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 “guess”. Since uncertainty is a key characteristic of everyday life, such a model can describe anything.
However, each time we use it, a different result will be obtained. Therefore, deterministic models are more often used. Although they are not as close to the real state of things, they always give the same result and make it easier to understand the situation, simplify it by introducing a complex of mathematical equations.
Key features
A stochastic model always includes one or more random variables. She seeks to reflect real life in all its manifestations. Unlike the deterministic model, the stochastic model 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 for the study of which is created.
- The outcome of each of the phenomena is uncertain. Therefore, the model includes probabilities. The accuracy of the general 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 a fairly subjective indicator.
For example, consider a coin flip situation. At first glance it seems that the likelihood that a “tails” will fall out 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 perfect balance of the coin. This means that you need to use a stochastic model. There are always parameters that we do not know. In real life, the cause always determines the effect, but there is also some degree of uncertainty. The choice between using deterministic and stochastic models depends on what we are ready 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 similar to introducing uncertainty into the calculation. Many scientists even admitted that this is already a stochastic model.
Lothar Breyer gracefully 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 just beginning to understand. This is the so-called deterministic chaos. " The new theory sounds very believable, so many modern scientists are its supporters. However, it still remains poorly developed, and it is rather difficult to apply in statistical calculations. Therefore, stochastic or deterministic models are often used.
Building
The stochastic mathematical model begins with the choice of the space of elementary outcomes. So in statistics they call a list of possible results of the studied process or event. Then the researcher determines the probability of each of the elementary outcomes. This is usually done based on a specific technique.
However, probabilities are still a fairly subjective parameter. The researcher then determines which events are most interesting to solve the problem. After that, he simply determines their probability.
Example
Consider the process of constructing the simplest stochastic model. Suppose we roll a die. If “six” or “one” appears, then our winnings will be ten dollars. The process of constructing a stochastic model in this case will look as follows:
- We define the space of elementary outcomes. The cube has six faces, so “one”, “two”, “three”, “four”, “five” and “six” can fall out.
- The probability of each outcome will be equal to 1/6, no matter how much we throw the dice.
- Now we need to determine the outcomes that interest us. This is the loss of the face with the number six or one.
- Finally, we can determine the likelihood of an event of interest to us. She is 1/3. We summarize the probabilities of both elementary events of interest to us: 1/6 + 1/6 = 2/6 = 1/3.
