Continuous-time Process Research Articles

Stability analysis of non-linear sampled data systems with time varying sample period and delay in the feedback loop

In almost all practical applications of control, technological and economical considerations impose limits on communication speed, frequency of communication, and frequency of actuator adjustment. Such limits turned the analysis of sampled data systems into a flourishing field. Water systems pose a particular challenge: the systems are networks of canals and reservoirs spread over large areas, and the actuators are relatively large and exposed to the elements. In this study, a theorem on the local exponential stability of sampled data systems with variable control time step and variable delay in the communication between the non-linear continuous time process and the non-linear discrete time controller is presented. To illustrate the application of the theorem, it is applied to a simple water system.

Controlling Fake News by Collective Tagging: A Branching Process Analysis

The spread of fake news on online social networks (OSNs) has become a matter of concern. These platforms are also used for propagating important authentic information. Thus, there is a need for mitigating fake news without significantly influencing the spread of real news. We leverage users’ inherent capabilities of identifying fake news and propose a warning-based control mechanism to curb this spread. Warnings are based on previous users’ responses that indicate the authenticity of the news. We use population-size dependent continuous-time multi-type branching processes to describe the spreading under the warning mechanism. We also have new results towards these branching processes. The (time) asymptotic proportions of the individual populations are derived using stochastic approximation tools. Using these, relevant type 1, type 2 performances are derived and an appropriate optimization problem is solved. The proposed mechanism effectively controls fake news, with negligible influence on the propagation of authentic news. We validate performance measures using Monte Carlo simulations on network connections provided by Twitter data.

Model of item monitoring system under unreliable supervision

Aim . The conducted research aims to develop an analytical model of item dependability for situations of technical state monitoring with constant inspection frequency and subject to inspection errors and failures of various types. The primary purpose of the model is the calculation and prediction of dependability indicators that depend on specified conditions. Methods . The model is based on the Markovian process theory. Models of two types are used, i.e. the continuous-time discrete process model and semi-Markovian model. The mathematical operations involved in the model implementation were performed in matrix form. An items’ operation is presented in the form of recurrent cycles separated from each other by the recovery state. A continuous-time model allows obtaining state probabilities within the periods between inspections, mean active state times and state probabilities at the end of a period. The probabilities of entering states at the end of a period are a priori for the semi-Markovian model, using which the mean numbers of active states within one cycle were obtained. Results . The mean up and down time within a cycle were calculated using mean state frequency and mean time of active state. Based on those parameters, formulas were obtained for calculating the availability and non-availability coefficients. Out of the above model follows that the dependability indicators depend on the frequencies of explicit and hidden failures, inspection frequency and inspection errors. The paper sets forth the calculation data for the mean cycle duration and non-availability coefficient under various failure rates and various probabilities of inspection errors. It is shown that the mean cycle duration significantly depends on the probability of inspection errors of the I kind and practically does not depend on the probability of inspection errors of the II kind. However, the non-availability coefficient practically does not depend on the probability of inspection errors of the I kind, yet there is a strong dependence on the probability of inspection errors of the II kind. Conclusions . The presented model allows calculating and predicting dependability indicators taking into consideration explicit and hidden failures, as well as the monitoring system parameters. While designing new and improving the maintenance procedures of existing systems, the effect of various factors on the dependability level should to be taken into consideration.

Corrigendum to: Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

AbstractWe would like to correct the statement of Lemma 4.1 in [BDK+18].

Persistence as an Optimal Hedging Strategy

Bacteria invest in a slow-growing subpopulation, called persisters, to ensure survival in the face of uncertainty. This hedging strategy is remarkably similar to financial hedging, where diversifying an investment portfolio protects against economic uncertainty. We provide a new, to our knowledge, theoretical foundation for understanding cellular hedging by unifying the study of biological population dynamics and the mathematics of financial risk management through optimal control theory. Motivated by the widely accepted role of volatility in the emergence of persistence, we consider several models of environmental volatility described by continuous-time stochastic processes. This allows us to study an emergent cellular hedging strategy that maximizes the expected per capita growth rate of the population. Analytical and simulation results probe the optimal persister strategy, revealing results that are consistent with experimental observations and suggest new opportunities for experimental investigation and design. Overall, we provide a new, to our knowledge, way of conceptualizing and modeling cellular decision making in volatile environments by explicitly unifying theory from mathematical biology and finance.

Joint UAV Access and GEO Satellite Backhaul in IoRT Networks: Performance Analysis and Optimization

With the growing demand for communications in remote and dispersed areas, Internet-of-Remote Things (IoRT) networks with joint unmanned aerial vehicle (UAV) access and geostationary orbit (GEO) satellite backhaul hold great promise to provide sufficient access services to Internet-of-Things (IoT) users and devices. As the fundamental of the performance optimization of IoRT networks, the performance analysis sheds light on the relationship between the network performance (i.e., backlog, delay, and throughput) and access scale (i.e., the numbers of UAVs and UAV users). Aiming at the challenges brought by the complex network structure (i.e., two-level queuing network along with the converged traffic), we introduce the stochastic network calculus-based min-plus convolution and the leftover service to mathematically describe the complex structure. For the analytical challenges of the continuous-time arrival process and heterogeneous two-level link capacities, we innovatively prove their supermartingale features and further derive the closed-form expressions of the network backlog and delay bounds based on the martingale theory. To pursue higher throughput while guaranteeing delay performance, we formulate a mixed-integer optimization problem of the access scale that contains a nondifferentiable variable derived from a transcendental equation. For the tractability, we propose a three-directional iterative (TDI) algorithm to search the optimal solution of the optimization problem. Simulation results verify the tightness of our performance bounds in contrast to the standard bound and the effectiveness of the proposed algorithm.

Optimum life-cycle maintenance strategies of deteriorating highway bridges subject to seismic hazard by a hybrid Markov decision process model

Highway bridges are a class of important infrastructure components in transportation networks. Highway bridges will inevitably deteriorate over time due to various external impacts, such as traffic loads and corrosion. In seismic-prone areas, seismic hazard is another critical factor that threatens the normal operation of ageing highway bridges. To ensure adequate safety of ageing highway bridges under potential earthquake shocks, it is imperative to perform proper maintenances during their service life. This paper aims at optimizing the life-cycle maintenance strategies for highway bridges under the combinatorial threats of progressive deterioration and sudden earthquakes. Driven by the distinct characteristics of the two threats, we have proposed a novel hybrid Markov decision process model that integrates a discrete-time Markov decision process model for dealing with progressive deterioration and a continuous-time Markov decision process model for dealing with sudden earthquakes into a unified framework. The proposed model is further reduced into the form of a discrete-time Markov decision process model with an equivalent state transition matrix and an equivalent discount factor, which can be easily solved by dynamic programming. A ternary state space that collectively describes corrosion initiation, corrosion damage cumulation, and seismic damage cumulation is defined. A gamma process and a Poisson process are adopted to model bridge corrosion and earthquake occurrences, respectively. To demonstrate the application of the proposed framework, a numerical example is conducted, in which the optimizations with respect to the maintenance strategy, inspection interval, and design performance level of a hypothetical highway bridge are carried out successively.

Computation of time probability distributions for the occurrence of uncertain future events

The determination of the time at which an event may take place in the future is a well-studied problem in a number of science and engineering disciplines. Indeed, for more than fifty years, researchers have tried to establish adequate methods to characterize the behaviour of dynamic systems in time and implement predictive decision-making policies. Most of these efforts intend to model the evolution in time of nonlinear dynamic systems in terms of stochastic processes; while defining the occurrence of events in terms of first-passage time problems with thresholds that could be either deterministic or probabilistic in nature. The random variable associated with the occurrence of such events has been determined in closed-form for a variety of specific continuous-time diffusion models, being most of the available literature motivated by physical phenomena. Unfortunately, literature is quite limited in terms of rigorous studies related to discrete-time stochastic processes, despite the tremendous amount of digital information that is currently being collected worldwide. In this regard, this article provides a mathematically rigorous formalization for the problem of computing the probability of occurrence of uncertain future events in both discrete- and continuous-time stochastic processes, by extending the notion of thresholds in first-passage time problems to a fully probabilistic notion of “uncertain events” and “uncertain hazard zones”. We focus on discrete-time applications by showing how to compute those probability measures and validate the proposed framework by comparing to the results obtained with Monte Carlo simulations; all motivated by the problem of fatigue crack growth prognosis.

Deep learning of the role of interleukin IL-17 and its action in promoting cancer

Abstract In breast cancer patients, metastasis remains a major cause of death. The metastasis formation process is given by an interaction between the cancer cells and the microenvironment that surrounds them. In this article, we develop a mathematical model that analyzes the role of interleukin IL-17 and its action in promoting cancer and in facilitating tissue metastasis in breast cancer, using a dynamic analysis based on a stochastic process that accounts for the local and global action of this molecule. The model uses the Ornstein–Uhlembeck and Markov process in continuous time. It focuses on the oncological expansion and the interaction between the interleukin IL-17 and cell populations This analysis tends to clarify the processes underlying the metastasis expansion mechanism both for a better understanding of the pathological event and for a possible better control of therapeutic strategies. IL-17 is a proinflammatory interleukin that acts when there is tissue damage or when there is a pathological situation caused by an external pathogen or by a pathological condition such as cancer. This research is focused on the role of interleukin IL-17 which, especially in the case of breast cancer, turns out to be a dominant “communication pin” since it interconnects with the activity of different cell populations affected by the oncological phenomenon. Stochastic modeling strategies, specially the Ornstein-Uhlenbeck process, with the aid of numerical algorithms are elaborated in this review. The role of IL-17 is discussed in this manuscript at all the stages of cancer. It is discussed that IL-17 also acts as “metastasis promoter” as a result of its proinflammatory nature. The stochastic nature of IL-17 is discussed based on the evidence provided by recent literature. The resulting dynamical analysis can help to select the most appropriate therapeutic strategy. Cancer cells, in the case of breast cancer, have high level of IL-17 receptors (IL-17R); therefore the interleukin itself has direct effects on these cells. Immunotherapy research, focused on this cytokine and interlinked with the stochastic modeling, seems to be a promising avenue.

An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods. Supplementary materials for this article are available online.

Towards a Classification of Behavioural Equivalences in Continuous-time Markov Processes

Bisimulation is a concept that captures behavioural equivalence of states in a transition system. In [Linan Chen, Florence Clerc, and Prakash Panangaden, Bisimulation for feller-dynkin processes, in: Proceedings of the Thirty-Fifth Conference on the Mathematical Foundations of Programming Semantics, Electronic Notes in Theoretical Computer Science 347 (2019) 45–63.], we proposed two equivalent definitions of bisimulation on continuous-time stochastic processes where the evolution is a flow through time. In the present paper, we develop the theory further: we introduce different concepts that correspond to different behavioural equivalences and compare them to bisimulation. In particular, we study the relation between bisimulation and symmetry groups of the dynamics. We also provide a game interpretation for two of the behavioural equivalences. We then compare those notions to their discrete-time analogues.

Cutoff phenomenon for the maximum of a sampling of Ornstein–Uhlenbeck processes

In this article we study the so-called cutoff phenomenon in the total variation distance when n→∞ for the family of continuous-time stochastic processes indexed by n∈N, Ztn≔maxj∈{1,…,n}Xtjt≥0,where X1,…,Xn is a sample of n ergodic Ornstein–Uhlenbeck processes driven by stable noise of index α. It is not hard to see that for each n∈N, Ztn converges in the total variation distance to a limiting distribution Z∞n, as t goes by. Using the asymptotic theory of extremes; in the Gaussian case we prove that the total variation distance between the distribution of Ztn and its limiting distribution Z∞n converges to a universal function in a constant time window around the cutoff time, a fact known as profile cutoff in the context of stochastic processes. On the other hand, in the heavy-tailed case we prove that there is not cutoff.

A lattice approach to evaluate participating policies in a stochastic interest rate framework

To achieve the accurate evaluation and management of the risk affecting long-term life insurance contracts, the insurer cannot leave aside the consideration of stochastic dynamics not only for the company’s assets but also for the interest rate. The aim of this paper is to provide, in such a framework, a flexible method for evaluating participating policies, life insurance products that combine financial and demographic risks, and provide benefits linked to the company’s asset returns. Participating policies embedding not only a minimum guaranteed bonus rate but also a surrender option are analyzed. The method is flexible in that it allows the insurer to choose the most appropriate dynamics, both for the interest rate and the company’s asset, among the ones widely diffused in finance. Lattice-based procedures are used to discretize the continuous time processes, and to provide a comprehensive evaluation method.

Using Semi-Markov Chains to Solve Semi-Markov Processes

This article provides a novel method to solve continuous-time semi-Markov processes by algorithms from discrete-time case, based on the fact that the Markov renewal function in discrete-time case is a finite series. Bounds of approximate errors due to discretization for the transition function matrix of the continuous-time semi-Markov process are investigated. This method is applied to a reliability problem which refers to the availability analysis of the system subject to sequential cyber-attacks. Two cases where sojourn times follow exponential and Weibull distributions are considered and computed in order to verify and illustrate the proposed method.

Higher order asymptotics for large deviations — Part II

We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly-dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a [Formula: see text]-dimensional compact manifold admit these asymptotic expansions of all orders.

The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations

In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression.

Statistical causality and separable processes

In this paper we consider the statistical concept for continuous-time stochastic processes, which is based on Granger’s definition of causality. We, also, show that separability is directly related to causality concepts. More precisely, we provide necessary conditions, in terms of statistical causality, for the space Lp(Ω,G∞,P) to be separable. The concept of statistical causality is related to the notion separability of stochastic processes, too.

Diffusion Approximation of Branching Processes in Semi-Markov Environment

We consider continuous-time Markov branching processes in semi-Markov random environment and obtain diffusion approximation results for the near critical case. The problem of semi-Markov environment, presented here, is new and more interesting than the Markov case, since it includes many particular interesting cases: Markov, renewal, etc. The particular case of the Markov random environment of continuous-time branching process diffusion approximation results are obtained.

The continuous-time limit of score-driven volatility models

We provide general conditions under which a class of discrete-time volatility models driven by the score of the conditional density converges in distribution to a stochastic differential equation as the interval between observations goes to zero. We show that the form of the diffusion limit depends on: (i) the link function, (ii) the conditional second moment of the score, (iii) the normalization of the score. Interestingly, the properties of the stochastic differential equation are strictly entangled with those of the discrete-time counterpart. Score-driven models with fat-tailed densities lead to continuous-time processes with finite volatility of volatility, as opposed to fat-tailed models with a GARCH update, for which the volatility of volatility is explosive. We examine in simulations the implications of such results on approximate estimation and filtering of diffusion processes. An extension to models with a time-varying conditional mean and to conditional covariance models is also developed.

Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes.

In the present paper, we consider the nonparametric regression model with random design based on (mathbf{X}_mathrm{t},mathbf{Y}_mathrm{t})_{mathrm{t}ge 0} a mathbb {R}^{d}times mathbb {R}^{q}-valued strictly stationary and ergodic continuous time process, where the regression function is given by m(mathbf{x},psi ) = mathbb {E}(psi (mathbf{Y}) mid mathbf{X} = mathbf{x})), for a measurable function psi : mathbb {R}^{q} rightarrow mathbb {R}. We focus on the estimation of the location {varvec{Theta }} (mode) of a unique maximum of m(cdot , psi ) by the location widehat{{varvec{Theta }}}_mathrm{T} of a maximum of the Nadaraya–Watson kernel estimator widehat{m}_mathrm{T}(cdot , psi ) for the curve m(cdot , psi ). Within this context, we obtain the consistency with rate and the asymptotic normality results for widehat{{varvec{Theta }}}_mathrm{T} under mild local smoothness assumptions on m(cdot , psi ) and the design density f(cdot ) of mathbf{X}. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.