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We now review the main points of mean-field approximation (MFA) in its application to multicomponent stochastic reaction-diffusion systems.

We present the chemical reaction model under study — brusselator. We write the kinetic equations of reaction supplementing them with terms that describe the diffusion of the intermediate components and the fluctuations of the concentrations of the initial products. We simulate the fluctuations as random Gaussian homogeneous and spatially isotropic fields with zero means and spatial correlation functions with a non-trivial structure. The model parameter values correspond to a spatially-inhomogeneous ordered state in the deterministic case.

In the MFA we derive single-site two-dimensional nonlinear self-consistent Fokker–Planck equation in the Stratonovich's interpretation for spatially extended stochastic brusselator, which describes the dynamics of probability distribution density of component concentration values of the system under consideration. We find the noise intensity values appropriate to two types of Fokker–Planck equation solutions: solution with transient bimodality and solution with the multiple alternation of unimodal and bimodal types of probability density. We study numerically the probability density dynamics and time behavior of variances, expectations, and most probable values of component concentrations at various noise intensity values and the bifurcation parameter in the specified region of the problem parameters.

Beginning from some value of external noise intensity inside the ordered phase disorder originates existing for a finite time, and the higher the noise level, the longer this disorder “embryo” lives. The farther away from the bifurcation point, the lower the noise that generates it and the narrower the range of noise intensity values at which the system evolves to the ordered, but already a new statistically steady state. At some second noise intensity value the intermittency of the ordered and disordered phases occurs. The increasing noise intensity leads to the fact that the order and disorder alternate increasingly.

Thus, the scenario of the noise induced order–disorder transition in the system under study consists in the intermittency of the ordered and disordered phases.

The currently performed mathematical and computer modeling of thermal processes in technical systems is based on an assumption that all the parameters determining thermal processes are fully and unambiguously known and identified (i.e., determined). Meanwhile, experience has shown that parameters determining the thermal processes are of undefined interval-stochastic character, which in turn is responsible for the intervalstochastic nature of thermal processes in the electronic system. This means that the actual temperature values of each element in an technical system will be randomly distributed within their variation intervals. Therefore, the determinative approach to modeling of thermal processes that yields specific values of element temperatures does not allow one to adequately calculate temperature distribution in electronic systems. The interval-stochastic nature of the parameters determining the thermal processes depends on three groups of factors: (a) statistical technological variation of parameters of the elements when manufacturing and assembling the system; (b) the random nature of the factors caused by functioning of an technical system (fluctuations in current and voltage; power, temperatures, and flow rates of the cooling fluid and the medium inside the system); and (c) the randomness of ambient parameters (temperature, pressure, and flow rate). The interval-stochastic indeterminacy of the determinative factors in technical systems is irremediable; neglecting it causes errors when designing electronic systems. A method that allows modeling of unsteady interval-stochastic thermal processes in technical systems (including those upon interval indeterminacy of the determinative parameters) is developed in this paper. The method is based on obtaining and further solving equations for the unsteady statistical measures (mathematical expectations, variances and covariances) of the temperature distribution in an technical system at given variation intervals and the statistical measures of the determinative parameters. Application of the elaborated method to modeling of the interval-stochastic thermal process in a particular electronic system is considered.

The effectiveness of communication and data transmission systems (CSiPS), which are an integral part of modern systems in almost any field of science and technology, largely depends on the stability of the frequency of the generated signals. The signals generated in the CSiPD can be considered as processes, the frequency of which changes under the influence of a combination of external influences. Changing the frequency of the signals leads to a decrease in the signal-tonoise ratio (SNR) and, consequently, a deterioration in the characteristics of the signal-to-noise ratio, such as the probability of a bit error and bandwidth. It is most convenient to consider the description of such changes in the frequency of signals as random processes, the apparatus of which is widely used in the construction of mathematical models describing the functioning of systems and devices in various fields of science and technology. Moreover, in many cases, the characteristics of a random process, such as the distribution law, mathematical expectation, and variance, may be unknown or known with errors that do not allow us to obtain estimates of the signal parameters that are acceptable in accuracy. The article proposes an algorithm for solving the problem of determining the characteristics of a random process (signal frequency) based on a set of samples of its frequency, allowing to determine the sample mean, sample variance and the distribution law of frequency deviations in the general population. The basis of this algorithm is the comparison of the values of the observed random process measured over a certain time interval with a set of the same number of random values formed on the basis of model distribution laws. Distribution laws based on mathematical models of these systems and devices or corresponding to similar systems and devices can be considered as model distribution laws. When forming a set of random values for the accepted model distribution law, the sample mean value and variance obtained from the measurement results of the observed random process are used as mathematical expectation and variance. The feature of the algorithm is to compare the measured values of the observed random process ordered in ascending or descending order and the generated sets of values in accordance with the accepted models of distribution laws. The results of mathematical modeling illustrating the application of this algorithm are presented.

We consider a portfolio with call option and the corresponding underlying asset under the standard assumption that stock-market price represents a random variable with lognormal distribution. Minimizing the variance hedging risk of the portfolio on the date of maturity of the call option we find a fraction of the asset per unit call option. As a direct consequence we derive the statistically fair lookback call option price in explicit form. In contrast to the famous Black–Scholes theory, any portfolio cannot be regarded as  risk-free because no additional transactions are supposed to be conducted over the life of the contract, but the sequence of independent portfolios will reduce risk to zero asymptotically. This property is illustrated in the experimental section using a dataset of daily stock prices of 37 leading US-based companies for the period from April 2006 to January 2013.