Stochastic Theory Research Articles

Square-mean S-Asymptotically $$\omega $$-Periodic Solutions for Some Stochastic Delayed Integrodifferential Inclusions

AbstractIn this work, by the resolvent operator theory, stochastic analysis theory and the fixed-point technique in Fréchet spaces, we present sufficient conditions for the existence of square-mean S-asymptotically $$\omega $$ ω -periodic mild solutions of abstract stochastic integrodifferential inclusions with infinite delay in separable Hilbert spaces and provide an example to illustrate the abstract results.

Effect of Variation in Viscosity on Static and Dynamic Characteristics of Rough Porous Journal Bearings with Micropolar Fluid Squeeze Film Lubrication

In the present study, an effort was made to determine the effects of a porous matrix with different viscosities on the dynamic and static behaviors of rough short journal bearings taking into account the action of a squeezing film under varying loads without journal rotation. The micropolar fluid was regarded as a lubricant that contained microstructure additives in both the porous region and the film region. By applying Darcy’s law for micropolar fluids through a porous matrix and stochastic theory related to uneven surfaces, a standardized Reynolds-type equation was extrapolated. Two scenarios with a stable and an alternating applied load were analyzed. The impacts of variations in viscosity, the porous medium, and roughness on a short journal bearing were examined. We inspected the dynamic and static behaviors of the journal bearing. We found that the velocity of the journal center with a micropolar fluid decreased when there was a cyclic load, and the impact of variations in the viscosity and porous matrix diminished the load capacity and pressure in the squeeze film and increased the velocity of the journal center.

Nonlinear conductivity of aqueous electrolytes: Beyond the first Wien effect.

The conductivity of strong electrolytes increases under high electric fields, a nonlinear response known as the first Wien effect. Here, using molecular dynamics simulations, we show that this increase is almost suppressed in moderately concentrated aqueous electrolytes due to the alignment of the water molecules by the electric field. As a consequence of this alignment, the permittivity of water decreases and becomes anisotropic, an effect that can be measured in simulations and reproduced by a model of water molecules as dipoles. We incorporate the resulting anisotropic interactions between the ions into a stochastic density field theory and calculate ionic correlations as well as corrections to the Nernst-Einstein conductivity, which are in qualitative agreement with the numerical simulations.

Research on passenger flow control at metro transfer stations based on real-time flow calculation of streamlines

PurposeThe volume of passenger traffic at metro transfer stations serves as a pivotal metric for the orchestration of crowd flow management. Given the intricacies of crowd dynamics within these stations and the recurrent instances of substantial passenger influxes, a methodology predicated on stochastic processes and the principle of user equilibrium is introduced to facilitate real-time traffic flow estimation within transfer station streamlines.Design/methodology/approachThe synthesis of stochastic process theory with streamline analysis engenders a probabilistic model of intra-station pedestrian traffic dynamics. Leveraging real-time passenger flow data procured from monitoring systems within the transfer station, a gradient descent optimization technique is employed to minimize the cost function, thereby deducing the dynamic distribution of categorized passenger flows. Subsequently, adhering to the tenets of user equilibrium, the Frank–Wolfe algorithm is implemented to allocate the intra-station categorized passenger flows across various streamlines, ascertaining the traffic volume for each.FindingsUtilizing the Xiaozhai Station of the Xi’an Metro as a case study, the Anylogic simulation software is engaged to emulate the intra-station crowd dynamics, thereby substantiating the efficacy of the proposed passenger flow estimation model. The derived solutions are instrumental in formulating a crowd control strategy for Xiaozhai Station during the peak interval from 17:30 to 18:00 on a designated day, yielding crowd management interventions that offer insights for the orchestration of passenger flow and operational governance within metro stations.Originality/valueThe construction of an estimation methodology for the real-time streamline traffic flow augments the model’s dataset, supplanting estimated values derived from surveys or historical datasets with real-time computed traffic data, thereby enhancing the precision and immediacy of crowd flow management within metro stations.

An approach for regime-switching stochastic control problems with memory and terminal conditions

In this research article, we focus on a stochastic optimal control problem with two types of terminal constraints. These specific conditions provide real-valued and stochastic Lagrange multipliers. Our model evolves according to a Markov regime-switching jump diffusion model with memory. In this context, the memory is represented by a Stochastic Differential Delay Equation. We present two theorems for each constraint within the general formulation of stochastic optimal control theory in a Lagrangian environment. We approach to this task from a theoretical perspective and provide mild technical assumptions, which make our theorems applicable for a broad class of stochastic control problems as well as for a wide range of disciplines such as engineering, biology, operations research, medicine, computer science and economics. In this work, we apply Stochastic Maximum Principle to demonstrate an optimal dividend policy corresponding to a time-delayed wealth process of a company. Moreover, we determine the real-valued Lagrange multiplier of this control problem explicitly.

Longitudinal Fluvial Dispersion of Coarse Particles: Insights From Field Observations and Model Simulations

AbstractIn this study, we use field observations augmented with model simulations to examine gravel dispersion over nine years (2007–2015) in Halfmoon Creek. The observations of flow, entrainment, and dispersion were used to develop a forward model utilizing the Einstein‐Hubbell‐Sayre (EHS) compound Poisson process. The observed mean virtual velocity of the tracer population slows down with cumulative excess energy after the 2010 large event. The forward model deviates from the observations in representation of tails, overpredicts mean displacements, and shows a narrower spatial distribution. The heavy‐tailed resting times indicate prolonged immobilization of some grains, suggesting the preferential movement of other most mobile grains. As such, 34% of most mobile grains constitute 50% of the total entrainments. The consideration of preferential movement explains the longitudinal spread but still overpredicts the displacement after the 2010 event. The model was then explored to consider additional transport‐related mechanisms causing deviations, such as reduction in virtual velocity, entrainment probability, and morphological trapping of meander bends, which helps to adequately recreate the observed dispersive behavior. The available historical flow records used for simulating dispersive behavior over multiple decades reveal an abrupt increase in displacements for exceptionally large events, suggesting the exhumation of deeply buried grains back in transport. The simulation results highlight the need for tracer studies with large sample sizes and improved recovery rates for longer time frames experiencing floods of widely varying magnitudes. Such models, inspired by Einstein's stochastic theory can be valuable for various river research applications.

Guaranteed Cost Control of Impulsive Nonlinear Stochastic Systems With Mixed Time Delays

ABSTRACTThis article mainly addresses the guaranteed cost control (GCC) problem for a class of impulsive nonlinear stochastic systems with mixed time delays, where both distributed time delay as well as discrete time delay are considered. Utilizing the Hamilton–Jacobi inequalities technique, Lyapunov–Krasovskii functional (LKF) and stochastic control theory, a new delay‐dependent sufficient condition is acquired to ensure the asymptotic stability in probability (ASP) of the closed‐loop system (CLS). Moreover, the guaranteed cost controller is designed and the upper bound of cost function is given. At the same time, three corollaries are presented for the special circumstance without impulse, distributed time delays or time‐varying time delay, respectively. Finally, the feasibility of the theoretical results is verified by an example.

Online adaptive critic designs with tensor product B-splines and incremental model techniques

As an effective optimal control scheme in the field of reinforcement learning, adaptive dynamic programming (ADP) has attracted extensive attention in recent decades. Neural networks (NNs) are commonly employed in ADP algorithms to realize nonlinear function approximation. However, the learning process with NN approximation typically requires a substantial amount of training data and places a heavy computational burden. To improve learning efficiency and alleviate computational load, this paper develops a novel model-free ADP approach based on multivariate splines and incremental control techniques, aimed at achieving online learning of nonlinear control. By utilizing input and output data, an incremental linear approximation model is identified online without any prior knowledge of system dynamics. To improve nonlinear approximation capabilities and lessen computational demands, tensor product B-splines, instead of NNs, are integrated into the critic module to approximate the value function, and the recursive least squares temporal difference (RLS-TD) algorithm is employed to update the weights of spline bases. Convergence analysis of the proposed control scheme is conducted based on the dual-timescale stochastic approximation theory. To illustrate the effectiveness and feasibility of the proposed control scheme, numerical simulations are performed in solving the online nonlinear control problem within the context of the inverted pendulum system.

Complex stochastic optimal control foundation of quantum mechanics

Abstract Recent studies have extended the use of the stochastic Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies often assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers when finding optimal controls. This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the stochastic movement of quantum particles within the framework of stochastic optimal control theory. We obtain the complex diffusion coefficient in the stochastic equation of motion using the Cauchy-Riemann theorem, considering that the particle’s stochastic movement is described by two perfectly correlated real and imaginary stochastic processes. During the development of the covariant form of the HJB equation, we demonstrate that if the temporal stochastic increments of the two processes are perfectly correlated, then the spatial stochastic increments must be perfectly anti-correlated, and vice versa. The diffusion coefficient we derive has a form that enables the linearization of the HJB equation. The method for linearizing the HJB equation, along with the subsequent derivation of the Dirac equation, was developed in our previous work [V. Yordanov, Scientific Reports 14, 6507 (2024)]. These insights deepen our understanding of quantum dynamics and enhance the application of stochastic optimal control theory to quantum mechanics.

Efficient Convex PCA with Applications to Wasserstein GPCA and Ranked Data

Convex PCA, which was introduced by Bigot et al. modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this article is 3-fold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.

Seismic Response Analysis of Hydraulic Tunnels Under the Combined Effects of Fault Dislocation and Non-Uniform Seismic Excitation

Hydraulic tunnels are prone to pass through faults and high-intensity earthquake areas, which will cause serious damage under fault dislocation and earthquake action. Fault dislocation and seismic excitation are often considered separately in previous studies. For tectonic earthquakes with higher frequency in seismic phenomena, fault dislocation and ground motion are often associated, and fault dislocation is usually the cause of earthquake occurrence, so it is limiting to consider the two separately. Moreover, strong earthquake records show that there will be significant differences in the mainland vibration within 50 m. The uniform ground motion inputs in previous studies are not suitable for long hydraulic tunnels. This paper begins with the simulation of non-uniform stochastic seismic excitations that consider spatial correlation. Based on stochastic vibration theory, multiple multi-point acceleration time-history curves that can reflect traveling wave effects, coherence effects, attenuation effects, and non-stationary characteristics are synthesized. Furthermore, a fault velocity function is introduced to account for the velocity effect of fault dislocation. Finally, numerical analyses of the response patterns of the tunnel lining under four different conditions are conducted based on an actual engineering project. The results indicate the following: (a) the maximum lining response values occur under the combined effects of fault dislocation and non-uniform seismic excitation, indicating its importance in the seismic resistance of the tunnel. (b) Compared to uniform seismic excitation, the peak displacement of the tunnel under non-uniform seismic excitation increases by up to 6.42%, and the peak maximum principal stress increases by up to 28%. Additionally, longer tunnels exhibit a noticeable delay effect in axial deformation during an earthquake. (c) Under non-uniform seismic excitation, the larger the fault dislocation magnitude, the greater the peak displacement and peak maximum principal stress at the monitoring points of the lining. The simulation results show that the extreme response values primarily occur at the crown and haunches of the tunnel, which require special attention. The research can provide valuable references for the seismic design of cross-fault tunnels.

Unified uniaxial compressive constitutive model for C20–C120 concrete based on stochastic damage theory

The concrete uniaxial compressive stress-strain model is essential for analyzing reinforced concrete members and structures. However, models applicable to C20–C120 concrete are still relatively rare. Moreover, the previous empirical model construction methods may not reflect the physical mechanisms and randomness of concrete failure well. To this end, the database was constructed by collecting and analyzing uniaxial compressive stress-strain curve test results from different scholars. The cubic compressive strength in the database ranges from 9.9 MPa to 137.3 MPa. The adaptability of the stochastic damage model was validated using the database of test curves. Subsequently, stochastic damage models for C20–C120 concrete were calibrated based on concrete compressive strength grouping curves, and practical analysis models were suggested. The results indicate that the calculated curves closely match the test curves when the plastic strain evolution parameters η1,c and η2,c in the stochastic models are taken as 0.001–0.30 and 0.19–0.25, respectively. The adaptability of the stochastic damage model is satisfactory, but the plastic strain evolution laws vary for different concrete compressive strengths, influenced by mix proportions and the probability of failure of each component on the meso-scale. When η1,c is calculated by the empirical formula proposed in this paper and η2,c is approximated as 0.22, the calculated curves for C20–C120 concrete are consistent with the mean values of the test curves. The proposed practical analysis models provide comparable accuracy to stochastic damage models, eliminating the need for iterative calculations, making them convenient for engineering applications.

Double-loop importance sampling for McKean–Vlasov stochastic differential equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with the solution to the d-dimensional McKean–Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting P-particle system, which is a set of P coupled d-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a P×d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P \ imes d$$\\end{document}-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in (dos Reis et al. 2023), generating a d-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of OTOLr-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \ extrm{TOL}_{\ extrm{r}}^{-4}\\right) $$\\end{document} with a significantly reduced constant to achieve a prescribed relative error tolerance TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document}. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given TOLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{TOL}_{\ extrm{r}}$$\\end{document} compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

Min–max group consensus of discrete-time multi-agent systems under directed random networks

This paper studies the min–max group consensus of discrete-time multi-agent systems under a directed random graph, where the presence of each directed edge is randomly determined by a probability and independent of the presence of other edges. Firstly, we propose a min–max consensus protocol without memory, and give the necessary and sufficient conditions to ensure that the multi-agent system can achieve the min–max group consensus in the sense of almost sure and mean square, respectively. Secondly, we design a novel consensus protocol with memory and a behavior mechanism. Using the stochastic analysis theory and the extremal algebra, some necessary and sufficient conditions are obtained for achieving the min–max group consensus in the sense of almost sure and mean square, respectively. It is shown that the protocol with memory can solve the loss problem of the maximum and minimum initial states. Finally, the effectiveness of the two group consensus protocols and the behavior mechanism is verified by four numerical simulations.

Digital screener of socio-motor agency balancing motor autonomy and motor control.

Dyadic social interactions evoke complex dynamics between two agents that, while exchanging unequal levels of body autonomy and motor control, may find a fine balance to synergize, take turns, and gradually build social rapport. To study the evolution of such complex interactions, we currently rely exclusively on subjective pencil and paper means. Here, we complement this approach with objective biometrics of socio-motor behaviors conducive to socio-motor agency. Using a common clinical test as the backdrop of our study to probe social interactions between a child and a clinician, we demonstrate new ways to streamline the detection of social readiness potential in both typically developing and autistic children by uncovering a handful of tasks that enable quantification of levels of motor autonomy and levels of motor control. Using these biometrics of autonomy and control, we further highlight differences between males and females and uncover a new data type amenable to generalizing our results to any social setting. The new methods convert continuous dyadic bodily biorhythmic activity into spike trains and demonstrate that in the context of dyadic behavioral analyses, they are well characterized by a continuous Gamma process that can classify individual levels of our thus defined socio-motor agency during a dyadic exchange. Finally, we apply signal detection processing tools in a machine learning approach to show the validity of the streamlined version of the digitized ADOS test. We offer a new framework that combines stochastic analyses, non-linear dynamics, and information theory to streamline and facilitate scaling the screening and tracking of social interactions with applications to autism.

Minimizing the penalized goal-reaching probability with multiple dependent risks

We consider a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who wishes to minimize the probability that the value of the wealth process reaches a low barrier before a high goal. We assume that the insurer can purchase per-loss reinsurance for every class of insurance business and invest its surplus in a risk-free asset and a risky asset. Using the technique of stochastic control theory and solving the associated Hamilton-Jacobi-Bellman (HJB) equation, we derive the robust optimal investment-reinsurance strategy and the associated value function. We conclude that the robust optimal investment-reinsurance strategy coincides with the one without model ambiguity, but the value function differs. We also illustrate our results by numerical examples.

Математична модель оптимального планування виробничого процесу

In the first half of the last century, due to the increasing complexity of production processes, there was a need for their more efficient organization. During this period, the foundations of mathematical modeling of economic processes were laid. Modern mathematical models of optimal planning integrate artificial intelligence, machine learning methods and large databases, take into account uncertainty and risks in production processes using stochastic dependencies and probability theory methods in the models. This makes it possible to model even more complex systems and take into account more factors, such as fluctuations in demand, changes in production supply chains, etc. Today, optimal planning models are the basis of enterprise management systems (ERP) and are used in various industries: from manufacturing to logistics and energy. This article considers an economic-mathematical model for planning the optimal production process under certain assumptions about the economic state. That is, the profit of the enterprise, which can produce different types of products, for each type of these products depends on the economic state of the country. It is determined what share in the total production of the enterprise will be occupied by a certain type of product in order to obtain maximum profit. The profit of the enterprise depends on the state of the economy, therefore the expected profit is characterized by the mathematical expectation of profit. For an optimal production plan, you need to strive for the best ratio between expected profit and risk (mean square deviation), that is, it is necessary to find the maximum of the function that characterizes this ratio and is a function of n unknowns. To find the extremum of this function, we find partial derivatives and obtain n nonlinear equations with n unknowns. Performing some transformations, we reduce this system to a system of n linear equations. If the non-negativity conditions are not imposed on the variables, then when solving the system, it may happen that some variables will take negative values. This means that in order to obtain optimal profit, it is not recommended to manufacture the corresponding type of products

Theoretical analysis of effect of MHD, couple stress and slip velocity on squeeze film lubrication of rough Triangular plates

In this study, Christensen's stochastic theory is utilized on rough surfaces' hydrodynamic lubricating effect to examine how surface roughness, couple stress fluid, slip velocity, magnetohydrodynamic (MHD), and the triangular surface interact. Modified Reynolds equation is derived analytically using combining theories of Stokes couple stresses, Lorentz forces, and Christensen's stochastic hypothesis about hydrodynamic lubrication. Pressure, load carrying capacity and squeeze film time expression is derived mathematically using Reynolds equation that accounts for surface roughness and coupling stress. For various parameters including rough parameter, slip velocity, Hartmann number, and couple stress, the lubricating characteristics are analysed graphically. Squeeze film time, load carrying capacity, and pressure are enhanced by an increase in slip velocity, couplestress and magnetic field. The lubrication characteristics decreases (increases) on squeeze film pressure, load carrying capacity and squeeze film time through increasing values of the longitudinal (transverse) roughness parameter.

The Dynamic Behavior of a Stochastic SEIRM Model of COVID-19 with Standard Incidence Rate

This paper studies the dynamic behavior of a stochastic SEIRM model of COVID-19 with a standard incidence rate. The existence of global solutions for dynamic system models is proven by integrating stochastic process theory and the concept of stopping times, together with the contradiction method. Moreover, we construct appropriate Lyapunov functions to analyze system stability and apply Dynkin’s formula and Fatou’s lemma to handle stopping times and expectations of stochastic processes. Notably, the extinction study provides mathematical proof that under the given system dynamics, the total population does not grow indefinitely but tends to stabilize over time. The properties of the diffusion matrix are harnessed to guarantee the system’s stationary distribution. Conclusively, numerical simulations confirm the model’s extinction outcomes.

Optimization of the power–transportation coupled power distribution network based on stochastic user equilibrium

With the increasing penetration of electric vehicles (EVs) in road traffic, the spatial and temporal stochasticity of the travel pattern and charging demand of EVs as a mode of transportation and an electrical load have generated different degrees of congestion impacts on both the power grid and the transportation network. Based on this, this paper proposes a power–transportation-coupled distribution network optimization strategy based on stochastic user traffic equilibrium theory. First, a stochastic user equilibrium-mixed traffic flow allocation model that expresses user non-completely rational path selection behavior is established, and the traffic flow equilibrium solution is obtained using an improved method of successive algorithm and mapped to charging loads. Second, a distribution network power flow optimization model under the coupled power–transportation architecture is established to optimize the operation state of the distribution network by combining distributed resources such as energy storage, demand response loads, wind power, photovoltaic, and gas turbines.