"Stochastic" is a word that physicists, mathematicians and other scientists use to describe processes that have an element of chance. Its origin is ancient Greek. Translated, it means "able to guess."
Meaning of the word “stochastic”
"Stochastic" is a concept that is used in many different fields of science. It means randomness, randomness, uncertainty of something. In the ethics of Aristotle (his sculptural portrait is presented above), the concept of “stochastic” is a definition that refers to the ability to guess. Obviously, mathematicians used it on the basis that the element of chance appears just when necessary to guess. The word "stochastic" is a concept that is defined in the New International Dictionary as "conjectural."
Thus, it can be noted that the technical meaning of this concept does not exactly correspond to its vocabulary (lexical) meaning. Some authors use the expression "stochastic process" as a synonym for the term "random process".
Stochasticity in mathematics
The use of this term in mathematics is currently widespread. For example, there is such a concept in probability theory as the stochastic process. Its result cannot be determined by the initial state of this system.
The use in mathematics of the concept of "stochasticity" is attributed to the works of Vladislav Bortskevich. It was he who used the term in the meaning of "put forward hypotheses." In mathematics, especially in such a section of this science as probability theory, the field of random research plays an important role. There is, for example, such a thing as a stochastic matrix. The columns or rows of this matrix add up to one.
Stochastic Mathematics (Financial)
This section of mathematics analyzes financial structures operating in conditions of uncertainty. It is designed to find the most rational methods of managing financial assets and structures, taking into account factors such as stochastic evolution, risk, time, etc.
In science, it is customary to distinguish the following structures and objects that are used in financial mathematics as a whole:
- firms (for example, companies);
- individuals;
- intermediary structures (pension funds, banks);
- financial markets.
The main object of study of stochastic financial mathematics is precisely the last of them. This section is based on such disciplines as statistics of random processes, theory of random processes, etc.
Currently, even people far from science, it is well known from numerous news and publications in the media that the values of the so-called global financial indices (for example, the Dow Jones index), stock prices are changing randomly. L. Bachelier made the first attempt to describe using mathematics the evolution of stock prices. His stochastic method is based on probability theory. L. Bachelier's dissertation, which presents this attempt, was published in 1900. The scientist has proven the formula currently known as the fair value formula for call options. It reflects the stochastic probability.
Important ideas that subsequently led to the emergence of an effective market theory were presented in the work of M. Kendall, published in 1953. This paper addresses the issue of stock price dynamics. The researcher describes it using stochastic processes.
Stochasticity in Physics
Thanks to physicists E. Fermi, S. Ulam, N. Metropolis and D.Neumann is widely used Monte Carlo method. Its name comes from a casino located in the same city in a country such as Monaco. It was here that Uncle Ulam borrowed money for the game. Using the nature of repetition and chance to study processes is similar to what happens in a casino.
When applying this modeling method, a probabilistic analogue is first searched. Prior to this, modeling was carried out in the opposite direction: it was used to verify the result of the deterministic problem obtained earlier. Although similar approaches existed before the discovery of the Monte Carlo method, they were not popular and general.
Enrico Fermi in 1930 applied stochastic techniques to calculate the properties of the neutron, which had just been discovered at that time. Monte Carlo methods were later used when working on the Manhattan project, although at that time the capabilities of computers were significantly limited. For this reason, they became widespread only after computers appeared.
Stochastic signals
Regular and stochastic signals have different waveforms. If we re-measure the latter, we get oscillations that have a new shape, which is different from the previous one, but shows a certain similarity in essential features. An example of a stochastic signal is the recording of sea wave oscillations.
Why is it necessary to talk about these rather unusual signals? The fact is that in the study of automatic systems, they are even more common than predicted.
Stochasticity and Artificial Intelligence
Stochastic artificial intelligence programs work using probabilistic methods. Algorithms such as stochastic optimization or neural networks can be cited as an example. The same applies to simulated annealing and genetic algorithms. In all these cases, stochasticity can be contained in the problem as such or in planning something under the condition of uncertainty. The deterministic environment for a modeling agent is simpler than stochastic.
So, as we see, the concept of interest to us is used in many fields of science. We have listed and characterized only the main areas of its application. The study of all these processes, you see, is very important and relevant. That is why the concept of interest to us is likely to be used for a long time in science.
