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 of everyday life, such a model can describe anything.
However, each time we apply it, the result will be different. 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.
Key Features
A stochastic model always includes one or morerandom variables. She seeks to reflect real life in all its manifestations. Unlike the deterministic model, the stochastic one 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 it was created to study.
- 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 series of random events, 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. 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. Always isparameters that we do not 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 give up - ease of analysis or realism.
In chaos theory
Recently, the concept of which model is called stochastic has become even more vague. 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 plume 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 mathematical modelbegins 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. This is usually done based on a specific methodology.
However, probabilities are still quite a subjective parameter. The researcher then determines which events are most interesting for solving the problem. After that, he simply determines their probability.
Example
Let's 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:
- 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 outcome will be 1/6, no matter how many times we roll the die.
- Now we need to determine the outcomes we are interested in. This is a drop of a face 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 simulation is often used in gambling. But it is also indispensable in economic forecasting, as it allowsdeeper than deterministic, understand the situation. 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.
Simulation 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 are not taken into account, this can have disastrous consequences.
