The construction of a stochastic model includes the development, quality assessment and study of the system behavior using equations that describe the process under study.
To do this, by conducting a special experiment with a real system, background information. In this case, methods of experiment planning, processing of results, as well as criteria for evaluating the obtained models based on such sections are used. mathematical statistics as dispersion, correlation, regression analysis, etc.
The methods for constructing a statistical model describing the technological process (Fig. 6.1) are based on the concept of a "black box". Multiple measurements of input factors are possible for it: x 1 ,x 2 ,…,x k and output parameters: y 1 ,y 2 ,…,y p, according to the results of which dependencies are established:
In statistical modeling, following the formulation of problem (1), the least important factors are screened out from a large number of input variables that affect the course of process (2). The input variables selected for further research make up a list of factors x 1 ,x 2 ,…,x k in (6.1), by controlling which it is possible to control the output parameters y n. The number of model outputs should also be reduced as much as possible to reduce the cost of experimentation and data processing.
When developing a statistical model, its structure (3) is usually set arbitrarily, in the form of convenient-to-use functions approximating experimental data, and then refined based on an assessment of the adequacy of the model.
The polynomial form of the model is most commonly used. So, for a quadratic function:
(6.2)
where b 0 , b i , b ij , b ii are the regression coefficients.
Usually, we first restrict ourselves to the simplest linear model, for which in (6.2) b ii =0, b ij =0. In case of its inadequacy, the model is complicated by the introduction of terms that take into account the interaction of factors x i ,x j and (or) quadratic terms .
In order to maximize the extraction of information from the ongoing experiments and reduce their number, experiments are planned (4) i.e. selection of the number and conditions for conducting experiments necessary and sufficient to solve the problem with a given accuracy.
To build statistical models, two types of experiments are used: passive and active. Passive experiment It is carried out in the form of long-term observation of the course of an uncontrolled process, which makes it possible to collect an extensive range of data for statistical analysis. V active experiment it is possible to control the conditions of the experiments. When it is carried out, the most effective is the simultaneous variation of the magnitude of all factors according to a certain plan, which makes it possible to identify the interaction of factors and reduce the number of experiments.
Based on the results of the experiments (5), the regression coefficients (6.2) are calculated and their statistical significance is estimated, which completes the construction of the model (6). The measure of the adequacy of model (7) is the variance, i.e. standard deviation of the calculated values from the experimental ones. The obtained variance is compared with the admissible one with the achieved accuracy of the experiments.
Series "Economics and Management"
6. Kondratiev N.D. Large conjuncture cycles and the theory of foresight. - M.: Economics, 2002. 768 p.
7. Kuzyk B.N., Kushlin V.I., Yakovets Yu.V. Forecasting, strategic planning and national programming. M.: Publishing House "Economics", 2008. 573 p.
8. Lyasnikov N.V., Dudin M.N. Modernization of the innovation economy in the context of the formation and development of the venture market // Social sciences. M.: Publishing house "MII Nauka", 2011. No. 1. S. 278-285.
9. Sekerin V.D., Kuznetsova O.S. Development of an innovation project management strategy // Bulletin of the Moscow State Academy of Business Administration. Series: Economy. - 2013. No. 1 (20). - S. 129 - 134.
10. Yakovlev V.M., Senin A.S. There is no alternative to the innovative type of development of the Russian economy // Actual issues of innovative economics. M.: Publishing House "Science"; Institute of Management and Marketing of the Russian Academy of Arts and Sciences under the President of the Russian Federation, 2012. No. 1(1).
11. Baranenko S.P., Dudin M.N., Ljasnikov N.V., Busygin KD. Using environmental approach to innovation-oriented development of industrial enterprises // American Journal of Applied Sciences.- 2014.- Vol. 11, No.2, - P. 189-194.
12. Dudin M.N. A systematic approach to determining the modes of interaction of large and small businesses // European Journal of Economic Studies. 2012. Vol. (2), no. 2, pp. 84-87.
13. Dudin M.N., Ljasnikov N.V., Kuznecov A.V., Fedorova I.Ju. Innovative Transformation and Transformational Potential of Socio-Economic Systems // Middle East Journal of Scientific Research, 2013. Vol. 17, No. 10. P. 1434-1437.
14. Dudin M.N., Ljasnikov N.V., Pankov S.V., Sepiashvili E.N. Innovative foresight as the method for management of strategic sustainable development of the business structures // World Applied Sciences Journal. - 2013. - Vol. 26, No. 8. - P. 1086-1089.
15. Sekerin V. D., Avramenko S. A., Veselovsky M. Ya., Aleksakhina V. G. B2G Market: The Essence and Statistical Analysis // World Applied Sciences Journal 31 (6): 1104-1108, 2014
Construction of a one-parameter, stochastic model of the production process
Ph.D. Assoc. Mordasov Yu.P.
University of Mechanical Engineering, 8-916-853-13-32, [email protected] gi
Annotation. The author has developed a mathematical, stochastic model of the production process, depending on one parameter. The model has been tested. For this, a simulation model of the production, machine-building process was created, taking into account the influence of random disturbances-failures. Comparison of the results of mathematical and simulation modeling confirms the expediency of applying the mathematical model in practice.
Key words: technological process, mathematical, simulation model, operational control, approbation, random perturbations.
The costs of operational management can be significantly reduced by developing a methodology that allows you to find the optimum between the costs of operational planning and the losses that result from the discrepancy between planned indicators and indicators of real production processes. This means finding the optimal duration of the signal in the feedback loop. In practice, this means a reduction in the number of calculations of calendar schedules for launching assembly units into production and, due to this, saving material resources.
The course of the production process in mechanical engineering is probabilistic in nature. The constant influence of continuously changing factors does not make it possible to predict for a certain perspective (month, quarter) the course of the production process in space and time. In statistical scheduling models, the state of a part at each specific point in time should be given in the form of an appropriate probability (probability distribution) of its being at different workplaces. However, it is necessary to ensure the determinism of the final result of the enterprise. This, in turn, implies the possibility, using deterministic methods, to plan certain terms for parts to be in production. However, experience shows that various interrelations and mutual transitions of real production processes are diverse and numerous. When developing deterministic models, this creates significant difficulties.
An attempt to take into account all the factors that affect the course of production makes the model cumbersome, and it ceases to function as a planning, accounting and regulation tool.
More simple method building mathematical models of complex real processes that depend on a large number of different factors, which are difficult or even impossible to take into account, is the construction of stochastic models. In this case, when analyzing the principles of functioning of a real system or when observing its individual characteristics, probability distribution functions are built for some parameters. In the presence of high statistical stability of the quantitative characteristics of the process and their small dispersion, the results obtained using the constructed model are in good agreement with the performance of the real system.
The main prerequisites for building statistical models of economic processes are:
Excessive complexity and associated economic inefficiency of the corresponding deterministic model;
Large deviations of the theoretical indicators obtained as a result of the experiment on the model from the indicators of actually functioning objects.
Therefore, it is desirable to have a simple mathematical apparatus that describes the impact of stochastic disturbances on the global characteristics of the production process (commercial output, volume of work in progress, etc.). That is, to build a mathematical model of the production process, which depends on a small number of parameters and reflects the total influence of many factors of a different nature on the course of the production process. The main task that a researcher should set himself when building a model is not passive observation of the parameters of a real system, but the construction of such a model that, with any deviation under the influence of disturbances, would bring the parameters of the displayed processes to a given mode. That is, under the action of any random factor, a process must be established in the system that converges to a planned solution. At present, in automated control systems, this function is mainly assigned to a person, who is one of the links in the feedback chain in the management of production processes.
Let us turn to the analysis of the real production process. Usually, the duration of the planning period (the frequency of issuing plans to workshops) is selected based on the traditionally established calendar time intervals: shift, day, five days, etc. They are guided mainly by practical considerations. The minimum duration of the planning period is determined by the operational capabilities of the planned bodies. If the production and dispatching department of the enterprise copes with the issuance of adjusted shift tasks to the shops, then the calculation is made for each shift (that is, the costs associated with the calculation and analysis of planned targets are incurred every shift).
To determine the numerical characteristics of the probability distribution of random
A series of "Economics and Management" disturbances will build a probabilistic model of a real technological process of manufacturing one assembly unit. Here and hereinafter, the technological process of manufacturing an assembly unit means a sequence of operations (works for the manufacture of these parts or assemblies), documented in the technology. Each technological operation of manufacturing products in accordance with the technological route can be performed only after the previous one. Consequently, the technological process of manufacturing an assembly unit is a sequence of events-operations. Under the influence of various stochastic reasons, the duration of an individual operation may change. In some cases, the operation may not be completed during the validity of this shift job. It is obvious that these events can be decomposed into elementary components: performance and non-performance of individual operations, which can also be put in correspondence with the probabilities of performance and non-performance.
For a specific technological process, the probability of performing a sequence consisting of K operations can be expressed by the following formula:
PC5 \u003d k) \u003d (1-pk + 1) PG \u003d 1P1, (1)
where: P1 - the probability of performing the 1st operation, taken separately; r is the number of the operation in order in the technological process.
This formula can be used to determine the stochastic characteristics of a specific planning period, when the range of products launched into production and the list of works that must be performed in a given planning period, as well as their stochastic characteristics, which are determined empirically, are known. In practice, only certain types of mass production, which have a high statistical stability of characteristics, satisfy the listed requirements.
The probability of performing one single operation depends not only on external factors, but also on the specific nature of the work performed and on the type of assembly unit.
To determine the parameters of the above formula, even with a relatively small set of assembly units, with small changes in the range of manufactured products, a significant amount of experimental data is required, which causes significant material and organizational costs and makes this method for determining the probability of uninterrupted production of products hardly applicable.
Let us subject the obtained model to the study for the possibility of its simplification. The initial value of the analysis is the probability of failure-free execution of one operation of the technological process of manufacturing products. In real production conditions, the probabilities of performing operations of each type are different. For a specific technological process, this probability depends on:
From the type of operation performed;
From a specific assembly unit;
From products manufactured in parallel;
from external factors.
Let us analyze the influence of fluctuations in the probability of performing one operation on the aggregated characteristics of the production process of manufacturing products (the volume of commercial output, the volume of work in progress, etc.) determined using this model. The aim of the study is to analyze the possibility of replacing in the model of various probabilities of performing one operation with an average value.
The combined effect of all these factors is taken into account when calculating the average geometric probability of performing one operation of the averaged technological process. Analysis modern production shows that it fluctuates slightly: practically within 0.9 - 1.0.
A clear illustration of how low the probability of performing one operation
walkie-talkie corresponds to a value of 0.9, is the following abstract example. Let's say we have ten pieces to make. The technological processes of manufacturing each of them contain ten operations. The probability of performing each operation is 0.9. Let us find the probabilities of lagging behind the schedule for a different number of technological processes.
A random event, which consists in the fact that a specific technological process of manufacturing an assembly unit will fall behind the schedule, corresponds to the underperformance of at least one operation in this process. It is the opposite of an event: the execution of all operations without failure. Its probability is 1 - 0.910 = 0.65. Since schedule delays are independent events, the Bernoulli probability distribution can be used to determine the probability of schedule delay for a different number of processes. The calculation results are shown in Table 1.
Table 1
Calculation of the probabilities of lagging behind the schedule of technological processes
to C^o0.35k0.651O-k Sum
The table shows that with a probability of 0.92, five technological processes will fall behind the schedule, that is, half. The mathematical expectation of the number of technological processes lagging behind the schedule will be 6.5. This means that, on average, 6.5 assembly units out of 10 will lag behind the schedule. That is, on average, from 3 to 4 parts will be produced without failures. The author is unaware of examples of such a low level of labor organization in real production. The considered example clearly shows that the imposed restriction on the value of the probability of performing one operation without failures does not contradict practice. All of the above requirements are met by the production processes of machine-assembly shops of machine-building production.
Thus, to determine the stochastic characteristics of production processes, it is proposed to construct a probability distribution for the operational execution of one technological process, which expresses the probability of performing a sequence of technological operations for manufacturing an assembly unit through the geometric average probability of performing one operation. The probability of performing K operations in this case will be equal to the product of the probabilities of performing each operation, multiplied by the probability of not performing the rest of the technological process, which coincides with the probability of not performing the (K + T)-th operation. This fact is explained by the fact that if any operation is not performed, then the following ones cannot be executed. The last entry differs from the rest, as it expresses the probability of complete passage without failure of the entire technological process. The probability of performing K of the first operations of the technological process is uniquely related to the probability of not performing the remaining operations. Thus, the probability distribution has the following form:
PY=0)=p°(1-p),
Р(§=1) = р1(1-р), (2)
P(^=1) = p1(1-p),
P(t=u-1) = pn"1(1 - p), P(t=n) = pn,
where: ^ - random value, the number of performed operations;
p is the geometric mean probability of performing one operation, n is the number of operations in the technological process.
The validity of the application of the obtained one-parameter probability distribution is intuitively evident from the following reasoning. Let's assume that we have calculated the geometric mean of the probability of performing one 1 operation on a sample of n elements, where n is large enough.
p = USHT7P7= tl|n]t=1p!), (3)
where: Iy - the number of operations that have the same probability of execution; ] - index of a group of operations that have the same probability of execution; m - the number of groups consisting of operations that have the same probability of execution;
^ = - - relative frequency of occurrence of operations with the probability of execution p^.
In law big numbers, with an unlimited number of operations, the relative frequency of occurrence in a sequence of operations with certain stochastic characteristics tends in probability to the probability of this event. Whence it follows that
for two sufficiently large samples = , then:
where: t1, t2 - the number of groups in the first and second samples, respectively;
1*, I2 - the number of elements in the group of the first and second samples, respectively.
It can be seen from this that if the parameter is calculated for a large number of tests, then it will be close to the parameter P calculated for this rather large sample.
Attention should be paid to the different proximity to the true value of the probabilities of performing a different number of process operations. In all elements of the distribution, except for the last one, there is a factor (I - P). Since the value of the parameter P is in the range of 0.9 - 1.0, the factor (I - P) fluctuates between 0 - 0.1. This multiplier corresponds to the multiplier (I - p;) in the original model. Experience shows that this correspondence for a particular probability can cause an error of up to 300%. However, in practice, one is usually interested not in the probabilities of performing any number of operations, but in the probability of complete execution without failures of the technological process. This probability does not contain a factor (I - P), and, therefore, its deviation from the actual value is small (practically no more than 3%). For economic tasks, this is a fairly high accuracy.
The probability distribution of a random variable constructed in this way is a stochastic dynamic model of the manufacturing process of an assembly unit. Time participates in it implicitly, as the duration of one operation. The model allows you to determine the probability that after a certain period of time (the corresponding number of operations) the production process of manufacturing an assembly unit will not be interrupted. For mechanical assembly shops of machine-building production, the average number of operations of one technological process is quite large (15 - 80). If we consider this number as a base number and assume that, on average, in the manufacture of one assembly unit, a small set of enlarged types of work is used (turning, locksmith, milling, etc.),
then the resulting distribution can be successfully used to assess the impact of stochastic disturbances on the course of the production process.
The author conducted a simulation experiment built on this principle. To generate a sequence of pseudo random variables, evenly distributed on the segment 0.9 - 1.0, a pseudo-random number generator was used, described in the work. The software of the experiment is written in the COBOL algorithmic language.
In the experiment, products of generated random variables are formed, simulating the real probabilities of the complete execution of a specific technological process. They are compared with the probability of performing the technological process, obtained using the geometric mean value, which was calculated for a certain sequence of random numbers of the same distribution. The geometric mean is raised to a power equal to the number of factors in the product. Between these two results, the relative difference in percent is calculated. The experiment is repeated for a different number of factors in the products and the number of numbers for which the geometric mean is calculated. A fragment of the results of the experiment is shown in Table 2.
table 2
Simulation experiment results:
n is the degree of the geometric mean; k - the degree of the product
n to Product Deviation to Product Deviation to Product Deviation
10 1 0,9680 0% 7 0,7200 3% 13 0,6277 -7%
10 19 0,4620 -1% 25 0,3577 -1% 31 0,2453 2%
10 37 0,2004 6% 43 0,1333 4% 49 0,0888 6%
10 55 0,0598 8% 61 0,0475 5% 67 0,0376 2%
10 73 0,0277 1% 79 0,0196 9% 85 0,0143 2%
10 91 0,0094 9% 97 0,0058 0%
13 7 0,7200 8% 13 0,6277 0% 19 0,4620 0%
13 25 0,3577 5% 31 0,2453 6% 37 0,2004 4%
13 43 0,1333 3% 49 0,0888 8% 55 0,0598 8%
13 61 0,0475 2% 67 0,0376 8% 73 0,0277 2%
13 79 0,0196 1% 85 0,0143 5% 91 0,0094 5%
16 1 0,9680 0% 7 0,7200 9%
16 13 0,6277 2% 19 0,4620 3% 25 0,3577 0%
16 31 0,2453 2% 37 0,2004 2% 43 0,1333 5%
16 49 0,0888 4% 55 0,0598 0% 61 0,0475 7%
16 67 0,0376 5% 73 0,0277 5% 79 0,0196 2%
16 85 0,0143 4% 91 0,0094 0% 97 0,0058 4%
19 4 0,8157 4% 10 0,6591 1% 16 0,5795 -9%
19 22 0,4373 -5% 28 0,2814 5% 34 0,2256 3%
19 40 0,1591 6% 46 0,1118 1% 52 0,0757 3%
19 58 0,0529 4% 64 0,0418 3% 70 0,0330 2%
19 76 0,0241 6% 82 0,0160 1% 88 0,0117 8%
19 94 0,0075 7% 100 0,0048 3%
22 10 0,6591 4% 16 0,5795 -4% 22 0,4373 0%
22 28 0,2814 5% 34 0,2256 5% 40 0,1591 1%
22 46 0,1118 1% 52 0,0757 0% 58 0,0529 8%
22 64 0,0418 1% 70 0,0330 3% 76 0,0241 5%
22 82 0,0160 4% 88 0,0117 2% 94 0,0075 5%
22 100 0,0048 1%
25 4 0,8157 3% 10 0,6591 0%
25 16 0,5795 0% 72 0,4373 -7% 28 0,2814 2%
25 34 0,2256 9% 40 0,1591 1% 46 0,1118 4%
25 52 0,0757 5% 58 0,0529 4% 64 0,0418 2%
25 70 0,0330 0% 76 0,0241 2% 82 0,0160 4%
28 4 0,8157 2% 10 0,6591 -2% 16 0,5795 -5%
28 22 0,4373 -3% 28 0,2814 2% 34 0,2256 -1%
28 40 0,1591 6% 46 0,1118 6% 52 0,0757 1%
28 58 0,0529 4% 64 0,041 8 9% 70 0,0330 5%
28 70 0,0241 2% 82 0,0160 3% 88 0,0117 1%
28 94 0,0075 100 0,0048 5%
31 10 0,6591 -3% 16 0,5795 -5% 22 0,4373 -4%
31 28 0,2814 0% 34 0,2256 -3% 40 0,1591 4%
31 46 0,1118 3% 52 0,0757 7% 58 0,0529 9%
31 64 0,0418 4% 70 0,0330 0% 76 0,0241 6%
31 82 0,0160 6% 88 0,0117 2% 94 0,0075 5%
When setting up this simulation experiment, the goal was to explore the possibility of obtaining, using the probability distribution (2), one of the enlarged statistical characteristics of the production process - the probability of performing one technological process of manufacturing an assembly unit consisting of K operations without failures. For a specific technological process, this probability is equal to the product of the probabilities of performing all its operations. As the simulation experiment shows, its relative deviations from the probability obtained using the developed probabilistic model do not exceed 9%.
Since the simulation experiment uses a more inconvenient than real probability distribution, the practical discrepancies will be even smaller. Deviations are observed both in the direction of decreasing and in the direction of exceeding the value obtained from the average characteristics. This fact suggests that if we consider the deviation of the probability of failure-free execution of not a single technological process, but several, then it will be much less. Obviously, it will be the smaller, the more technological processes will be considered. Thus, the simulation experiment shows a good agreement between the probability of performing without failures of the technological process of manufacturing products with the probability obtained using a one-parameter mathematical model.
In addition, simulation experiments were carried out:
To study the statistical convergence of the probability distribution parameter estimate;
To study the statistical stability of the mathematical expectation of the number of operations performed without failures;
To analyze methods for determining the duration of the minimum planning period and assessing the discrepancy between planned and actual indicators of the production process, if the planned and production periods do not coincide in time.
Experiments have shown good agreement between the theoretical data obtained through the use of techniques and the empirical data obtained by simulation on
Series "Economics and Management"
Computer of real production processes.
Based on the application of the constructed mathematical model, the author has developed three specific methods for improving the efficiency of operational management. For their approbation, separate simulation experiments were carried out.
1. Methodology for determining the rational volume of the production task for the planning period.
2. Methodology for determining the most effective duration of the operational planning period.
3. Evaluation of the discrepancy in the event of a mismatch in time between the planned and production periods.
Literature
1. Mordasov Yu.P. Determining the duration of the minimum operational planning period under the action of random disturbances / Economic-mathematical and simulation modeling using computers. - M: MIU im. S. Ordzhonikidze, 1984.
2. Naylor T. Machine simulation experiments with models of economic systems. -M: Mir, 1975.
The transition from concentration to diversification is an effective way to develop the economy of small and medium-sized businesses
prof. Kozlenko N. N. University of Mechanical Engineering
Annotation. This article considers the problem of choosing the most effective development of Russian small and medium-sized businesses through the transition from a concentration strategy to a diversification strategy. The issues of the expediency of diversification, its advantages, criteria for choosing the path of diversification are considered, a classification of diversification strategies is given.
Key words: small and medium businesses; diversification; strategic fit; competitive advantages.
An active change in the parameters of the macro environment (changes in market conditions, the emergence of new competitors in related industries, an increase in the level of competition in general) often leads to non-fulfillment of the planned strategic plans of small and medium-sized businesses, loss of financial and economic stability of enterprises due to a significant gap between the objective conditions for the activities of small businesses. enterprises and the level of technology of their management.
The main conditions for economic stability and the possibility of maintaining competitive advantages are the ability of the management system to respond in a timely manner and change internal production processes (change the assortment taking into account diversification, rebuild production and technological processes, change the structure of the organization, use innovative marketing and management tools).
A study of the practice of Russian small and medium-sized enterprises of production type and service has revealed the following features and basic cause-and-effect relationships regarding the current trend in the transition of small enterprises from concentration to diversification.
Most SMBs start out as small, one-size-fits-all businesses serving local or regional markets. At the beginning of its activity, the product range of such a company is very limited, its capital base is weak, and its competitive position is vulnerable. Typically, the strategy of such companies focuses on sales growth and market share, as well as
480 rub. | 150 UAH | $7.5 ", MOUSEOFF, FGCOLOR, "#FFFFCC",BGCOLOR, "#393939");" onMouseOut="return nd();"> Thesis - 480 rubles, shipping 10 minutes 24 hours a day, seven days a week and holidays
Demidova Anastasia Vyacheslavovna Method for constructing stochastic models of one-step processes: dissertation ... Candidate of Physical and Mathematical Sciences: 05.13.18 / Demidova Anastasia Vyacheslavovna; [Place of defense: Peoples' Friendship University of Russia].- Moscow, 2014.- 126 p.
Introduction
Chapter 1. Review of works on the topic of the dissertation 14
1.1. Overview of population dynamics models 14
1.2. Stochastic population models 23
1.3. Stochastic Differential Equations 26
1.4. Information on stochastic calculus 32
Chapter 2 One-Step Process Modeling Method 39
2.1. One step processes. Kolmogorov-Chapman equation. Basic kinetic equation 39
2.2. Method for modeling multidimensional one-step processes. 47
2.3. Numerical simulation 56
Chapter 3 Application of the method of modeling one-step processes 60
3.1. Stochastic models of population dynamics 60
3.2. Stochastic models of population systems with various inter- and intraspecific interactions 75
3.3. Stochastic model of the spread of network worms. 92
3.4. Stochastic models of peer-to-peer protocols 97
Conclusion 113
Literature 116
Stochastic differential equations
One of the objectives of the dissertation is the task of writing a stochastic differential equation for a system so that the stochastic term is associated with 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, one can write an equivalent stochastic differential equation in the form of the Langevin equation.
Section 1.4. contains the basic information necessary to indicate the relationship 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, on the basis of this theory, a method for modeling one-step processes is formulated.
Section 2.1 provides basic information from the theory of random one-step processes.
One-step processes are understood as Markov processes with continuous time, taking values in the region of integers, the transition matrix of which allows only transitions between adjacent sections.
We consider a multidimensional one-step process Х() = (i(),2(), ...,n()) = ( j(), = 1, ) , (0.1) Є , where is the length of the time interval on which the X() process is specified. The set G \u003d (x, \u003d 1, Є NQ x NQ1 is the set of discrete values that a random process can take.
For this one-step process, the probabilities of transitions per unit time s+ and s from state Xj to state Xj__i and Xj_i, respectively, are introduced. In this case, it is considered that the probability of transition from state x to two or more steps per unit of time is very small. Therefore, we can say that the state vector Xj 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 temporal evolution occurs as a result of the interaction of system elements, it is convenient to describe using the main kinetic equation (another name is the master 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, with the help of 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 the Langevin equations satisfy this definition. However, they are directly related to other equations, namely the Fokker-Planck equation and the basic 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 it is possible to 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 writing for the system under study the interaction scheme, 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 main kinetic equation.
Section 2.3. the Runge-Kutta method for the numerical solution of stochastic differential equations is considered, which is used in the third chapter to illustrate the results obtained.
The third chapter presents an illustration of the application of the method of constructing stochastic models described in the second chapter, using the example of systems describing the dynamics of the growth of interacting populations, such as "predator-prey", symbiosis, competition and their modifications. The aim is to write them as stochastic differential equations and to investigate 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 on 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 those already well known.
The analysis of the obtained equations showed that to study the deterministic behavior of the system, one can use the drift vector A of the obtained 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 of the equations have a periodic form and the phase volume is preserved, while the introduction of stochastics into the model gives a monotonous 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.
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 model of the interaction of three populations.
Information on stochastic calculus
The development of the theory of random processes led to a transition in the study of natural phenomena from deterministic representations and models of population dynamics to probabilistic ones and, as a result, the emergence of a large number of works devoted to stochastic modeling in mathematical biology, chemistry, economics, etc.
When considering deterministic population models, such important points as the random influences 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 that reflect these moments 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, they can be used to reveal qualitatively new aspects of population behavior.
Stochastic models of changes in population states can be described using random processes. Under some assumptions, we can assume that the behavior of the 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 the processes of population dynamics, 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 paper.
N. N. Kalinkin in his works to illustrate the processes occurring in systems with interacting elements uses interaction schemes 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 paper considers probabilistic population models, for the construction of which the apparatus of birth-death processes is used, and the resulting systems of differential-difference equations are dynamic equations for random processes. The paper also considers methods for finding solutions to these equations.
You can find many articles devoted to the construction of stochastic models that take into account various factors influencing the dynamics of changes in the number of populations. So, for example, in the articles a model of the dynamics of the size of a biological community is built and analyzed, in which individuals consume food resources containing harmful substances. And in the model of population evolution, the article takes into account the factor of settling 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. birth-death processes.
One can consider the “predator-prey” model as a realization of birth-death processes. In this interpretation, they can be used for models in many fields of science. In the 1970s, M. Doi proposed a method for studying such models based on creation-annihilation operators (by analogy with second quantization). Here you can mark the work. In addition, this method is now being actively 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. Here you can mark the work.
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 the stochastic term is added from general considerations about the behavior of the system and is intended to describe random environmental effects. 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. The most used and well-known example of a stochastic differential equation (SDE) is an equation with a term that describes white noise and can be viewed as a Wiener process Wt, t 0.
Stochastic differential equations are an important and widely used mathematical tool in the study and modeling of dynamic systems that are subject to various random perturbations.
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 studied the movement of plant pollen in a liquid. The first rigorous explanation of this phenomenon was independently given by A. Einstein and M. Smoluchowski. It is worth noting the collection of articles in which the works of A. Einstein and M. Smoluchowski on Brownian motion are collected. These studies have made a significant contribution to the development of the theory of Brownian motion and its experimental verification. A. Einstein created a molecular kinetic theory for the quantitative description of Brownian motion. The obtained formulas were confirmed by the experiments of J. Perrin in 1908-1909.
Method for modeling multidimensional one-step processes.
To describe the evolution of systems with interacting elements, there are two approaches - this is the construction of deterministic or stochastic models. Unlike deterministic, stochastic models allow taking into account the probabilistic nature of the processes occurring in the systems under study, as well as the effects 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 from one state to another is associated with the interaction of system elements. An example is models that describe the growth dynamics of interacting populations, such as "predator-prey", symbiosis, competition and their modifications. The aim is to write down for such systems SDE and to investigate the influence of the introduction of the stochastic part on the behavior of the solution of the equation describing the deterministic behavior.
Chemical kinetics
The systems of equations that arise when describing systems with interacting elements are in many ways similar to systems of differential equations that describe the kinetics of chemical reactions. Thus, for example, the Lotka-Volterra system was originally deduced by Lotka as a system describing some hypothetical chemical reaction, and only later Volterra deduced it as a system describing the "predator-prey" model.
Chemical kinetics describes chemical reactions with the help of the so-called stoichiometric equations - equations reflecting the quantitative ratios of the reactants and products of a chemical reaction and having the following general form: where the natural numbers mі and U are called stoichiometric coefficients. This is a symbolic record of a chemical reaction in which ti molecules of the reagent Xi, ni2 molecules of the reagent Xh, ..., tr molecules of the reagent Xp, having entered into the reaction, form u molecules of the substance Yї, u molecules of the substance I2, ..., nq molecules of the substance Yq, respectively .
In chemical kinetics, it is believed that a chemical reaction can occur only with the direct interaction of reagents, and the rate of a chemical reaction is defined as the number of particles formed per unit time per unit volume.
The main postulate of chemical kinetics is the law of mass action, which says 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 any substance over time as a result of a chemical reaction:
Further, it is proposed to use the basic ideas of chemical kinetics to describe systems whose evolution in time occurs as a result of the interaction of the elements of this system with each other, making the following main changes: 1. not the reaction rates are considered, but the transition probabilities; 2. it is proposed that the probability of a transition from one state to another, which is the result of an interaction, is proportional to the number of possible interactions of this type; 3. To describe the system in this method, the main kinetic equation is used; 4. deterministic equations are replaced by stochastic ones. A similar approach to the description of such systems can be found in the works. To describe the processes occurring in the simulated system, it is supposed to use, as noted above, Markov one-step processes.
Consider a system consisting of types of different elements that can interact with each other in various ways. Denote by an element of the -th type, where = 1, and by - the number of elements of the -th type.
Let (), .
Let's assume that the file consists of one part. Thus, in one step of interaction between the new node that wants to download the file and the node that distributes 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 enter the system with intensity, and distributing nodes can leave it with intensity. Then the interaction scheme and the vector r will look like:
A stochastic differential equation in the Langevin form can be obtained 100 using the corresponding formula (1.15). Because the drift vector A fully describes the deterministic behavior of the system, you can get a system of ordinary differential equations that describe the dynamics of the number of new customers and seeds:
Thus, depending on the choice of parameters, the singular point can have a different character. Thus, for /3A 4/I2, the singular point is a stable focus, and for the inverse relation, it is a stable node. In both cases, the singular point is stable, since the choice of coefficient values, changes in system variables can occur along one of two trajectories. If the 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 vibrationless mode (see Fig. 3.13). The phase portraits of the system for each of the two cases are shown, respectively, in graphs (3.14) and (3.15).
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1. An example of building a stochastic process model
In the course of a bank's operation, it is very often necessary to solve the problem of choosing an asset vector, i.e. bank's investment portfolio, and the uncertain parameters that must be taken into account in this task are primarily related to 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 obligations.
For problems of this class, the fundamental issue is the construction of a model of the stochastic process of price changes, since the operation researcher, of course, has only a finite series of observations of realizations of random variables - prices. Next, one of the approaches to solving this problem is presented, which is being developed at the Computing Center of the Russian Academy of Sciences in connection with solving control problems for 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 - expressed as a percentage of yields during the current session. If a paper of the type at the end of the session is bought at the price and sold at the end of the session at the price, then.
Yields are random variables formed as follows. The existence of basic returns is assumed - 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 a certain scale factor equal to (), and is a random variable that has the meaning of a deviation from the base value and is determined similarly:
where are also standard normally distributed random variables.
It is assumed that some operating party, hereinafter referred to as the operator, manages its capital invested in securities (at any moment in paper of exactly one type) for some time, selling them at the end of the current session and immediately buying other securities with the proceeds. Management, 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 securities. We will consider various hypotheses about this awareness and, accordingly, various control algorithms. We will assume that the researcher of the operation develops and optimizes the control algorithm using the available series of observations of the process, i.e., using information about the closing prices at exchange sessions, and also, possibly, about the values, at a certain time interval 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 under conditions when the algorithms are tuned and evaluated on the same series of observations. To estimate the theoretical mathematical expectation, the Monte Carlo method is used by “sweeping” the control over a sufficiently large generated series, i.e. by a matrix of dimensions, where the columns correspond to the realizations of values and by sessions, and the number is determined by computational capabilities, but provided that the matrix elements are at least 10000. It is necessary that the “polygon” be the same in all experiments. The available series of observations simulates the generated dimension matrix, where the values in the cells have the same meaning as above. The number and values in this matrix will vary in the future. Matrices of both types are formed by means of a procedure for generating random numbers, simulating the implementation of random variables, and calculating the desired elements of the matrices using these implementations and formulas (1) - (3).
Evaluation of control efficiency on a series of observations is made according to 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 located during the session. In addition, we will also calculate the monthly efficiency. The number 22 roughly corresponds to the number of trading sessions per month.
Computational experiments and analysis of results
Hypotheses
Exact knowledge by the operator of future returns.
The index is chosen as. This option gives an upper estimate for all possible control algorithms, even if additional information (taking into account some additional factors) allows us to refine the price forecast model.
Random control.
The operator does not know the law of pricing and conducts operations by random selection. Theoretically, in this model, the mathematical expectation of the result of operations is the same as if the operator invested not in one paper, but equally in all. With zero mathematical expectations of the values, the mathematical expectation of the value is equal to 1. Calculations according to 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 accurate knowledge of the profitability model, all its parameters and the observed value .
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 by formulas (1) - (3) the mathematical values.
where, according to (2), . (6)
Control with knowledge of the structure of the yield 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 values influencing the formation that precede the values of these parameters (the memory depth of Markov processes). It also does not know whether the coefficients are the same or different for different values. Let's consider different variants of 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
Here, for the sake of completeness, a free term has been added. However, this term can be excluded either for meaningful reasons or by statistical methods. Therefore, to simplify the calculations, we further exclude free terms when setting the parameters from consideration and formula (7) takes the form:
Depending on whether the researcher assumes the same or different coefficients 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 according to the observed values for all securities together. In cases 4.m. 2 coefficients are adjusted for each security separately, while the researcher works under the hypothesis that the coefficients are different for different and, for example, in case 4.2.2. values are determined by the modified formula (3)
First setup method- the classical method of least squares. Let's consider it on the example of setting the coefficients at in options 4.3.
According to formula (8),
It is required to find such values of the coefficients in order to minimize the sample variance for implementations on a known series of observations, an array, provided that the mathematical expectation of the values is determined by formula (9).
Here and in what follows, the sign "" indicates the realization of a random variable.
The minimum of the quadratic form (10) is reached at the only point where all partial derivatives are equal to zero. From here we obtain a system of three algebraic linear equations:
the solution of which gives the desired values of the coefficients.
After the coefficients are verified, the choice of controls is carried out in the same way as in case 3.
Comment. In order to facilitate the work on programs, it is accepted to 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 set to zero.
The second setting method consists in choosing the values of the parameters so as to maximize the estimate from formula (4). This task is analytically and computationally hopelessly difficult. Therefore, here we can only talk about methods of some improvement of the criterion value relative to the starting point. The starting point can be taken from the least squares values and then computed around these values on a grid. In this case, the sequence of actions is as follows. First, the grid is calculated on the parameters (square or cube) with the remaining parameters fixed. Then for cases 4.m. 1, the grid is calculated on the parameters, and for cases 4.m. 2 on the parameters with the remaining parameters fixed. In case 4.m. 2 further parameters are also optimized. When all parameters are exhausted by this process, the process is repeated. Repetitions are made until the new cycle gives an improvement in the criterion values compared to the previous one. So that the number of iterations does not turn out to be too large, we apply next move. Inside each block of calculations on a 2- or 3-dimensional parameter space, a rather 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, but if the best point is internal, then a new grid is built around this point with a smaller step, but with the same total number of points, and so on some, but a reasonable number of times.
Management under unobserved and without taking into account the dependence between the yields of different securities.
This means that the researcher of the operation does not notice the relationship between different securities, knows nothing about the existence and tries to predict the behavior of each security separately. Consider, as usual, three cases when the researcher models the process of generating returns as a Markov process with depths 1, 2, and 3:
The coefficients for predicting the expected return are not important, and the coefficients are adjusted in two ways, described in paragraph 4. The controls are chosen in the same way as it was done above.
Note: As well as for choosing a control, for the least squares method it makes sense to write a single procedure with a maximum number of variables - 3. If the variables are adjustable, say, then for the solution of a linear system, a formula is written that includes only constants, is defined through , and through and. In cases where there are less than three variables, the values of extra variables are set to zero.
Although the calculations in different variants are carried out in a similar way, the number of variants is quite large. When the preparation of 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.
Management under unobserved taking into account the dependence between the yields of different securities.
This series of experiments imitates the manipulations that were performed in the GKO problem. We assume that the researcher knows practically nothing about the mechanism of formation of returns. He has only a series of observations, a matrix. From substantive considerations, 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 securities yields from session to session, he makes the assumption that at each moment of time the points whose coordinates are the numbers of securities and yields (in reality, these were the maturities of securities and their prices) are grouped near a certain curve (in the case of GKO - parabolas).
Here - the point of intersection of the theoretical line with the y-axis (base return), and - its slope (which should be equal to 0.05).
By constructing the theoretical lines in this way, the researcher of the operation can calculate the values - the deviations of the values 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 the values from the currently known values, . Insofar as
to predict the values, the researcher needs to introduce a hypothesis about the formation of the values, and. Using the matrix, the researcher can establish a significant correlation between the values of and. You can accept the hypothesis of a linear relationship between the quantities from: . From meaningful considerations, the coefficient is immediately assumed to be equal to zero, and the least squares method is sought in the form:
Further, as above, and are modeled by means of a Markov process and are described by formulas similar to (1) and (3) with different number variables depending on the memory depth of the Markov process in the considered version. (here it is determined not by formula (2), but by formula (16))
Finally, as above, two ways of adjusting the parameters by the least squares method are implemented, and estimates are made by directly maximizing the criterion.
Experiments
For all the described options, the criteria scores were calculated for different matrices. (matrices with the number of rows 1003, 503, 103, and about a hundred matrices were implemented for each dimension option). According to the results of calculations for each dimension, the mathematical expectation and dispersion of values, and their deviation from the values, were estimated for each of the prepared options.
As shown by the first series of computational experiments with a small number of adjustable parameters (about 4), the choice of the tuning method does not significantly affect the value of the criterion in the problem.
2. Classification of modeling tools
stochastic simulation bank algorithm
The classification of modeling methods and models can be carried out according to the degree of detail of the models, according to the nature of the features, according to the scope of application, etc.
Consider one of the most common classifications of models by means of modeling, this aspect is the most important in the analysis of various phenomena and systems.
material in the case when the study is conducted on models, the connection of which with the object under study exists objectively, is of a material nature. Models in this case are built by the researcher or selected by him from the surrounding world.
By means of modeling, modeling methods are divided into two groups: material modeling methods and ideal modeling methods. Modeling is called material in the case when the study is conducted on models, the connection of which with the object under study exists objectively, is of a material nature. Models in this case are built by the researcher or selected by him from the surrounding world. In turn, in material modeling, one 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 under study. Models in this case are geometrically similar to the objects of study (any layouts).
Models used in physical modeling designed to reproduce the dynamics of processes occurring in the object under study. Moreover, the commonality of the 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 of various types. For example, the study of aircraft based on experiments in a wind tunnel.
analog modeling is associated with the use of material models that have a different physical nature, but described by the same mathematical relations as the object under study. It is based on the analogy in the mathematical description of the model and the object (the study of mechanical vibrations with the help of an electrical system described by the same differential equations, but more convenient for experiments).
In all cases of material modeling, the model is a material reflection of the original object, and the study consists in the material impact on the model, that is, in the experiment with the model. Material modeling by its nature is an experimental method and is not used in economic research.
It is fundamentally different from material modeling perfect modeling, based on an ideal, conceivable connection between the object and the model. Ideal modeling methods are widely used in economic research. They can be conditionally divided into two groups: formalized and non-formalized.
V formalized In modeling, systems of signs or images serve as a model, together with which the rules for their transformation and interpretation are set. If systems of signs are used as models, then modeling is called iconic(drawings, graphs, diagrams, formulas).
An important type of sign modeling is mathematical modeling, based on the fact that various studied objects and phenomena 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 attributed to formalized modeling, when the rules for the interaction of objects used in the model are clearly fixed (for example, in an ideal gas, a collision of two molecules is considered as a collision of balls, and the result of a collision is thought of by everyone in the same way). Models of this type are widely used in physics, they are called "thought experiments".
Non-formalized modeling. It includes such an analysis of problems of various types, when the model is not formed, but instead of it, some accurately 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 non-formalized modeling, when a thinking individual has some image of the object of study, which can be interpreted as a non-formalized model of reality.
The study of economic objects for a long time was carried out only on the basis of such uncertain ideas. At present, the analysis of non-formalized models remains the most common means of economic modeling, namely, every person who makes an economic decision without the use of mathematical models is 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 turn out to 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 in decision-making in the economy.
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The stochastic model describes the situation when there is uncertainty. In other words, the process is characterized by some degree of randomness. The adjective "stochastic" itself comes from the Greek word "guess". Since uncertainty is a key characteristic Everyday life, then such a model can describe anything.
However, each time we apply it, the result will be different. Therefore, more commonly used deterministic models. 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 seeks to reflect real life in all its manifestations. Unlike stochastic, it does not aim to simplify everything and reduce 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 for which it was created.
- The outcome of each of the phenomena 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 seems to be a succession for others - processes 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 quite subjective.
For example, consider a coin toss situation. At first glance, it looks like there is a 50% chance of getting tails. Therefore, a deterministic model must be used. However, in reality, it turns out that much depends on the dexterity of the hands of the players and the perfection of the balancing of the coin. This means that a stochastic model must be used. There are always parameters that we do not know. V 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 give up - simplicity of analysis or realism.
In chaos theory
Recently, 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 give different results with a slight change in the initial parameters. This is like an introduction to the calculation of uncertainty. Many scientists have even admitted that this is already a stochastic model.
Lothar Breuer elegantly explained everything with the help of poetic images. He wrote: “A mountain brook, a beating heart, an epidemic of smallpox, a column of rising smoke - all this is an example of a dynamic phenomenon, which, as it seems, is sometimes 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.” The new theory sounds very plausible, which is why many modern scientists are its supporters. However, it still remains little developed, and it is rather difficult to apply it in statistical calculations. Therefore, stochastic or deterministic models are often used.
Building
Stochastic begins with the choice of the space of elementary outcomes. So in statistics they call the list of possible results of the process or event being studied. The researcher then determines the probability of each of the elementary outcomes. Usually this is done on the basis of a certain technique.
However, the probabilities are still quite a subjective parameter. The researcher then determines which events are most interesting for solving the problem. After that, it simply determines their probability.
Example
Consider the process of building the simplest stochastic model. Suppose we roll a die. If "six" or "one" falls out, then 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 one, two, three, four, five, and six can come up.
- The probability of each of the outcomes will be equal to 1/6, no matter how much we roll the die.
- Now we need to determine the outcomes of interest to us. This is the loss of a face with the number "six" or "one".
- Finally, we can determine the probability of the event of interest to us. 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 simulation is often used in gambling. But it is also indispensable in economic forecasting, as it allows you to understand the situation deeper than deterministic ones. Stochastic models in economics are often used in making investment decisions. They allow you to make assumptions about the profitability of investments in certain assets or their groups.
Modeling makes financial planning more efficient. With its help, investors and traders optimize the distribution of their assets. Using stochastic modeling always has advantages in the long run. In some industries, refusal or inability to apply it can even lead to the bankruptcy of the enterprise. This is due to the fact that in real life new important parameters appear daily, and if they do not, it can have disastrous consequences.
