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Stochastic process model. Construction of a stochastic model Types of mathematical models

Stochastic process model. Construction of a stochastic model Types of mathematical models

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1. An example of constructing a stochastic process model

In the process of functioning of a bank, very often the need arises to solve the problem of choosing a vector of assets, i.e. investment portfolio of the bank, and the uncertain parameters that must be taken into account in this task are primarily associated with the uncertainty of asset prices (securities, real investments, etc.). As an illustration, we can give an example with the formation of a portfolio of government short-term liabilities.

For problems of this class, the fundamental question is the construction of a model of the stochastic process of price changes, since at the disposal of the operation researcher, naturally, there is only a finite series of observations of realizations of random variables - prices. Next, we outline one of the approaches to solving this problem, which is being developed at the Computing Center of the Russian Academy of Sciences in connection with solving problems of control of stochastic Markov processes.

Are being considered M types of securities, i=1,… , M, which are traded at special exchange sessions. Securities are characterized by values - yields expressed as a percentage during the current session. If a security of the type at the end of the session is bought at a price and sold at the end of a session at a price, then.

Yields are random variables formed as follows. It is assumed that there are basic returns - random variables that form a Markov process and are determined by the following formula:

Here, are constants, and are standard normally distributed random variables (i.e. with zero mathematical expectation and unit variance).

where is some scale factor equal to (), and - random value, which has the meaning of deviation from the base value and is defined similarly:

where are also standard normally distributed random variables.

It is assumed that some operating party, hereinafter called the operator, manages its capital invested in securities (at any moment in exactly one type of security), selling them at the end of the current session and immediately buying other securities with the proceeds. The management and selection of purchased securities is carried out according to an algorithm that depends on the operator’s awareness of the process that forms the yield of the securities. We will consider various hypotheses about this awareness and, accordingly, various control algorithms. We will assume that the operation researcher develops and optimizes the control algorithm using the available series of observations of the process, i.e., using information about closing prices at exchange sessions, and also, possibly, about values over a certain period of time corresponding to the sessions with numbers. The purpose of the experiments is to compare estimates of the expected efficiency of various control algorithms with their theoretical mathematical expectation in conditions where the algorithms are configured and evaluated on the same series of observations. To estimate the theoretical mathematical expectation, the Monte Carlo method is used by “running” the control over a sufficiently voluminous generated series, i.e. according to a matrix of dimensions, where the columns correspond to realizations of values and by sessions, and the number is determined by computing capabilities, but provided that there are at least 10,000 elements of the matrix. It is necessary that the “polygon” be the same in all experiments performed. The existing series of observations is simulated by a generated dimensional matrix, where the values in the cells have the same meaning as above. The number and values in this matrix will vary further. Matrices of both types are formed through the procedure of generating random numbers, simulating the implementation of random variables, and calculating the required matrix elements using these implementations and formulas (1) - (3).

Management efficiency assessment for a number of observations is made using the formula

where is the index of the last session in the series of observations, and is the number of bonds selected by the algorithm at the step, i.e. the type of bonds in which, according to the algorithm, the operator’s capital will be held during the session. In addition, we will also calculate monthly efficiency. The number 22 approximately corresponds to the number of trading sessions per month.

Computational experiments and analysis of results

Hypotheses

Accurate knowledge by the operator of future profitability.

The index is selected as. This option gives an upper bound for all possible control algorithms, even if Additional Information(taking into account some additional factors) will make it possible to clarify the price forecast model.

Random control.

The operator does not know the law of pricing and carries out transactions at random. Theoretically, in this model, the mathematical expectation of the result of operations coincides with the same as if the operator invested capital not in one security, but in all equally. With zero mathematical expectations of values, the mathematical expectation of a value is equal to 1. Calculations based on this hypothesis are useful only in the sense that they allow, to some extent, to control the correctness of the written programs and the generated matrix of values.

Management with exact knowledge of the profitability model, all its parameters and observable values .

In this case, the operator at the end of the session, knowing the values for both sessions, and, and in our calculations, using rows and matrices, calculates mathematical expectations of values using formulas (1) - (3) and selects for purchase the paper with the largest of these values of quantities.

where, according to (2), . (6)

Management with knowledge of the structure of the return model and the observed value , but unknown coefficients .

We will assume that the researcher of the operation not only does not know the values of the coefficients, but also does not know the number of quantities influencing the formation, the previous values of these parameters (memory depth of Markov processes). He also does not know whether the coefficients are the same or different different meanings. Let's consider different options for the researcher's actions - 4.1, 4.2, and 4.3, where the second index denotes the researcher's assumption about the memory depth of the processes (the same for and). For example, in case 4.3, the researcher assumes that it is formed according to the equation

A dummy term has been added here for completeness. However, this term can be excluded either from substantive considerations or by statistical methods. Therefore, to simplify the calculations, we further exclude free terms when setting parameters from consideration and formula (7) takes the form:

Depending on whether the researcher assumes the coefficients to be the same or different for different values, we will consider subcases 4.m. 1 - 4.m. 2, m = 1 - 3. In cases 4.m. 1 coefficients will be adjusted based on the observed values for all securities together. In cases 4.m. 2, the coefficients are adjusted for each paper separately, while the researcher works under the hypothesis that the coefficients are different for different ones, for example, in case 4.2.2. values are determined by the modified formula (3)

First setup method- classic method least squares. Let's consider it using the example of setting the coefficients in options 4.3.

According to formula (8),

It is required to find such values of the coefficients as to minimize the sample variance for realizations on well-known series observations, array, provided that the mathematical expectation of the values is determined by formula (9).

Here and in what follows, the sign “” indicates the implementation of a random variable.

Minimum quadratic form(10) is achieved at a single point at which all partial derivatives are equal to zero. From here we obtain a system of three algebraic linear equations:

the solution of which gives the required values of the coefficients.

After the coefficients are verified, the selection of controls is carried out in the same way as in case 3.

Comment. In order to facilitate the work on programs, it is customary to immediately write the control selection procedure described for Hypothesis 3, focusing not on formula (5), but on its modified version in the form

In this case, in the calculations for cases 4.1.m and 4.2.m, m = 1, 2, the extra coefficients are reset to zero.

Second setup method consists in choosing parameter values so as to maximize the estimate from formula (4). This problem is analytically and computationally hopelessly complex. Therefore, here we can only talk about techniques for some improvement in the value of the criterion relative to the starting point. You can take the values obtained using the least squares method as a starting point, and then calculate around these values on a grid. In this case, the sequence of actions is as follows. First, the grid is calculated using parameters (square or cube) with other parameters fixed. Then for cases 4.m. 1, the grid is calculated using the parameters, and for cases 4.m. 2 on parameters with other parameters fixed. In the case of 4.m. 2, then the parameters are also optimized. When all parameters are exhausted by this process, the process is repeated. Repetitions are carried out until the new cycle provides an improvement in the criterion values compared to the previous one. To prevent the number of iterations from being too large, we apply the following technique. Inside each block of calculations on a 2- or 3-dimensional parameter space, a fairly coarse grid is first taken, then, if the best point is on the edge of the grid, then the square (cube) under study is shifted and the calculation is repeated, if the best point is internal, then a new mesh is built around this point with a smaller step, but with the same total number points, and so on a certain but reasonable number of times.

Control under the unobservable and without taking into account the dependence between the yields of different securities.

This means that the transaction researcher does not notice the dependence between different securities, knows nothing about the existence and tries to predict the behavior of each security separately. Let us consider, as usual, three cases when the researcher models the process of generating returns in the form of a Markov process of depth 1, 2, and 3:

The coefficients for forecasting the expected profitability are not important, and the coefficients are adjusted in two ways, described in paragraph 4. Controls are selected in the same way as was done above.

Note: Just like for control selection, for the least squares method it makes sense to write a single procedure with a maximum number of variables - 3. If the custom variables are, say, then for from the solution linear system a formula is written out, which includes only constants, defined through, and through and. In cases where there are fewer than three variables, the values of the extra variables are reset to zero.

Although calculations in various options are carried out in a similar way, the number of options is quite large. When preparing tools for calculations in all of the above options turns out to be difficult, the issue of reducing their number is considered at the expert level.

Control under the unobservable taking into account the dependence between the yields of different securities.

This series of experiments simulates the manipulations that were performed in the GKO task. We assume that the researcher knows practically nothing about the mechanism by which returns are formed. He only has a series of observations, a matrix. For substantive reasons, he makes an assumption about the interdependence of the current yields of different securities, grouped around a certain basic yield, determined by the state of the market as a whole. Considering the graphs of security yields from session to session, he makes the assumption that at each moment in time the points whose coordinates are the security numbers and yields (in reality, these were the maturities of the securities and their prices) are grouped near a certain curve (in the case of GKOs - parabolas).

Here is the point of intersection of the theoretical straight line with the y-axis (basic profitability), and is its slope (what should be equal to 0.05).

Having constructed theoretical straight lines in this way, the operation researcher can calculate values - deviations of quantities from their theoretical values.

(Note that here they have a slightly different meaning than in formula (2). There is no dimensional coefficient, and deviations are considered not from the base value, but from the theoretical straight line.)

The next task is to predict values based on the values known at the moment, . Because the

to predict values, the researcher needs to introduce a hypothesis about the formation of values, and. Using the matrix, the researcher can establish a significant correlation between the quantities and. One can accept the hypothesis about linear dependence between values from: . For substantive reasons, the coefficient is immediately set to zero, and is found using the least squares method in the form:

Further, as above, they are modeled using a Markov process and described by formulas similar to (1) and (3) with a different number of variables depending on the memory depth of the Markov process in the variant under consideration. (here determined not by formula (2), but by formula (16))

Finally, as above, two methods of setting parameters using the least squares method are implemented, and estimates are made by directly maximizing the criterion.

Experiments

For all described options, criteria estimates were calculated using different matrices. (matrices with the number of rows 1003, 503, 103 and for each dimension option about one hundred matrices were implemented). Based on the calculation results for each dimension, the mathematical expectation and dispersion of the values, and their deviation from the values, were estimated for each of the prepared options.

As the first series of computational experiments showed with a small number of adjustable parameters (about 4), the choice of the adjustment method does not have a significant impact on the value of the criterion in the problem.

2. Classification of modeling tools

stochastic simulation bank algorithm

Classification of modeling methods and models can be carried out according to the degree of detail of the models, the nature of the features, the scope of application, etc.

Let's consider one of the common classifications of models according to modeling tools; this aspect is the most important when analyzing various phenomena and systems.

material in the case when the research is carried out on models, the connection of which with the object under study exists objectively and is of a material nature. In this case, models are built by the researcher or selected from the surrounding world.

Based on modeling tools, modeling methods are divided into two groups: material methods and ideal modeling methods. Modeling is called material in the case when the research is carried out on models, the connection of which with the object under study exists objectively and is of a material nature. In this case, models are built by the researcher or selected from the surrounding world. In turn, in material modeling we can distinguish: spatial, physical and analog modeling.

In spatial modeling models are used that are designed to reproduce or display the spatial properties of the object being studied. The models in this case are geometrically similar to the objects of study (any layouts).

Models used in physical modeling are designed to reproduce the dynamics of processes occurring in the object being studied. Moreover, the commonality of processes in the object of study and the model is based on the similarity of their physical nature. This modeling method is widely used in engineering when designing technical systems. various types. For example, the study of aircraft based on wind tunnel experiments.

Analog modeling is associated with the use of material models that have different physical nature, but described by the same mathematical relationships as the object being studied. It is based on an analogy in the mathematical description of the model and the object (the study mechanical vibrations using an electrical system described by the same differential equations, but more convenient in conducting experiments).

In all cases of material modeling, the model is a material reflection of the original object, and the research consists of a material impact on the model, that is, an experiment with the model. Material modeling by its nature is an experimental method and is not used in economic research.

Fundamentally different from material modeling perfect modeling, based on an ideal, conceivable connection between an object and a model. Ideal modeling methods are widely used in economic research. They can be divided into two groups: formalized and informal.

IN formalized In modeling, the model is a system of signs or images, along with which the rules for their transformation and interpretation are specified. If sign systems are used as models, then modeling is called iconic(drawings, graphs, diagrams, formulas).

An important type of sign modeling is math modeling, based on the fact that various objects and phenomena under study can have the same mathematical description in the form of a set of formulas, equations, the transformation of which is carried out on the basis of the rules of logic and mathematics.

Another form of formalized modeling is figurative, in which models are built on visual elements (elastic balls, fluid flows, trajectories of bodies). The analysis of figurative models is carried out mentally, so they can be classified as formalized modeling, when the rules for the interaction of objects used in the model are clearly fixed (for example, in ideal gas the collision of two molecules is considered as a collision of balls, and the result of the collision is thought of by everyone in the same way). Models of this type are widely used in physics; they are commonly called “thought experiments.”

Unformalized modeling. This includes such an analysis of problems of various types, when a model is not formed, and instead of it, some precisely not fixed mental representation of reality is used, which serves as the basis for reasoning and decision-making. Thus, any reasoning that does not use a formal model can be considered unformalized modeling, when a thinking individual has some image of the object of study, which can be interpreted as an unformalized model of reality.

For a long time, the study of economic objects was carried out only on the basis of such vague ideas. Currently, the analysis of informal models remains the most common means of economic modeling, namely, every person making an economic decision without using mathematical models forced to be guided by one or another description of the situation based on experience and intuition.

The main disadvantage of this approach is that the solutions may be ineffective or erroneous. For a long time, apparently, these methods will remain the main means of decision-making not only in most everyday situations, but also when making decisions in the economy.

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Stochastic differential equations

One of the objectives of the dissertation is the problem of writing a stochastic differential equation for a system so that the stochastic term is related to the structure of the system under study. One possible solution to this problem is to obtain the stochastic and deterministic parts from the same equation. For these purposes, it is convenient to use the basic kinetic equation, which can be approximated by the Fokker-Planck equation, for which, in turn, the equivalent stochastic differential equation can be written in the form of the Langevin equation.

Section 1.4. contains the basic information necessary to indicate the connection between the stochastic differential equation and the Fokker-Planck equation, as well as the basic concepts of stochastic calculus.

The second chapter provides basic information from the theory of random processes and, based on this theory, formulates a method for modeling one-step processes.

Section 2.1 provides basic information from the theory of random one-step processes.

One-step processes are understood as continuous-time Markov processes taking values in the range of integers, the transition matrix of which allows only transitions between adjacent sections.

We consider a multidimensional one-step process X() = (i(),2(), ...,n()) = ( j(), = 1, ) , (0.1) varying along the segment, i.e. Є, where is the length of the time interval in which process X() is specified. The set G = (x, = 1, Є NQ x NQ1 is a set of discrete values that a random process can take.

For a given one-step process, the probabilities of transitions per unit time s+ and s from the state Xj to the state Xj__i and Xj_i are introduced, respectively. It is believed that the probability of transition from state x to two or more steps per unit time is very small. Therefore, we can say that the vector Xj of the state of the system changes in steps of length Г( and then, instead of transitions from x to Xj+i and Xj_i, we can consider transitions from X to X + Гі and X - Гі, respectively.

When modeling systems in which time evolution occurs as a result of the interaction of system elements, it is convenient to describe it using the main kinetic equation (another name is the control equation, and in the English literature it is called the Master equation).

Next, the question arises of how to obtain a description of the system under study, described by one-step processes, using a stochastic differential equation in the form of the Langevin equation from the basic kinetic equation. Formally, only equations containing stochastic functions should be classified as stochastic equations. Thus, only Langevin's equations satisfy this definition. However, they are related directly to other equations, namely the Fokker-Planck equation and the fundamental kinetic equation. Therefore, it seems logical to consider all these equations together. Therefore, to solve this problem, it is proposed to approximate the main kinetic equation by the Fokker-Planck equation, for which we can write an equivalent stochastic differential equation in the form of the Langevin equation.

Section 2.2 formulates a method for describing and stochastic modeling of systems described by multidimensional one-step processes.

In addition, it is shown that the coefficients for the Fokker-Planck equation can be obtained immediately after recording the interaction scheme for the system under study, the state change vector r and expressions for the transition probabilities s+ and s-, i.e. at practical application With this method there is no need to write down the basic kinetic equation.

In section 2.3. the Runge-Kutta method for numerical solution stochastic differential equations, which is used in the third chapter to illustrate the results obtained.

The third chapter provides an illustration of the application of the method of constructing stochastic models described in the second chapter, using the example of systems that describe the growth dynamics of interacting populations, such as “predator-prey”, symbiosis, competition and their modifications. The goal is to write them in the form of stochastic differential equations and to study the effect of introducing stochastics on the behavior of the system.

In section 3.1. The application of the method described in the second chapter is illustrated using the example of the “predator-prey” model. Systems with the interaction of two types of populations of the “predator-prey” type have been widely studied, which makes it possible to compare the results obtained with already well-known ones.

Analysis of the resulting equations showed that to study the deterministic behavior of the system, it is possible to use the drift vector A of the resulting stochastic differential equation, i.e. The developed method can be used to analyze both stochastic and deterministic behavior. In addition, it was concluded that stochastic models provide a more realistic description of the behavior of the system. In particular, for the “predator-prey” system in the deterministic case, the solutions to the equations have a periodic form and the phase volume is preserved, while the introduction of stochastics into the model gives a monotonic increase in the phase volume, which indicates the inevitable death of one or both populations. In order to visualize the results obtained, numerical simulation was carried out.

In section 3.2. The developed method is used to obtain and analyze various stochastic models of population dynamics, such as the “predator-prey” model taking into account interspecific competition among prey, symbiosis, competition and the interaction model of three populations.

Information on stochastic calculus

The development of the theory of random processes led to the transition to research natural phenomena from deterministic concepts and models of population dynamics to probabilistic ones and, as a consequence, the emergence large number works devoted to stochastic modeling in mathematical biology, chemistry, economics, etc.

When considering deterministic population models, such important points as the random influence of various factors on the evolution of the system remain uncovered. When describing population dynamics, one should take into account the random nature of reproduction and survival of individuals, as well as random fluctuations that occur in the environment over time and lead to random fluctuations in system parameters. Therefore, probabilistic mechanisms reflecting these points should be introduced into any model of population dynamics.

Stochastic modeling allows a more complete description of changes in population characteristics, taking into account both all deterministic factors and random effects that can significantly change the conclusions from deterministic models. On the other hand, with their help it is possible to identify qualitatively new aspects of population behavior.

Stochastic models of changes in population states can be described using random processes. Under certain assumptions, we can assume that the behavior of a population given its present state does not depend on how this state was achieved (i.e., with a fixed present, the future does not depend on the past). That. To model population dynamics processes, it is convenient to use Markov birth-death processes and the corresponding control equations, which are described in detail in the second part of the work.

N. N. Kalinkin in his works uses interaction schemes to illustrate the processes occurring in systems with interacting elements and, on the basis of these schemes, builds models of these systems using the apparatus of branching Markov processes. The application of this approach is illustrated by the example of modeling processes in chemical, population, telecommunication and other systems.

The work examines probabilistic population models, for the construction of which the apparatus of birth-death processes is used, and the resulting systems of differential-difference equations represent dynamic equations for random processes. The paper also discusses methods for finding solutions to these equations.

You can find many articles devoted to the construction of stochastic models taking into account various factors influencing the dynamics of changes in population numbers. For example, in the articles a model of the population dynamics of a biological community in which individuals consume food resources containing harmful substances. And in the model of population evolution, the article takes into account the factor of settlement of representatives of populations in their habitats. The model is a system of self-consistent Vlasov equations.

It is worth noting the works that are devoted to the theory of fluctuations and the application of stochastic methods in the natural sciences, such as physics, chemistry, biology, etc. In particular, the mathematical model of changes in the number of populations interacting according to the “predator-prey” type is built on the basis of multidimensional Markov birth-death processes.

One can consider the “predator-prey” model as the implementation of birth-death processes. In this interpretation, it is possible to use them for modes in many fields of science. In the 70s, M. Doi proposed a method for studying such models based on creation–annihilation operators (by analogy with secondary quantization). Works can be noted here. In addition, this method is now actively being developed in the group of M. M. Gnatich.

Another approach to modeling and studying models of population dynamics is associated with the theory of optimal control. Works can be noted here.

It can be noted that most of the works devoted to the construction of stochastic models of population processes use the apparatus of random processes to obtain differential-difference equations and subsequent numerical implementation. In addition, stochastic differential equations in the Langevin form are widely used, in which a stochastic term is added from general considerations about the behavior of the system and is intended to describe random influences environment. Further study of the model is their qualitative analysis or finding solutions using numerical methods.

Stochastic Differential Equations Definition 1. A stochastic differential equation is a differential equation in which one or more terms represent a stochastic process. Most used and good famous example stochastic differential equation (SDE) is an equation with a term that describes White noise and it can be considered as a Wiener process Wt, t 0.

Stochastic differential equations are an important and widely used mathematical apparatus when studying and modeling dynamic systems that are subject to various random disturbances.

The beginning of stochastic modeling of natural phenomena is considered to be the description of the phenomenon of Brownian motion, which was discovered by R. Brown in 1827, when he conducted research on the movement of plant pollen in a liquid. The first rigorous explanation of this phenomenon was given independently by A. Einstein and M. Smoluchowski. It is worth noting a collection of articles that contains the works of A. Einstein and M. Smoluchowski on Brownian motion. These studies made a significant contribution to the development of the theory of Brownian motion and its experimental verification. A. Einstein created the molecular kinetic theory for a quantitative description of Brownian motion. The resulting formulas were confirmed by the experiments of J. Perrin in 1908-1909.

A method for modeling multidimensional one-step processes.

There are two approaches to describe the evolution of systems with interacting elements - the construction of deterministic or stochastic models. Unlike deterministic models, stochastic models make it possible to take into account the probabilistic nature of the processes occurring in the systems under study, as well as the influences of the external environment that cause random fluctuations in the model parameters.

The subject of study are systems, the processes occurring in which can be described using one-step processes and those in which the transition of their state to another is associated with the interaction of system elements. An example would be models describing the growth dynamics of interacting populations, such as “predator-prey”, symbiosis, competition and their modifications. The goal is to write down SDEs for such systems and study the effect of introducing a stochastic part on the behavior of the solution to the equation describing deterministic behavior.

Chemical kinetics

The systems of equations that arise when describing systems with interacting elements are in many ways close to the systems of differential equations that describe the kinetics of chemical reactions. For example, the Lotka-Volterra system was originally developed by Lotka as a system describing some hypothetical chemical reaction, and only later was it developed by Volterra as a system describing the predator-prey model.

Chemical kinetics describes chemical reactions using so-called stoichiometric equations - equations reflecting the quantitative ratios of reagents and products chemical reaction and having the following general form: Where integersТі and Ш are called stoichiometric coefficients. This is a symbolic record of a chemical reaction in which thi molecules of the reagent Xi, ni2 molecules of the reagent Xh, ..., 3 molecules of the reagent Xp, upon entering into the reaction form n molecules of the substance Yi, n molecules of the substance I2, ..., nq molecules of the substance Yq, respectively .

In chemical kinetics, it is believed that a chemical reaction can only occur through direct interaction of reagents, and the rate of a chemical reaction is defined as the number of particles formed per unit time in a unit volume.

The main postulate of chemical kinetics is the law of mass action, which states that the rate of a chemical reaction is directly proportional to the product of the concentrations of reactants in powers of their stoichiometric coefficients. Therefore, if we denote by XI and y I the concentrations of the corresponding substances, then we have an equation for the rate of change in the concentration of a substance over time as a result of a chemical reaction:

Next, it is proposed to use the basic ideas of chemical kinetics to describe systems, the evolution in time of which occurs as a result of the interaction of elements of a given system with each other, introducing the following basic changes: 1. not reaction rates are considered, but transition probabilities; 2. it is proposed that the probability of transition from one state to another, which is a consequence of interaction, is proportional to the number possible interactions of this type; 3. to describe the system in this method, the basic kinetic equation is used; 4. deterministic equations are replaced by stochastic ones. A similar approach to describing such systems can be found in the works. To describe the processes occurring in the simulated system, it is proposed to use, as noted above, Markov one-step processes.

Consider a system consisting of types of different elements that can interact with each other different ways. Let us denote by an element of the -type, where = 1, and by the number of elements of the -type.

Let (), .

Let's make the assumption that the file consists of one part. Thus, in one step of interaction between a new node wanting to download a file and a node distributing the file, the new node downloads the entire file and becomes the distribution node.

Let is the designation of the new node, is the distributing node, and is the interaction coefficient. New nodes can come into the system with intensity, and distributing nodes can leave it with intensity. Then the interaction diagram and vector r will look like:

A stochastic differential equation in Langevin form can be obtained using the corresponding formula (1.15). Because drift vector A completely describes the deterministic behavior of the system; we can obtain a system of ordinary differential equations that describe the dynamics of the number of new clients and seeds:

Thus, depending on the choice of parameters singular point may have a different character. Thus, for /ZA 4/I2, the singular point is a stable focus, and for the opposite ratio, it is a stable node. In both cases, the singular point is stable, since the choice of coefficient values and changes in system variables can occur along one of two trajectories. If a singular point is a focus, then damped oscillations in the numbers of new and distributing nodes occur in the system (see Fig. 3.12). And in the nodal case, the approximation of numbers to stationary values occurs in a non-oscillation mode (see Fig. 3.13). Phase portraits of the system for each of the two cases are depicted, respectively, in graphs (3.14) and (3.15).

In the later chapters of this book, stochastic processes are almost always represented using linear differential systems driven by white noise. This representation of a stochastic process usually takes the following form. Let's pretend that

a - white noise. By choosing such a representation of the stochastic process V, it can be modeled. The use of such models can be justified as follows.

a) Stochastic phenomena associated with the influence of rapidly changing fluctuations on an inertial differential system are often encountered in nature. A typical example of white noise affecting a differential system is thermal noise in an electronic circuit.

b) As will be seen from what follows, in linear control theory only the average value and are almost always considered. covariance of a Stochastic process. For a linear model, it is always possible to approximate any experimentally obtained characteristics of the mean value and covariance matrix with arbitrary accuracy.

c) Sometimes the problem arises of modeling a stationary stochastic process with a known spectral energy density. In this case, it is always possible to generate a stochastic process as a process at the output of a linear differential system; in this case, the matrix of spectral densities of energy approximates with arbitrary accuracy the matrix of spectral densities of energy of the initial stochastic process.

Examples 1.36 and 1.37, as well as Problem 1.11, illustrate the modeling method.

Example 1.36. First order differential system

Suppose that the measured covariance function of a stochastic scalar process known to be stationary is described by the exponential function

This process can be modeled at as the state of a first-order differential system (see example 1.35)

where is intensity white noise - a stochastic quantity with zero mean and variance.

Example 1.37. Mixing tank

Consider the mixing tank from Example 1.31 (Section 1.10.3) and calculate the output dispersion matrix for it variable example 1.31 it was assumed that concentration fluctuations in fluxes are described by exponentially correlated noise and thus can be modeled as a solution to a first-order system driven by white noise. Let's now add to differential equation mixing tank equation models stochastic processes We get

Here is the scalar white noise intensity so that

to obtain the variance of the process equal, let us assume that for the process we use a similar model. Thus, we obtain a system of equations

The stochastic model describes a situation where there is uncertainty. In other words, the process is characterized by some degree of randomness. The adjective “stochastic” itself comes from the Greek word “to guess.” Because uncertainty is a key characteristic Everyday life, then such a model can describe anything.

However, each time we use it, we will get a different result. Therefore, deterministic models are more often used. Although they are not as close as possible to the real state of affairs, they always give the same result and make it easier to understand the situation, simplify it by introducing a set of mathematical equations.

Main features

A stochastic model always includes one or more random variables. She strives to reflect real life in all its manifestations. Unlike stochastic, it does not have the goal of simplifying everything and reducing it to known values. Therefore, uncertainty is its key characteristic. Stochastic models are suitable for describing anything, but they all have the following common features:

Any stochastic model reflects all aspects of the problem it was created to study.
The outcome of each event is uncertain. Therefore, the model includes probabilities. The correctness of the overall results depends on the accuracy of their calculation.
These probabilities can be used to predict or describe the processes themselves.

Deterministic and stochastic models

For some, life appears to be a series of processes for others, in which the cause determines the effect. In fact, it is characterized by uncertainty, but not always and not in everything. Therefore, it is sometimes difficult to find clear differences between stochastic and deterministic models. Probabilities are a fairly subjective indicator.

For example, consider a coin toss situation. At first glance, it seems that the probability of landing “tails” is 50%. Therefore, a deterministic model must be used. However, in reality it turns out that a lot depends on the sleight of hand of the players and the perfection of balancing the coin. This means that you need to use a stochastic model. There are always parameters that we don't know. IN real life The cause always determines the effect, but there is also a certain degree of uncertainty. The choice between using deterministic and stochastic models depends on what we are willing to sacrifice - ease of analysis or realism.

In chaos theory

IN Lately the concept of which model is called stochastic has become even more blurred. This is due to the development of the so-called chaos theory. It describes deterministic models that can produce different results with slight changes in the initial parameters. This is like an introduction to uncertainty calculation. Many scientists even admitted that this is already a stochastic model.

Lothar Breuer explained everything gracefully with poetic imagery. He wrote: “A mountain stream, a beating heart, an epidemic of smallpox, a column of rising smoke - all this is an example of a dynamic phenomenon that sometimes seems to be characterized by chance. In reality, such processes are always subject to a certain order, which scientists and engineers are only just beginning to understand. This is the so-called deterministic chaos." New theory sounds very plausible, which is why many modern scientists are its supporters. However, it still remains poorly developed and is quite difficult to apply in statistical calculations. Therefore, stochastic or deterministic models are often used.

Construction

Stochastic begins with the choice of the space of elementary outcomes. This is what statistics call a list of possible results of the process or event being studied. The researcher then determines the probability of each of the elementary outcomes. This is usually done based on a specific methodology.

However, probabilities are still a rather subjective parameter. The researcher then determines which events seem most interesting to solve the problem. After that, he simply determines their probability.

Example

Let's consider the process of constructing the simplest stochastic model. Let's say we're rolling a dice. If “six” or “one” comes up, our winnings will be ten dollars. The process of building a stochastic model in this case will look like this:

Let us define the space of elementary outcomes. The die has six sides, so the rolls can be “one”, “two”, “three”, “four”, “five” and “six”.
The probability of each outcome will be 1/6, no matter how many times we roll the dice.
Now we need to determine the outcomes we are interested in. This is the fall of the edge with the number “six” or “one”.
Finally, we can determine the probability of the event we are interested in. It is 1/3. We sum up the probabilities of both elementary events of interest to us: 1/6 + 1/6 = 2/6 = 1/3.

Concept and result

Stochastic modeling is often used in gambling. But it is also indispensable in economic forecasting, since it allows us to understand the situation more deeply than deterministic ones. Stochastic models in economics are often used when making investment decisions. They allow you to make assumptions about the profitability of investments in certain assets or groups of assets.

Modeling makes financial planning more effective. With its help, investors and traders optimize the allocation of their assets. Using stochastic modeling always has benefits in the long run. In some industries, refusal or inability to apply it can even lead to bankruptcy of the enterprise. This is due to the fact that in real life, new important parameters appear every day, and if they do not exist, they can have catastrophic consequences.

As the name suggests, this type of model is focused on describing systems that exhibit statistically regular random behavior, and time in them can be considered as discrete quantity. The essence of time discretization is the same as in discrete-deterministic models. Models of systems of this kind can be built on the basis of two formalized description schemes. Firstly, these are finite-difference equations, among the variables of which functions that define random processes are used. Secondly, they use probabilistic automata.

An example of constructing a discrete-stochastic system. Let there be some production system, the structure of which is shown in Fig. 3.8. Within this system, a homogeneous material flow moves through the stages of storage and production.

Suppose, for example, that a raw material flow consists of metal ingots that are stored in an incoming warehouse. Then these blanks go to production, where they are used to produce some kind of product. Finished products are stored in the output warehouse, from where they are taken for further actions with them (transferred to the next phases of production or for sale). In general, such a production system converts material flows of raw materials, materials and semi-finished products into a flow of finished products.

Let the time step in this production system be equal to one (D? = 1). We will take a change in the operation of this system as one. We assume that the process of manufacturing a product lasts one time step.

Rice. 3.8, Production system diagram

The production process is controlled by a special regulatory body, which is given a product production plan in the form of a target production intensity (the number of products that must be produced per unit of time, in this case per shift). Let us denote this intensity dt. In fact, this is the speed of production. Let d t =a+ bt, i.e. is linear function. This means that with each subsequent shift the plan increases by bt.

Since we are dealing with a homogeneous material flow, we believe that on average the volume of raw materials entering the system per unit of time, the volume of production per unit of time, the volume of finished products leaving the system per unit of time should be equal dt.

The input and output flows for the regulatory body are uncontrollable, their intensity (or speed - the number of ingots or products per unit of time, respectively, coming into the system and leaving it) must be equal dt. However, during transportation the blanks may be lost, or some of them will be of poor quality, or for some reason more of them will arrive than needed, etc. Therefore we will assume that input stream has intensity:

x tin =d t +ξ t in,

where ξ 1 in is a uniformly distributed random variable from -15 to +15.

Approximately the same processes can occur with the output stream. Therefore, the output flow has the following intensity:

x t in y x =d t +ξ t out,

where ξ tout is a normally distributed random variable with zero mathematical expectation and variance equal to 15.

We will assume that in the production process there are accidents associated with workers not showing up for work, machine breakdowns, etc. These randomnesses are described by a normally distributed random variable with zero mathematical expectation and variance equal to 15. Let us denote it ξ t/ The production process lasts a unit of time, during which it is removed from the input warehouse xt raw materials, then these raw materials are processed and transferred to the output warehouse in the same unit of time. The regulatory body receives information about the operation of the system in three possible ways (they are marked with numbers 1, 2, 3 in Fig. 3.8). We believe that for some reason these methods of obtaining information are mutually exclusive in the system.

Method 1. The regulatory body receives only information about the state of the input warehouse (for example, about changes in inventories in the warehouse or about deviations in the volume of inventories from their standard level) and uses it to judge the speed of the production process (the speed of withdrawal of raw materials from the warehouse):

1) ( u t input - u t-1 input )- change in the volume of inventory in the warehouse (u t input - volume of raw materials in the input warehouse at the time t);

2) (ù- u t in) - deviation of the volume of raw materials in the input warehouse from the stock norm.

Way 2. The regulatory body receives information directly from production (x t - actual production intensity) and compares it with the target intensity (d t -x t).

Method 3. The regulatory body receives information as in method 1, but from the output warehouse in the form ( u t out - u t-1 out )- or (ù -u tout). He also judges the production process on the basis of indirect data - the increase or decrease in finished goods inventories.

To maintain a given output intensity dt, the regulatory body makes decisions yt,(or (y t - y t - 1)), aimed at changing the actual output intensity xt. As a solution, the regulatory body informs production of the intensity values at which it must work, i.e. x t = y t . The second option for the control solution is (y t -y t-1), those. the regulatory body tells production how much to increase or decrease production intensity (x t -x t-1).

Depending on the method of obtaining information and the type of variable describing the control action, the following quantities may influence decision making.

1. Decision basis (the value to which the actual production intensity should be equal if there were no deviations):

directive intensity of release at the moment t(d t);

rate of change of the directive intensity of release at the moment t(d t -d t-1).

2. Deviation amount:

deviation of actual output from the target (d t -x t);

deviation of actual output volume from planned volume

Σ d τ - Σ x τ

change in input inventory level ( ( u t input - u t-1 input) or output

(u t out - u t-1 out) warehouses;

deviation of the inventory level at the input (ù- u t input) or output ( ù -u t out) warehouses from the standard level.

In general, the management decision made by the regulatory body consists of the following components:

Examples of solutions:

y t = d t +y(d t-1 -x t-1);

y t = d t -y(ù -u tout)

By making decisions of various forms, the regulatory body strives to achieve the main goal - to bring the actual output intensity closer to the target one. However, he cannot always directly focus his decisions on the degree to which this goal is achieved. (d t - x t). The final results can be expressed in the achievement of local goals - stabilization of inventory levels at the input or output warehouse ( and t in(out) - and t-1 input(output)) or in bringing the inventory level in the warehouse closer to the standard (And-And in(out)). Depending on the goal achieved, the type of sign (+ or -) in front of the mismatch fraction used for regulation is determined in the control decision.

Let in our case the regulatory authority receive information about the state of the input warehouse (change in inventory level). It is known that in any management system there are delays in the development and implementation of solutions. IN in this example information about the state of the input warehouse arrives at the regulatory authority with a delay of one time step. This delay is called the delay in making a decision and means that by the time information is received from the regulatory body, the actual state of the inventory level at the input warehouse will already be different. Once the regulator has made a decision y t it will also take time (in our example this will be a unit of time) to bring the decision to the executor. This means that the actual production intensity is equal to yt, but to the decision that the governing body made a unit of time ago. This is a delay in the implementation of the decision.

To describe our production system we have the following equations:

xtBX =d t +ξ t in

xt out =dt+ξ tout;

y t = d t + y(u -u t-2 input)

x t = y t-1 + ξt

u t in - u t-1 input = xt in - xt

This system equations allows you to build a model of a production system in which the input variables will be dt,ξ t in, ξ t out, ξ t,a

day off - xt. This is so because an outside observer views our production as a system receiving raw materials at an intensity d t and producing products with intensity xt, subject to randomness ξ t in, ξ t out, ξ t. Having carried out all the substitutions in the resulting system of equations, we arrive at one dynamic equation that characterizes the behavior xt depending on the dt,ξ t in, ξ t out, ξ t.

The model discussed above did not contain restrictions on warehouse volumes and production capacity. If we assume that the capacity of the input warehouse is V in, the capacity of the output warehouse is V BX, and the production capacity is M, That new system The equations for such a nonlinear production system will be as follows:

xtBX=min((d t+ ξ t in),(V in - u t in)) - you cannot put more into the input warehouse than space allows;

x out =min((d t+ ξ t out),(V out - u t out)) - you cannot take more products from the output warehouse than are available there;

y t =d t + y(u t in -u t-1 input)

xtBX = min(( u tin, ( y t-1+ ξ t in), M,(V out - u t out)) - it is impossible to produce more products than ordered, the limiting factors are the number of available blanks and the availability of free space in the output warehouse;

u t in -u t-1 input = xtBX-xt