General Stochastic Model Research Articles

Computational analysis of the behavior of stochastic volatility models with financial applications

Volatility estimation is an important issue in certain aspects of the financial community, such as risk management and asset pricing. It is known that stock returns often exhibit volatility clustering and the tails of the distributions of these series are fatter than the normal distribution. As a response to the need of these issues, the high unconditional volatility of assets encourages the users to predict their price in an ever changing market environment. Our main focus in this paper is to study the behavior of returns and volatility dynamics of some general stochastic economic models. First, we apply the local polynomial kernel smoothing method based on nonparametric regression to estimate the mean and the variance of the returns. We then implement and develop an empirical likelihood procedure in terms of conditional variance on daily log returns for inference on the nonparametric stochastic volatility as well as to construct a confidence interval for the volatility function. It appears that the proposed algorithm is applicable to some popular financial models and represents a good fit for the behavior observed in the stock and cryptocurrency markets. Some numerical results in connection to real data on the S&P 500 index and highly volatile Bitcoin dataset are also illustrated.

Minimax and pointwise sequential changepoint detection and identification for general stochastic models

This paper considers the problem of joint change detection and identification assuming multiple composite post-change hypotheses. We propose a multihypothesis changepoint detection-identification procedure that controls the probabilities of false alarm and wrong identification. We show that the proposed procedure is asymptotically minimax and pointwise optimal, minimizing moments of the detection delay as probabilities of false alarm and wrong identification approach zero. The asymptotic optimality properties hold for general stochastic models with dependent and nonidentically distributed observations. We illustrate general results for detection-identification of changes in multistream Markov ergodic processes. We consider several examples, including an application to rapid detection-identification of COVID-19 in Italy. Our proposed sequential algorithm allows much faster detection of COVID-19 than standard methods.

Стохастическое моделирование каскадных сценариев возникновения и развития чрезвычайных ситуаций

Introduction. The development of methods for modeling the flows of events in the occurrence and development of emergency and critical states, designed to solve the problems of supporting the management of rapid response to incidents in conditions of incomplete information, as well as the generation of preventive algorithms for rapid response in emergency situations, seems to be a promising direction for improving information and analytical support of management in security systems. The purpose of the research is to create mathematical tools for predicting emergency and critical conditions in local incidents based on discrete-event modeling of the dynamics of emergency situations. Methods of stochastic modeling are based on the representation of a cascade scenario of the dynamics of the development of accidents and catastrophes by a Markov process using a three-parameter Weibull distribution to set the intensities of transitions between emergency and critical states of adjacent levels. Results and discussion. A general stochastic model is constructed that describes the dynamics of the occurrence and development of accidents and catastrophes of technogenic and natural origin initiated by local incidents according to a cascade scenario with a branching structure with a set of possible elementary states represented in the form of a stratified graph. The main stages of the emergence and development of emergency situations are considered within the framework of the proposed representation of the cascade model by a Markov process with discrete states and continuous time. The expediency of local modeling of individual stages is substantiated based on the description of transitions between states of adjacent levels. Local systems of differential equations corresponding to some of the most probable scenarios for the development of events in the event of incidents are given. Non-stationary solutions of differential equations of Kolmogorov-Chapman are obtained, with assignment of intensities of transitions by three-parameter Weibull distributions. Conclusions. The proposed stochastic model makes it possible to predict the dynamics of the development of emergency and critical states according to a cascade scenario, and also reflects the probability characteristics of the timeliness and effectiveness of possible impacts in a prompt response to incidents. Keywords: stochastic modeling of technogenic accidents, cascade development of accidents and catastrophes, Weibull distribution.

Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.

Reliability-based energy scheduling of active buildings subject to renewable energy and demand uncertainty

The increasing penetration of renewable energy sources (RESs) and the inherent volatility in demand profiles have added another layer of complexity to the management of energy resources in modern active buildings (ABs). Yet, three challenges have been neglected in previous studies: (1) There is no universal systematic method for identifying an AB’s general stochastic model; (2) No research has been conducted on the reliability-based design optimization for ABs’ energy supply; (3) Uncertain sources are not categorized based on their importance in regard to the optimization problem. This article aims to solve these challenges by proposing a probabilistic-based optimization approach for solving the reliability issue of energy supply in buildings with on-site renewable energy sources (RESs), taking into account the uncertainty associated with photovoltaic (PV) production and demand fluctuations. The suggested framework seeks to reduce the overall costs of the system while ensuring high energy supply reliability. The proposed methodology, when applied to a real-world case study, demonstrates a 60% increase in reliability of energy supply as compared to typical deterministic methodologies.

Service-oriented distributionally robust lane reservation

Most existing lane reservation studies usually consider a static transportation network with assuming constant link travel times. However, in reality the link travel times are highly uncertain due to various factors such as weather, accidents, road maintenance, intersections, etc. Moreover, the precise link travel time probability distribution is usually difficult to be obtained. This paper studies a new stochastic bus lane reservation problem with partial link travel time information, i.e., only the mean and covariance matrix are known. The objective is to maximize the bus service level measured by the probability of the event that all lines are jointly scheduled on time. For the problem, we formulate a service-oriented distributionally robust optimization model. Its complexity is shown to be NP-hard. To solve the problem, a sample average approximation (SAA)-based method is first adapted. Since the SAA-based approach is computational expensive, a new approximated mixed integer second-order cone programming (MI-SOCP)-based approach is developed. Computational results on a real-life case show that the proposed MI-SOCP-based approach can efficiently obtain satisfactory solutions of high quality. Besides, our results indicate that the proposed model and algorithm can provide better solutions with higher service level, as compared with general stochastic models with known distributions and without considering service levels.

An integrated degradation modeling framework considering model uncertainty and calibration

General path models and stochastic process models are two widely applied categories of probabilistic degradation models. The former explains the randomness of degradation data as normal distributed errors with zero mean. The latter describes degradation with stochastic processes such as Wiener process, Gamma process and Inverse Gaussian process. For general path models, a limitation is the assumption of normally distributed errors. For stochastic process models, model uncertainty with respect to the available stochastic processes should be considered, but the widely-applied model selection methods in consideration of model uncertainty are unable to warn when all the candidate models fit data poorly. Therefore, an integrated degradation modeling framework based on wavelet density estimation is proposed, which can calibrate the distribution of errors for general path models and deal with model uncertainty for stochastic process models. The proposed framework can select the best stochastic process if certain stochastic processes fit the degradation data well. Otherwise, all the candidate stochastic processes can be calibrated, which overcomes the drawback of model selection methods. The effectiveness and feasibility of the proposed framework are illustrated through a case study and a numerical example.

Endogenous stochastic arbitrage bubbles and the Black–Scholes model

This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in the Black–Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizes the deterministic arbitrage model obtained in the literature (Contreras et al., 2010). It is considered to be a generic stochastic dynamic for the arbitrage bubble, and a generalized Black–Scholes equation is then derived. The resulting equation is similar to that of the stochastic volatility models, but there are no undetermined parameters as the market price of risk.The proposed theory has asymptotic behaviors that are associated with the weak and strong arbitrage bubble limits. For the case where the arbitrage bubble’s volatility is zero (deterministic bubble), the weak limit corresponds to the usual Black–Scholes model. The strong limit case also give a Black–Scholes model, but the underlying asset’s mean value replaces the interest rate. When the bubble is stochastic, the theory also has weak and strong asymptotic limits that give rise to option price dynamics that are similar to the Black–Scholes model. Explicit formulas are derived for Gaussian and lognormal stochastic bubbles. Consequently, the Black–Scholes model can be considered to be a weak bubble limit of a more general stochastic model.

Fixed-dimension renormalization group analysis of conserved surface roughening.

Conserved surface roughening represents a special case of interface dynamics where the total height of the interface is conserved. Recently, it was suggested [Caballero etal., Phys. Rev. Lett. 121, 020601 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.020601] that the original continuum model known as the Conserved Kardar-Parisi-Zhang (CKPZ) equation is incomplete, as additional nonlinearity is not forbidden by any symmetry in d>1. In this work, we perform detailed field-theoretic renormalization group (RG) analysis of a general stochastic model describing conserved surface roughening. Systematic power counting reveals additional marginal interaction at the upper critical dimension, which appears also in the context of molecular beam epitaxy. Depending on the origin of the surface particle's mobility, the resulting model shows two different scaling regimes. If the particles move mainly due to the gravity, the leading dispersion law is ω∼k^{2}, and the mean-field approximation describing a flat interface is exact in any spatial dimension. On the other hand, if the particles move mainly due to the surface curvature, the interface becomes rough with the mean-field dispersion law ω∼k^{4}, and the corrections to scaling exponents must be taken into account. We show that the latter model consist of two subclasses of models that are decoupled in all orders of perturbation theory. Moreover, our RG analysis of the general model reveals that the universal scaling is described by a rougher interface than the CKPZ universality class. The universal exponents are derived within the one-loop approximation in both fixed d and ɛ-expansion schemes, and their relation is discussed. We point out all important details behind these two schemes, which are often overlooked in the literature, and their misinterpretation might lead to inconsistent results.

A General Wideband Non-Stationary Stochastic Channel Model for Intelligent Reflecting Surface-Assisted MIMO Communications

Intelligent reflecting surface (IRS), which is composed of a large number of low-cost passive elements, has the ability to reflect the incident signal independently with an adjustable phase and amplitude shifts, has been regarded as a key and emerging technology for achieving the cost-effectively spectrum and addressing energy issues in the next generation of wireless networks. In this paper, we propose a general wideband non-stationary channel model for IRS-assisted multiple-input multiple-output (MIMO) communication scenarios, which aims at capturing the underlying propagation characteristics of IRS-assisted communication systems. By properly adjusting the key system parameters, the proposed channel model can be used to describe various IRS-assisted communication scenarios. Furthermore, we separate the channel between the mobile transmitter (MT) and mobile receiver (MR) into the subchannel between the MT and IRS, the subchannel between the IRS and MR, and the subchannel between the MT and MR. The physical properties of each subchannel are investigated accordingly; then, we develop an equivalent channel model to study the key characteristics of the proposed IRS-assisted channel model, such as the time-varying spatial-temporal (ST) cross-correlation functions (CCFs), temporal auto-correlation functions (ACFs), and frequency CCFs. Finally, numerical results demonstrate that the proposed channel model is practical for describing the IRS-assisted MIMO wireless communication scenarios.

Stochastic reaction and diffusion systems in molecular communications: Recent results and open problems

Chemical reactions and diffusion are two basic mechanisms governing the dynamics of molecules in a fluid. As such, they play a critical role in molecular communication for channel modeling, design of detection rules, implementation of molecular circuits for computation, and modeling interactions with external biochemical systems. For finite numbers of information-carrying molecules, stochastic models naturally arise with the simplest example given by the Wiener process, often known as Brownian motion. Nevertheless, the Wiener process fails to be accurate when external forces, friction, and chemical reactions are present. Recently, there have been several contributions that tailor molecular communication systems to these more challenging channel conditions. In this paper, we first overview a general family of stochastic models of reaction and diffusion systems, including both Langevin diffusion and the reaction-diffusion master equation. These models form a basis for the use of these models as molecular communication channels, from which modulation and detection schemes can be developed. We survey recent results on the design of these schemes, with a focus on a recently developed approach which is robust to a wide range of channel models, known as equilibrium signaling. We then turn to the implementation of these detection schemes and related parameter estimation problems via stochastic molecular circuits, based on stochastic chemical reaction networks. Finally, interactions between molecular communication systems and stochastic biological systems as well as open problems are discussed. Our overarching goal is to highlight how the consideration of general stochastic models of reaction and diffusion can be utilized in order to widen the application of molecular communications both within engineered systems, and also as motivation for advances in the mathematical characterization of these models.

A product form for the general stochastic matching model

AbstractWe consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Bušić, Mairesse, and Weiss (2018) for the bipartite matching model.

CTMC integral equation method for American options under stochastic local volatility models

In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the 4/2 model and the α−hypergeometric models. They illustrate the high accuracy and efficiency of the method.

A Kernel-Density based Semi-Parametric stochastic degradation model with dependent increments

The degradation modeling of highly reliable industrial products is a significant issue for manufacturers, and stochastic process models have been widely applied to model degradation trends. However, they suffer from the two underlying assumptions: the degradation increments following a specific parametric distribution and mutually independent degradation increments. Hence some degradation trends cannot be well captured by these models. In this paper, we propose a general semi-parametric stochastic degradation model to fit the degradation data. The probability density function of the degradation increments is estimated by the adaptive kernel density estimation method, and the copula function is used to measure the dependence of the successive degradation increments. Increments are extrapolated by marginal conditional distributions. A simulation study is carried out where the degradation increments are generated under five distributions, and four degradation models are used to fit the data. The simulation results show that the proposed model can well fit the data generated from the existing stochastic process models as well as other models. Finally, several real datasets are used to verify the validity of the proposed method, which can generate more similar degradation paths to the real ones and thus can provide a more accurate lifetime prediction.

A general stochastic model for bivariate episodes driven by a gamma sequence

We propose a new stochastic model describing the joint distribution of (X,N), where N is a counting variable while X is the sum of N independent gamma random variables. We present the main properties of this general model, which include marginal and conditional distributions, integral transforms, moments and parameter estimation. We also discuss in more detail a special case where N has a heavy tailed discrete Pareto distribution. An example from finance illustrates the modeling potential of this new mixed bivariate distribution.

Time-dependent probability density function for general stochastic logistic population model with harvesting effort

We derive and analyze the time-dependent probability density function for the number of individuals in a population at a given time in a general logistic population model with harvesting effort using the Fokker–Planck equation. The time-dependent probability density function (obtained as the unique principal solution of the Fokker–Planck equation corresponding to certain initial value and boundary conditions) is used to describe how the distribution of the population process changes with time. We assume the environment is randomly varying and the population is subject to a continuous spectrum of disturbances, with fluctuations in the intrinsic growth rate and the harvesting effort. The randomness is expressed as independent white noise processes. The effect of changes in the intrinsic growth rate, harvesting effort, and noise intensities on the distribution is investigated. In addition, conditions for the existence of optimal harvesting policy are obtained using properties of the time-dependent probability density function. The results obtained in this work are validated using population and published parameters.

An Asymptotic Theory of Joint Sequential Changepoint Detection and Identification for General Stochastic Models

The paper addresses a joint sequential changepoint detection and identification/isolation problem for a general stochastic model, assuming that the observed data may be dependent and non-identically distributed, the prior distribution of the change point is arbitrary, and the post-change hypotheses are composite. The developed detection–identification theory generalizes the changepoint detection theory developed by Tartakovsky (2019) to the case of multiple composite post-change hypotheses when one has not only to detect a change as quickly as possible but also to identify (or isolate) the true post-change distribution. We propose a multi-hypothesis change detection–identification rule and show that it is nearly optimal, minimizing moments of the delay to detection as the probability of a false alarm and the probabilities of misidentification go to zero.

Dynamic analysis of a stochastic four species food-chain model with harvesting and distributed delay

A stochastic four species food-chain model is proposed in this paper. Here, artificial harvest in each species and the effect of time delay for interaction between species are considered, which makes the model more applicable in real situations. Specifically, we address the stochastic global dynamics behavior, including the existence of global positive solutions, stochastic ultimate boundedness, extinction with probability one, persistence in mean and global stability. The asymptotic stability in the probability distribution is obtained, and the criterion for the existence and non-existence of the optimal harvesting strategy is also derived. Furthermore, this paper can provide reference for the research of general n-species stochastic food-chain models.

Stochastic SIR Lévy Jump Model with Heavy-Tailed Increments

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Lévy jump process with heavy-tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita’s inequality instead of the Burkholder–Davis–Gundy inequality for continuous processes, and allows for the treatment of infinite Lévy measures by the definition of new threshold \(\mathcal {\bar{R}}_0\). An SIR model driven by a tempered stable process is presented as an example of application with the ability to model sudden disease outbreak, illustrated by numerical simulations. The results show that persistence and extinction are dependent not only on the variance of the processes increments, but also on the shapes of their distributions.

A general stochastic matching model on multigraphs

We extend the general stochastic matching model on graphs introduced in (Mairesse and Moyal, 2016), to matching models on multigraphs, that is, graphs with self-loops. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is `First Come, First Matched'.