Stochastic Modeling and Stochastic Process


What is stochastic modeling?

Stochastic modeling is a technique for presenting data or predicting results that takes into account a certain degree of randomness or unpredictability. The insurance industry, for example, relies heavily on stochastic modeling to predict the future condition of company balance sheets, as these can depend on unpredictable events that result in the payment of claims. Many other industries and fields of study can benefit from stochastic modeling, such as statistics, stock investing, biology, linguistics, and quantum physics.

Especially in the insurance world, stochastic modeling is crucial in determining which outcomes can be expected versus which are unlikely. Instead of using fixed variables, as in other mathematical models, a stochastic model incorporates random variations to predict future conditions and see what they might look like. Of course, the possibility of random variation implies that many could occur. For this reason, stochastic models are run not just once, but hundreds or even thousands of times. This larger collection of data not only expresses which outcomes are most likely, but also which ranges can be expected.

To understand the idea of ​​stochastic modeling, it may be helpful to consider that it is the opposite, in a way, of deterministic modeling. This second type of modeling is what most elementary mathematics consists of. The solution to a problem can generally only have one correct answer, and the graph of a function can only have a specific set of values. Stochastic modeling, on the other hand, is like varying a complicated math problem slightly to see how the solution is affected, and then doing it many times and in different ways. These slight variations represent the randomness or unpredictability of real world events and their effects.

Another real world application of stochastic modeling, besides insurance, is manufacturing. Manufacturing is considered a stochastic process due to the effect that unknown or random variables can have on the end result. For example, a factory that makes a certain product will always find that a small percentage of the products do not go as planned and cannot be sold. This can be due to a variety of factors, such as the quality of inputs, the working conditions of the production machinery, and the competence of employees, among others. The unpredictability of how these factors affect results can be modeled to predict a certain error rate in manufacturing, which can be planned in advance. 

 

Stochastic process

A stochastic process is a set of random variables that depends on a parameter or an argument. In time series analysis, that parameter is time. Formally, it is defined as a family of random variables Y indexed by time, t. Such that for each value of t, Y has a given probability distribution. 

In much simpler terms, a stochastic process is one that cannot be predicted. It moves randomly. Although, as we will see later, there are different types of stochastic processes. One of the most classic examples to refer to a stochastic process is the stock market.

Despite this, there are strategies that have amply demonstrated that the stock market is not a strictly stochastic process. However, in this case, we are referring to the stock market second by second. Not even the best predictive model in the world would be able to predict whether the stock market will rise or fall every second.

Examples of stochastic processes

Below are various examples of phenomena that constitute stochastic processes.

  • Electrocardiogram
  • Earthquakes
  • The weather
  • The concrete second of a match in which a player scores a goal
  • Number of people who say a specific word around the world

As we can see, they are totally random processes. It is impossible to know in what second a player will score a goal. Just as it is impossible to predict exactly what the weather will be like in an area at a certain point in time. And despite technological advances, it is still impossible to predict an earthquake. Thus, once introduced into stochastic processes, it is necessary to describe the types that exist.

Types of stochastic processes

There are two types of stochastic processes. The difference between them has to do with the predictability of a time series:

  • Stationary stochastic processes: It has a series of characteristics that make it, in a certain way, predictable.
  • Non-stationary stochastic processes: Generally speaking, it would be unpredictable.

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