Random Function Of Time Research Articles

ЗАДАЧА ОПТИМАЛЬНОГО СТОХАСТИЧЕСКОГО УПРАВЛЕНИЯ И ОЦЕНКИ СТОИМОСТИ КОРПОРАТИВНЫХ ОБЛИГАЦИЙ, ЗАВИСЯЩИХ ОТ ВРЕМЕНИ И СЛУЧАЙНЫХ СОСТОЯНИЙ ПАРАМЕТРОВ

In the optimal stochastic control problem, one estimates, using the utility indifference method, the bond price and the premium of a default swap contract (Credit default swap) when the model parameters (market interest rate, drift coef-ficient, and volatility of risky underlying assets) are random functions of time and state. Namely, the interest rate of the risk-free asset depends on time, and the prices of risky assets are described by linear homogeneous stochastic dif-ferential equations (SDEs) with multiplicative noise. To do this, the authors consider a portfolio with a risk-free asset and a risky asset with no default risk. For each such portfolio, the authors determine the amount of risky assets that maximizes the expected utility of its final wealth. This quantity allows one to solve the parabolic partial differential equations arising from the Hamilton-Jacobi-Bellman (HJB) equation by transforming them into ordinary differential equations using the method of variable separation to obtain the instantaneous value function of each portfolio. The authors derive the bond price and the default swap-contract premium (CDS) rate, which are the amounts that provide the same level of the expected utility by investing all of one’s wealth in a portfolio that does not contain these credit instruments, or by investing these amounts in credit instruments and the remainder of one’s wealth in the portfolio.

Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension dimP(E) of the target set E.

On the statistical background of quantum mechanics: generalities and a concrete example

We revisit our description of randomness in quantum processes that began in collaboration of Jean Ginibre. The calculations were performed on a worked example: the fluorescence of a single two-level atom pumped by a resonant laser field. This pump laser is described classically (by a function, not an operator). Our aim is first to built a Kolmogorov-type equation (K-equation) for the atomic state, so that the two parameters θ, φ that define this density matrix are random functions of time, therefore the atomic density matrix is a random density matrix. Such an approach, initiated for gas kinetics, was not yet applied to quantum phenomena, whereas it is especially tailored to very quick events well separated (in time) like the quantum jumps observed in spontaneous emission of photons by an atom. Here, we try to clarify the basis of our statistical approach leading to the K-equation below, and we present the main results deduced from it. We explain finally that our approach can be interpreted in terms of Everett’s theory of many-worlds, because at every emission a new history begins for the atom, with two nonoverlapping wave functions.

Selling price management of a highly liquid asset under given volume constraints

Goal: this paper considers the selling strategy of a highly liquid asset traded at market prices. The purpose of this paper is to develop a management that provides an improvement in the weighted average selling price. Methodology: it is assumed that market prices follow a diffusion process in which the drift and volatility coefficients are random functions of time. Under thegiven assumptions, using the stochastic differential equation theory we build a sales management in which only the prices of exchange transactions act as feedback. Findings: the management constructed in the present research allows the seller to hedge against sharp drops in market prices by artificially raising the weighted average selling price due to the implementation of a speculative trading strategy. Originality and contribution of the authors: in contrast to the management proposed earlier, this paper provides a lower limit on the minimum number of assets that must be traded, as well as constraints on the maximum possible number of assets allowed to be traded. Examples of virtual trading on real world exchanges are given, demonstrating the effect of the imposed restrictions on the values of the weighted average selling prices.

Using an Ensemble Filter to Improve the Representation of Temporal Source Variations in a Volcanic Ash Forecasting System

The use of ensemble models to forecast the dispersion and transport of airborne volcanic ash in operational contexts is increasingly being explored. The ensemble members are usually constructed to represent a priori uncertainty estimates in meteorological fields and volcanic ash source parameters. Satellite data can be used to further filter ensemble members within an analysis time window by rejecting poorly performing members, leading to improved forecasts. In this study, the ensemble filtering technique is used to improve the representation of temporal source variations. Ensemble members are initially created by representing the source time variations as random functions of time that are modulated by crude initial estimates of the variations estimated from satellite imagery. Ensemble filtering is then used to remove members whose fields match poorly with observations within a specified analysis time window that are represented by satellite retrievals of volcanic ash properties such as mass load, effective radius, and cloud top height. The filtering process leads to an ensemble with statistics in closer agreement with the observations. It is shown in the context of the 30 May 2014 Sangeang Api eruption case study that this method leads to significantly enhanced forecasting skill beyond the analysis time window—about 20% improvement on average—when compared to a system that assumes constant emission rates for the duration of the eruption, as is the case in many operational volcanic ash forecasting systems.

General principles and criteria of failure of solids on different scale-structural levels under static and alternating loading

Experimental and theoretical studies of the fracture regularities under monotonic and alternating loadings, including stress-corrosion fracture, revealed the main scale-structural levels of the brittle fracture (formation of a strip structure, nucleation of microcracks, microstructurally short non-propagating cracks and short propagating cracks, merging of short cracks with a macrocrack nucleation) and viscous fracture (an evolution of dislocation in slip bands, cellular structure formation, nucleation of a pit relief with micropores, merging of pores with a formation of meso-shear bands and initiation of a viscous crack, growing of a viscous crack under inelastic deformation) of crystal bodies. A transition from one level to another is determined by changes in the fracture mechanisms and occurs with varying degrees of probability. The necessity of considering the destruction as a hierarchical random step-by-step process on all scale-structural levels is substantiated. The fracture of plastic materials is considered as the sum of independent or dependent events, namely the initiation and growing of brittle cracks and evolution of pores through various mechanisms. Attaining of the limit states is determined by the statistics of the distribution of inhomogeneities throughout the entire body volume. The results of physical studies (using electron microscopy, X-ray structural analysis, etc.) provide determination of the geometry and density of defects, dislocations, pores, cracks at different levels. Macrocharacteristics of static, long-term and fatigue strength are determined using methods of mechanical tests. The regularities of the fracture evolution of engineering structures are determined using methods of non-destructive testing (ultrasonic analysis, acoustic emission analysis, magnetic flaw detection, etc.). A brief analytical review of the main known physical approaches — dislocation and energy structural theories, dislocation continuous theories, dilaton-frustron models, stochastic physical theories, and phenomenological approaches, including continuum damage theory and fracture mechanics — is presented. The analysis of gained data leads to the necessity of describing fracture in the framework of stochastic multilevel models. We consider the approach according to which the probability of reaching limit sates at each level under arbitrary multiaxial loading is determined by linear functionals of the loading process, kernels of the functionals being random functions of time. For simple proportional loading, the theory of brittle fatigue scale-structural failure is presented.

Method for determining and refining the interval of data collection of gas-dynamic processes by safety criterion

To solve the problems of controlling the gas-dynamic processes of production areas and air distribution in the ventilation network, their mathematical description is required, which can be obtained only on the basis of a detailed study and determination of the parameters and characteristics of gas-dynamic processes and regularities of air distribution. The mathematical description of the production ventilation system is complicated by the fact that the variables that determine the state of the control system are random functions of time. Therefore, the main attention should be paid to determining the probabilities of the characteristics of aerogasdynamic processes. The mathematical description of the ventilation of the production area into static and dynamic characteristics was the basis for the development of an optimal control algorithm for the production area. The control algorithm for the ventilation of the mining area consists of subalgorithms for controlling the parameters of the mine atmosphere, primary processing of the received information, and generating control actions. Their development is possible on the basis of research and mathematical description of gas-dynamic processes and methods for controlling the parameters of the mine atmosphere. Effective control of gas-dynamic processes in production areas is possible only when considering the entire ventilation system, due to the interconnectedness of aerodynamic parameters. The task of optimal air distribution control in the network is to ensure the required airflow rates for ventilation, determined as a result of solving the first problem with minimal energy costs for ventilation.

The Object According State Prediction to Diagnostic Data

The predicting methodology the state of the object based on diagnostic data is considered. With the selected parameter that determines the state of the object, it is measured in real time at a fixed sampling step. According to the measurement data, the value of this parameter is predicted in the future. This operation is implemented by an extrapolator of the l order - a l degree polynomial, built using the least squares method based on the previous measurements results. The changing process model of the diagnosed parameter is a random time function described by the stationary centered random component sum and a mathematical expectation deterministic change. The estimating prediction error method and the extrapolator parameters influence on its value are presented.

Weighted average price management of sales under the given minimum volume of assets obligatory for realization

In this paper, we consider the construction of a sales management strategy for a highly liquid asset trading at the market. The proposed strategy goal is to maximize the weighted average price of sales. It is assumed that the market price follows a geometric Brownian motion process in which drift and volatility coefficients are random functions of time. The important feature of the given management is that the volumes of assets in the succession of buy and sell executed bargains are calculated exclusively on the basis of the observed market prices rather than on the model coefficient values. In contrast to the management proposed earlier in this study, obligatory realization of some minimum volume of assets on the given time period is demanded. Examples of real-world markets trade demonstrating the imposed constraints effect on the weighted average price values obtained within the constructed management are given.

Correlation technique using of-the-shelf CW DFB laser for interrogation of fiber optic sensors

The paper presents a simple fiber-optic sensor technique based on methods of correlation optical time domain reflectometry. A correlation reflectometry technique can measure distribution of reflection coefficient along the optical fiber by calculating the correlation function between a probe optical signal (reference) and the signal returned back due to reflections or/and back-scattering from the fiber under the test. To obtain the best sensor performance, the probe signal power should be a truly random function of time. As an optical source we use a free-running DFB laser diode operating in a continuous wave regime without any external modulation. To generate the probe test signal, laser light is passed through an interferometer with an optical path difference much longer than the coherence length of the laser light. The light intensity at the interferometer output has a truly random fluctuations and its auto correlation function is suitable for correlation optical reflectometry. We present results of experimental verification of the techniques in different sensor configurations. Multipoint sensor using very low reflective fiber Bragg gratings with reflectivity of 0.01% printed in a long SMF-28 optical fiber was demonstrated.

Использование комплекса фотовидеофиксации нарушений правил дорожного движения для выделения детерминированной и стохастической составляющих интенсивности транспортного потока

The relevance of the manuscript is due to the need to process and analyze the information accumulated by the complexes of photo-video recording of traffic violations, which will further develop mathematical, computational and simulation models of road transport, solve problems of optimization and management of traffic flows, make management decisions to reduce the number of congestion and reduce the anthropogenic load on the environment. The object of the study is a part of a three-lane road with heavy one-way traffic, equipped with a software and technical complex that allows measuring the main characteristics of the traffic flow (vehicle speeds, including the average values on the controlled road part, driving time, etc.). The subject of the study is the traffic flow intensity during a 7-day time (from Monday to Sunday). The analysis of the obtained dependences allowed us to formulate a hypothesis about the presence of determin- istic and stochastic components in the traffic flow intensity, which is a random function of time, and the verification of which is the purpose of this study. Statistical processing of the obtained data is used as a theoretical and methodological approach, as well as the assumption that the traffic flow intensity can be represented by the sum of deterministic and stochastic components. The developed approach using the smoothing procedure allowed us to select both components, and this is a scientific novelty of the analysis performed. As a result of the study, it is shown that the deterministic component of the traffic flow intensity for working days is qualitatively different from the deterministic component for weekends. Statistical indicators of probabilistic distributions of traffic flow intensities and random components selected from them are determined. Estimates of the correspondence of the selected curves to the normal law of probability distribution are obtained using the Kolmogorov and Pearson criteria, which contradict each other. Practical significance consists in the use of a deterministic component for predicting traffic flows, controlling the operation of traffic lights, monitoring the operation of equipment, as well as in the reconstruction, design and construction of roads and road objects. The direction of further research is to obtain, statistically process and generalize data on the traffic flows intensity in other parts of the road network.

Exact solution to the first-passage problem for a particle with a dichotomous diffusion coefficient.

We consider the problem of first-passage time for reaching a boundary of a particle which diffuses in one dimension and is confined to the region x∈(0,L), with a diffusion coefficient that switches randomly between two states, having diffusivities that are different. Exact analytical expressions are found for the survival probability of the particle as a function of time. The survival probability has a multiexponential decay, and to characterize it, we use the average rate constant k, as well as the instantaneous rate r(t). Our approach can easily be extended to the case where the diffusion coefficient takes n different values. The model should be of interest to biological processes, in which a reactant searches for a target in a heterogeneous environment, making the diffusion coefficient a random function of time. The best example for this is a protein searching for a target site on the DNA.

An Approximate Method of Constructing a Switching Regression with Unknown Switch Points

The method is applicable to regressions whose variables are time series. The estimation method is based on the fact that these series are considered as observed values of continuous random functions of time. This property makes it possible to solve the estimation problem using gradient methods for optimization problems. Examples of using the proposed method are given.

The balance of the diamond-shaped plow in the horizontal plane

Diamond-shaped plow, opening the furrow with an inclined wall, reduces the area of the side support, which gives rise to the problem of maintaining a stable course of the plow. At the same time, the wider groove of the diamond-shaped case allows the energy-saturated tractor to move the right wheels on it and thereby shift the line of the plow thrust to the right, which in turn unloads the side support and reduces the traction resistance of the plow. Replacing field boards on all cases with a common field board installed on the last case changes the location of the balancing reaction of the furrow wall and its size and, as a result the plow’s traction resistance also changes. In the vertical plane of the projections, this corresponds to the displacement of the friction force of the field board along the line of action. In a diamond case, due to a different shape of the cut layer, the point of application of the resultant soil pressure on it is shifted in a horizontal plane closer to the toe of the plowshare compared to a conventional case. The studied dia-mond-shaped plow is hung on the tractor in a two-point scheme with the passage of the horizontal projection of the central rod through the point of attachment of the lower rods to the tractor. This point is the in-stantaneous center of rotation of the plow in the horizontal plane. When considering the plow scheme with field boards, identical on all cases (production), we will conditionally present it with one field board on the middle case, which facilitates the theoretical analysis. The continuous change in time of the forces acting on the working and additional bodies is a specific feature of the working conditions of agricultural units. On this basis, these efforts are considered as stationary random functions of time, and for studying dynamic objects under their influence, they use a statistical method based on the concepts of the theory of stationary random processes.

A new method for reconstruction of solar irradiance

The purpose of this research is to show how time series should be reconstructed using an example with the data on total solar irradiation (TSI) of the Earth and on sunspot numbers (SSN) since 1749. The traditional approach through regression equation(s) is designed for time-invariant vectors of random variables and is not applicable to time series, which present random functions of time. The autoregressive reconstruction (ARR) method suggested here requires fitting a multivariate stochastic difference equation to the target/proxy time series. The reconstruction is done through the scalar equation for the target time series with the white noise term excluded. The time series approach is shown to provide a better reconstruction of TSI than the correlation/regression method. A reconstruction criterion is introduced which allows one to define in advance the achievable level of success in the reconstruction. The conclusion is that time series, including the total solar irradiance, cannot be reconstructed properly if the data are not treated as sample records of random processes and analyzed in both time and frequency domains.

PARAMETRIC IDENTIFICATION OF STOCHASTIC SYSTEM BY NON-GRADIENT RANDOM SEARCHING

At this moment we know a great variety of identification objects, tasks and methods and its significance is constantly increasing in various fields of science and technology. The identification problem is dependent on a priori information about identification object, besides that the existing approaches and methods of identification are determined by the form of mathematical models (deterministic, stochastic, frequency, temporal, spectral etc.). The paper considers a problem for determination of system parameters (identification object) which is assigned by the stochastic mathematical model including random functions of time. It has been shown that while making optimization of the stochastic systems subject to random actions deterministic methods can be applied only for a limited approximate optimization of the system by taking into account average random effects and fixed structure of the system. The paper proposes an algorithm for identification of parameters in a mathematical model of the stochastic system by non-gradient random searching. A specific feature of the algorithm is its applicability practically to mathematic models of any type because the applied algorithm does not depend on linearization and differentiability of functions included in the mathematical model of the system. The proposed algorithm ensures searching of an extremum for the specified quality criteria in terms of external uncertainties and limitations while using random searching of parameters for a mathematical model of the system. The paper presents results of the investigations on operational capability of the considered identification method while using mathematical simulation of hypothetical control system with a priori unknown parameter values of the mathematical model. The presented results of the mathematical simulation obviously demonstrate the operational capability of the proposed identification method.

Diffusing Diffusivity: Survival in a Crowded Rearranging and Bounded Domain.

We consider a particle diffusing in a bounded, crowded, rearranging medium. The rearrangement happens on a time scale longer than the typical time scale of diffusion of the particle; as a result, effectively, the diffusion coefficient of the particle varies as a stochastic function of time. What is the probability that the particle will survive within the bounded region, given that it is absorbed the first time it hits the boundary of the region in which it diffuses? This question is of great interest in a variety of chemical and biological problems. If the diffusion coefficient is a constant, then analytical solutions for a variety of cases are available in the literature. However, there is no solution available for the case in which the diffusion coefficient is a random function of time. We discuss a class of models for which it is possible to find analytical solutions to the problem. We illustrate the method for a circular, two-dimensional region, but our methods are easy to apply to diffusion in arbitrary dimensions and spherical/rectangular regions. Our solution shows that if the dimension of the region is large, then only the average value of the diffusion coefficient determines the survival probability. However, for smaller-sized regions, one would be able to see the effects of the stochasticity of the diffusion coefficient. We also give generalizations of the results to N dimensions.

Analyses of dynamic response of a railway bogie

Dynamic behaviour of rail car body under operating conditions is a vital aspect for the railway coach maintenance department in identifying the effect of various surface and track irregularities. Harmonic and PSD response spectrum analyses of an Indian railway RCF bogie have been carried out using FE analysis in this paper. Major components of the RCF bogie are modelled using UNIGRAPHICSNX7.5 and have been translated into parasolid format, which is imported in ANSYS 12.1 finite element analysis package. Entire bogie is discretised using Solid92 tetrahedral elements. The primary and secondary suspension systems are modelled as Combination14 elements in ANSYS. Modal analysis is carried out to extract first few natural modes of vibration of the bogie. In harmonic analysis, sinusoidal excitation is fed as input to bottom wheel points to analyse the response of the bogie in terms of displacement at different salient locations. Vertical irregularities considered as a random function of time in the form of PSD have been given as base excitation input and the response spectrum analysis is carried out to identify the response in terms of PSD.

The stochastic mode of the Faraday instability of shallow fluid layers

The instability of a vertically oscillated layer of fluid is a classical problem whose history dates back to Faraday in the 19th century. We consider the stability of a shallow layer for which the oscillation, as expressed through the effective gravitational acceleration, is a random function of time. Using both theoretical linear stability analysis and high-resolution numerical simulations, including both individual realizations and ensemble calculations, of the nonlinear system of equations, we find that two different stochastic modes of instability exist. Both modes find their expression in finite amplitude oscillations of the free surface that exhibit sharp crests and broad troughs, or in other words, that resemble the classical Stokes wave. We demonstrate that a necessary condition for the first type of instability is snapshot, or instantaneous, instability. The subdominant instability resembles classical parametric resonance that can exist in a harmonically oscillated layer of fluid, and occurs even when the flow is always snapshot stable (or the gravitational acceleration is non-negative).

Critical phenomena far from equilibrium

Stationary regimes of active systems—those in which dissipation is compensated by pumping—are considered. Approaching the bifurcation point of such a regime leads to an increase in susceptibility, with soft modes making the dominant contribution. Weak noise, which is inherent to any real system, increases. Sufficiently close to bifurcation, the amplitude of random pulsations is comparable to the average value of the fluctuating quantity, as in the case of developed turbulence. The spectrum of critical pulsations is independent of the original noise. Numerical simulation of the neighborhood of a bifurcation point is considered unreliable because of the poor reproducibility of results. Due to the high susceptibility, calculation roundings result in ‘chaotic’ jumps of the solution in response to a smooth change in the parameters. It is therefore necessary in the simulation process to introduce a small random function of time, white noise. The solutions of the Langevin equations obtained in this way should be processed statistically. Their properties (except for the intensity of pulsations) are independent of the noise induced. Examples of the statistical description of bifurcations are given.