KMP Models and Harmonic Models

We study a class of stochastic models of mass transport on discrete vertex set V. For these models, a one-parameter family of homogeneous product measures ⊗i∈Vνθ is reversible. We prove that the set of mixtures of inhomogeneous product measures with equilibrium marginals, i.e., the set of measures of the form ∫(⨂i∈Vνθi)Ξ(∏i∈Vdθi)is left invariant by the dynamics in the course of time, and the “mixing measure” Ξ evolves according to a Markov process which we then call “the hidden parameter model”. This generalizes results from De Masi et al. (Preprint arXiv:2310.01672, 2023) to a larger class of models and on more general graphs. The class of models includes discrete and continuous generalized KMP models, as well as discrete and continuous harmonic models. The results imply that in all these models, the non-equilibrium steady state of their reservoir driven version is a mixture of product measures where the mixing measure is in turn the stationary state of the corresponding “hidden parameter model”. For the boundary-driven harmonic models on the chain {1,…,N} with nearest neighbor edges, we recover that the stationary measure of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution (cf. Carinci et al., Preprint arXiv:2307.14975, 2023), with a purely probabilistic proof based on a spatial Markov property of the hidden parameter model.

Skewness and option prices under stochastic volatility models: the role of shot-noise jumps

In this paper, we propose a novel option pricing model that incorporates both shot-noise effects in stock return jumps and stochastic volatility. The model aims to reflect the decaying effects on stock returns after the shock and attempts to provide a more reasonable structure for return jumps. We emphasize the theoretical nature of this paper and manage to derive closed-form expressions for the VIX index and the SKEW index under the new model. For the VIX index, we prove that it can be decomposed into a linear combination of the volatility part and the jump part. For the SKEW index, we obtain its analytical representation beyond traditional affine models and check its sensitivity with respect to model parameters. We also perform a calibration exercise using options data to compare our model with a classical stochastic volatility model with jumps. Several empirical observations are made.

The Causal Relation within Air–Sea Interaction as Inferred from Observations

Abstract The intricate cause–effect relations in air–sea interaction are investigated using a quantitative causal inference formalism. The formalism is first validated with a classical stochastic coupled model and then applied to the observational time series of sea surface temperature (SST) and air–sea turbulent surface heat flux (SHF). We identify an overall asymmetry of causality between the two variables, namely, the causality from SHF to SST is significantly larger than that from SST to SHF over most of the global oceans. Geographically, the coupling is strongest in the tropics and gets weaker substantially in the extratropics. In the midlatitude ocean, SST makes higher contributions to the SHF variability in frontal regions. We further show that the identified causality is space and time scale dependent. The dominance of SHF driving SST occurs at basin scales, whereas the dominance of SST driving SHF mostly occurs at scales smaller than 10°. The causalities in both directions get larger with increasing time scale and are less asymmetric at longer time scales. The seasonality of the causality is also discussed here.

Gaussian approximation and moderate deviations of Poisson shot noises with application to compound generalized Hawkes processes

Abstract In this article, we give explicit bounds on the Wasserstein and Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard normal distribution, using the results of Last et al. (Prob. Theory Relat. Fields165, 2016). Relying on the theory developed by Saulis and Statulevicius in Limit Theorems for Large Deviations (Kluwer, 1991) and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results of Hillairet et al. (ALEA19, 2022) and Khabou et al. (J. Theoret. Prob.37, 2024) regarding the normal approximation and those of Zhu (Statist. Prob. Lett.83, 2013) for moderate deviations.

A Study of Mixed Non-Motorized Traffic Flow Characteristics and Capacity Based on Multi-Source Video Data.

Mixed non-motorized traffic is largely unaffected by motor vehicle congestion, offering high accessibility and convenience, and thus serving as a primary mode of "last-mile" transportation in urban areas. To advance stochastic capacity estimation methods and provide reliable assessments of non-motorized roadway capacity, this study proposes a stochastic capacity estimation model based on power spectral analysis. The model treats discrete traffic flow data as a time-series signal and employs a stochastic signal parameter model to fit stochastic traffic flow patterns. Initially, UAVs and video cameras are used to capture videos of mixed non-motorized traffic flow. The video data were processed with an image detection algorithm based on the YOLO convolutional neural network and a video tracking algorithm using the DeepSORT multi-target tracking model, extracting data on traffic flow, density, speed, and rider characteristics. Then, the autocorrelation and partial autocorrelation functions of the signal are employed to distinguish among four classical stochastic signal parameter models. The model parameters are optimized by minimizing the AIC information criterion to identify the model with optimal fit. The fitted parametric models are analyzed by transforming them from the time domain to the frequency domain, and the power spectrum estimation model is then calculated. The experimental results show that the stochastic capacity model yields a pure EV capacity of 2060-3297 bikes/(h·m) and a pure bicycle capacity of 1538-2460 bikes/(h·m). The density-flow model calculates a pure EV capacity of 2349-2897 bikes/(h·m) and a pure bicycle capacity of 1753-2173 bikes/(h·m). The minimal difference between these estimates validates the effectiveness of the proposed model. These findings hold practical significance in addressing urban road congestion.

Relaxation and fluctuations of a mass- and dipole-conserving stochastic lattice gas

Han [] have recently introduced a classical stochastic lattice gas model which, in addition to particle conservation, also conserves the particles' dipole moment. Because of its intrinsic nonlinearity this model exhibits unusual macroscopic scaling behaviors, different from those of lattice gases that conserve only the number of particles. Here we investigate some basic relaxation and fluctuation properties of this model at large scales and at long times. These properties crucially depend on whether the total number of particles is infinite or finite. We find similarity solutions, describing relaxation of the dipole-conserving gas (DCG) in several standard settings. A major part of our effort is an extension to this model of the macroscopic fluctuation theory (MFT), previously developed for lattice gases where only the number of particles is conserved. We apply the MFT to the calculation of the variance of nonequilibrium fluctuations of the excess number of particles on the positive semi-axis when starting from an (either deterministic, or random) constant density at t=0. Using the MFT, we also identify the equilibrium Boltzmann-Gibbs distribution for the DCG. Finally, based on these results, we determine the probability distribution of, and the most probable density history leading to, a large deviation in the form of a macroscopic void of a given size in an initially uniform DCG at equilibrium. Published by the American Physical Society 2024

Two-dimensional quad-stable Gaussian potential stochastic resonance model for enhanced bearing fault diagnosis

In this study, a two-dimensional quad-stable Gaussian potential stochastic resonance model is explored for the first time. First, the structure of the proposed model is analyzed to have a broader potential field and verified to break through the severe output saturation inherent in the classical two-dimensional quad-stable stochastic resonance model. Then, we analyze the relationship between the structure and parameters of the model and derive the steady-state probability density and the mean first-passage time using adiabatic approximation theory to describe the specific process of the Brownian particle transitions. By combining the adiabatic approximation theory and the probability flow equation, the spectral amplification factor of the model is derived, and the effects of different parameters on the model performance are investigated. Further, a fourth-order Runge-Kutta algorithm was applied to evaluate the model performance in multiple dimensions. Finally, the model parameters were optimized using an adaptive genetic algorithm and applied to complex practical engineering detection. The experimental results show that the proposed model is superior and universal in fault diagnosis. Overall, this study provides important mathematical support for solving various engineering problems and demonstrates a wide range of practical applications.

Valuation of Currency Option Based on Uncertain Fractional Differential Equation

Uncertain fractional differential equations (UFDEs) are excellent tools for describing complicated dynamic systems. This study analyzes the valuation problems of currency options based on UFDE under the optimistic value criterion. Firstly, a new uncertain fractional currency model is formulated to describe the dynamics of the foreign exchange rate. Then, the pricing formulae of European, American, and Asian currency options are obtained under the optimistic value criterion. Numerical simulations are performed to discuss the properties of the option prices with respect to some parameters. Finally, a real-world example is provided to show that the uncertain fractional currency model is superior to the classical stochastic model.

Computation of random time-shift distributions for stochastic population models

Even in large systems, the effect of noise arising from when populations are initially small can persist to be measurable on the macroscale. A deterministic approximation to a stochastic model will fail to capture this effect, but it can be accurately approximated by including an additional random time-shift to the initial conditions. We present a efficient numerical method to compute this time-shift distribution for a large class of stochastic models. The method relies on differentiation of certain functional equations, which we show can be effectively automated by deriving rules for different types of model rates that arise commonly when mass-action mixing is assumed. Explicit computation of the time-shift distribution can be used to build a practical tool for the efficient generation of macroscopic trajectories of stochastic population models, without the need for costly stochastic simulations. Full code is provided to implement the calculations and we demonstrate the method on an epidemic model and a model of within-host viral dynamics.

Rare events in extreme value statistics of jump processes with power tails.

We study rare events in the extreme value statistics of stochastic symmetric jump processes with power tails in the distributions of the jumps, using the big -jump principle. The principle states that in the presence of stochastic processes with power tails statistics, if at a certain time a physical quantity takes on a value much larger than its typical value, this large fluctuation is realized through a single macroscopic jump that exceeds the typical scale of the process by several orders of magnitude. In particular, our estimation focuses on the asymptotic behavior of the tail of the probability distribution of maxima, a fundamental quantity in a wide class of stochastic models used in chemistry to estimate reaction thresholds, in climatology for earthquake risk assessment, in finance for portfolio management, and in ecology for the collective behavior of species. We determine the analytical form of the probability distribution of rare events in the extreme value statistics of three jump processes with power tails: Lévy flights, Lévy walks, and the Lévy-Lorentz gas. For the Lévy flights, we re-obtain through the big-jump approach recent analytical results, extending their validity. For the Lévy-Lorentz gas, we show that the topology of the disordered lattice along which the walker moves induces memory effects in its dynamics, which influences the extreme value statistics. Our results are confirmed by extensive numerical simulations.

FORECASTING THE VALUE OF IRON ORE RAW MATERIALS ON THE BASIS OF STATISTICAL TIME SERIES ANALYSIS

Objective. The objective of the present article is the development of evaluation of the effectiveness of forecasting the time series of cost indicators as stochastic using known methods. Methods. The following methods and techniques of cognition were used in the research process: theoretical generalization and comparison, analysis and synthesis, induction and deduction, generalization and systematization, statistical methods of time series analysis. Results. The article presents the stages of forming a statistical analysis of a time series. It has been established that in modern conditions of evaluating the effectiveness of economic research, a more thorough analysis of the dependence of value indicators on time is necessary. Their mathematical models are usually used to describe the behavior of physical objects. If a model based on physical laws can be obtained, such a model would be deterministic. At the same time, in practice, even such a model is not completely deterministic, since a number of unaccounted factors may participate in it. For such objects, it is not possible to offer a deterministic model that allows accurate calculation of the future behavior of the object. Nevertheless, it is possible to propose a model that allows you to calculate the probability that some future value will lie in a certain interval. Such a model is called stochastic. Time series models of commodity prices in the time domain are actually stochastic. An important class of stochastic models for describing time series are stationary models. They are based on the assumption that the process remains in equilibrium with respect to a constant average level, which is confirmed by studying the time series of the cost of goods in the time domain.Mathematical models of stochastic time series were built based on the study of the real dependence of indicators on time. In practical terms, this will improve the economic performance of the enterprise. For practical implementation, a stochastic time series of the cost indicator was constructed; an economic-mathematical model for the value indicator based on a time series was formed for the purpose of forecasting. The quality of the forecast is determined not only by the forecast error, but also by the number of parameters included in the model of the forecasting function. Analysis of the data shows that the smallest forecast error occurs for the analytical trend function. Along with this, the trend function has six parameters. If we take into account the number of parameters, then the best method will be the moving average, which has an error variance of 54 with one parameter.

Distributed Pseudo-Likelihood Method for Community Detection in Large-Scale Networks

This paper proposes a distributed pseudo-likelihood method (DPL) to conveniently identify the community structure of large-scale networks. Specifically, we first propose a block-wise splitting method to divide large-scale network data into several subnetworks and distribute them among multiple workers. For simplicity, we assume the classical stochastic block model. Then, the DPL algorithm is iteratively implemented for the distributed optimization of the sum of the local pseudo-likelihood functions. At each iteration, the worker updates its local community labels and communicates with the master. The master then broadcasts the combined estimator to each worker for the new iterative steps. Based on the distributed system, DPL significantly reduces the computational complexity of the traditional pseudo-likelihood method using a single machine. Furthermore, to ensure statistical accuracy, we theoretically discuss the requirements of the worker sample size. Moreover, we extend the DPL method to estimate degree-corrected stochastic block models. The superior performance of the proposed distributed algorithm is demonstrated through extensive numerical studies and real data analysis.

A novel term-structure-based Heston model for implied volatility surface

It remains a challenge for existing financial models to accurately capture the shapes of implied volatility (IV) of options with all maturities simultaneously. Inspired by Chen et al.’s empirical finding, ‘the shorter the option expiry, the higher the IV is’, we propose a novel and simple approach to solve this challenge, by introducing a term-structure-based correction (i.e. an exponential increase function of the options expiry) to the volatility of volatility (vol–vol) term of the classical Heston stochastic volatility model. We derive an approximate formula for the IV under the corrected model with the perturbation method and further apply the formula to predict the IVs of options written on the Shanghai Stock Exchange 50 ETF. Numerical experiments and empirical results show that the introduction of a term-structure-based correction function surely overcomes the deficiency of the classical Heston model in capturing the short-term IVs, thus improving notably its performance of IV forecasting.

What would be the impact on the rabies risk of reducing the waiting period before dogs are imported? A modelling study based on the European Union legislation.

Lyssavirus rabies (RABV) is responsible for a major zoonotic infection that is almost always lethal once clinical signs appear. Rabies can be (re)introduced into rabies-free areas through transboundary dog movements, thus compromising animal and human health. A number of measures have been implemented to prevent this happening, one of which is the waiting period (WP) after anti-rabies vaccination and serological testing. This WP ensures that antibodies assessed through the serological test are due to the vaccine, not to infection. Indeed, if antibodies are due to RABV infection, the dog should display clinical signs within this WP and would not therefore be imported. Within a framework of quantitative risk assessment, we used modelling approaches to evaluate the impact of this WP and its duration on the risk of introducing rabies via the importation of dogs into the European Union. Two types of models were used, a classical stochastic scenario tree model and an individual-based model, both parameterised using scientific literature or data specifically applicable to the EU. Results showed that, assuming perfect compliance, the current 3-month waiting period was associated with a median annual number of 0.04 infected dogs imported into the EU. When the WP was reduced, the risk increased. For example, for a 1-month WP, the median annual number of infected dogs imported was 0.17 or 0.15 depending on the model, which corresponds to a four-fold increase. This in silico study, particularly suitable for evaluating rare events such as rabies infections in rabies-free areas, provided results that can directly inform policymakers in order to adapt regulations linked to rabies and animal movements.

Bearing fault feature extraction based on MOMEDA and CS-Wood–Saxon stochastic resonance

Since rolling bearing is of great significance to ensure the safe and stable operation of rotating machinery, bearing fault feature extraction then demonstrates a hot topic of general interest in industry. In this work, we applied Multipoint Optimal Minimum Entroy Deconvolution Adjusted preprocessing algorithm to deal with the large amount of background noise containing in the collected bearing fault original signal. Then, the Wood–Saxon stochastic resonance nonlinear system model is adopted to solve the bearing fault feature extraction problem, which avoids the frequency interval and system parameters disadvantages in bistable stochastic resonance system. Furthermore, the parameter step and scale transform factor in the Wood–Saxon stochastic resonance nonlinear system is optimized adaptively by Cuckoo Search algorithm, in which way the output signal-to-noise of bearing fault signal is improved significantly. Therefore, the bearing fault feature can be extracted more effectively compared with the classical bistable stochastic resonance system model. Simulation and examples demonstrated that the proposed method can effectively reduce the noise in the signal and enhance the weak feature in bearing fault signal, so as to realize the accurate early bearing fault diagnosis.

Markov-switching threshold stochastic volatility models with regime changes

<abstract><p>This paper introduces a comprehensive class of models known as Markov-Switching Threshold Stochastic Volatility (MS-TSV) models, specifically designed to address asymmetry and the leverage effect observed in the volatility of financial time series. Extending the classical threshold stochastic volatility model, our approach expresses the parameters governing log-volatility as a function of a homogeneous Markov chain with a finite state space. The primary goal of our proposed model is to capture the dynamic behavior of volatility driven by a Markov chain, enabling the accommodation of both gradual shifts due to economic forces and sudden changes caused by abnormal events. Following the model's definition, we derive several probabilistic properties of the MS-TSV models, including strict (or second-order) stationarity, causality, ergodicity, and the computation of higher-order moments. Additionally, we provide the expression for the covariance function of the squared (or powered) process. Furthermore, we establish the limit theory for the Quasi-Maximum Likelihood Estimator (QMLE) and demonstrate the strong consistency of this estimator. Finally, a simulation study is presented to assess the performance of the proposed estimation method.</p></abstract>

A machine learning approach to deal with ambiguity in the humanitarian decision‐making

One of the major challenges for humanitarian organizations in response planning is dealing with the inherent ambiguity and uncertainty in disaster situations. The available information that comes from different sources in postdisaster settings may involve missing elements and inconsistencies, which can hamper effective humanitarian decision‐making. In this paper, we propose a new methodological framework based on graph clustering and stochastic optimization to support humanitarian decision‐makers in analyzing the implications of divergent estimates from multiple data sources on final decisions and efficiently integrating these estimates into decision‐making. To the best of our knowledge, the integration of ambiguous information into decision‐making by combining a cluster machine learning method with stochastic optimization has not been done before. We illustrate the proposed approach on a realistic case study that focuses on locating shelters to serve internally displaced people (IDP) in a conflict setting, specifically, the Syrian civil war. We use the needs assessment data from two different reliable sources to estimate the shelter needs in Idleb, a district of Syria. The analysis of data provided by two assessment sources has indicated a high degree of ambiguity due to inconsistent estimates. We apply the proposed methodology to integrate divergent estimates in making shelter location decisions. The results highlight that our methodology leads to higher satisfaction of demand for shelters than other approaches such as a classical stochastic programming model. Moreover, we show that our solution integrates information coming from both sources more efficiently thereby hedging against the ambiguity more effectively. With the newly proposed methodology, the decision‐maker is able to analyze the degree of ambiguity in the data and the degree of consensus between different data sources to ultimately make better decisions for delivering humanitarian aid.

Randomness accelerates the dynamic clearing process of the COVID-19 outbreaks in China

During the implementation of strong non-pharmaceutical interventions (NPIs), more than one hundred COVID-19 outbreaks induced by different strains in China were dynamically cleared in about 40 days, which presented the characteristics of small scale clustered outbreaks with low peak levels. To address how did randomness affect the dynamic clearing process, we derived an iterative stochastic difference equation for the number of newly reported cases based on the classical stochastic SIR model and calculate the stochastic control reproduction number (SCRN). Further, by employing the Bayesian technique, the change points of SCRNs have been estimated, which is an important prerequisite for determining the lengths of the exponential growth and decline phases. To reveal the influence of randomness on the dynamic zeroing process, we calculated the explicit expression of the mean first passage time (MFPT) during the decreasing phase using the relevant theory of first passage time (FPT), and the main results indicate that random noise can accelerate the dynamic zeroing process. This demonstrates that powerful NPI measures can rapidly reduce the number of infected people during the exponential decline phase, and enhanced randomness is conducive to dynamic zeroing, i.e. the greater the random noise, the shorter the average clearing time is. To confirm this, we chose 26 COVID-19 outbreaks in various provinces in China and fitted the data by estimating the parameters and change points. We then calculated the MFPTs, which were consistent with the actual duration of dynamic zeroing interventions.

A Bayesian multivariate spatial approach for illness-death survival models.

Illness-death models are a class of stochastic models inside the multi-state framework. In those models, individuals are allowed to move over time between different states related to illness and death. They are of special interest when working with non-terminal diseases, as they not only consider the competing risk of death but also allow us to study the progression from illness to death. The intensity of each transition can be modelled including both fixed and random effects of covariates. In particular, spatially structured random effects or their multivariate versions can be used to assess spatial differences between regions and among transitions. We propose a Bayesian methodological framework based on an illness-death model with a multivariate Leroux prior for the random effects. We apply this model to a cohort study regarding progression after an osteoporotic hip fracture in elderly patients. From this spatial illness-death model, we assess the geographical variation in risks, cumulative incidences and transition probabilities related to recurrent hip fracture and death. Bayesian inference is done via the integrated nested Laplace approximation.

Robust Statistical Approach for Determination of Graphite Nitridation Using Bayesian Model Comparison

A better estimation of surface reaction efficiency of semiconductor-grade graphite with atomic nitrogen, as well as the calibration error are calculated using Bayesian updating based on experimental data. Compared with a conventional deterministic model, the stochastic model approach is a powerful tool in the sense that the model is capable of taking into account underlying error correlations among the data quantities. In this paper, we investigate four different stochastic models (called “stochastic system model classes” herein) corresponding to different descriptions of modeling and measurement error structures, given one deterministic physical model. These stochastic system model classes differ in the covariance matrix structure that is used in the uncertainty model to represent uncertainties associated with the physical model and experimental measurements. For each model class, Bayesian inference is used to estimate the posterior probabilities of the physical model parameters as well as of the stochastic model parameters. Model comparison and selection are then applied based on two measures including Bayesian evidence and Bayesian information criterion, as well as the deviance information criterion. Both measures suggest the stochastic model class, which considers that a correlation between errors in two data quantities among different data points is the most plausible. With the stochastic model class, the range of uncertainty in surface reaction efficiency is estimated to be about two orders of magnitude at .