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## Stochastic processes pdf

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 at random. 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 on a second-by-second basis. Not even the best predictive model in the world would be able to predict whether the stock market will go up or down every second.

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

### Stochastic definition

In probability theory, a stochastic process is a mathematical concept used to represent time-varying random quantities or to characterize a succession of random (stochastic) variables that evolve as a function of another variable, usually time. Each of the random variables in the process has its own probability distribution function and may or may not be correlated with each other.

The stochastic process can be understood as a uniparametric family of random variables indexed by means of time t. Stochastic processes allow to deal with dynamic processes in which there is some randomness.

### Stochastic processes with applica

A stationary stochastic process, or simply a stationary process in mathematics, is one in which the mean and variance do not change with time. This is technically “second order stationarity” or “weak stationarity”.

We can classify random processes based on many different criteria. One of the important questions we can ask about a random process is whether it is a stationary process. Intuitively, a random process {X( t ) , t ∈ J} is stationary if its statistical properties do not change with time. For example, for a stationary process, X( t ) and X( t + Δ )have the same probability distributions.

More generally, for a stationary process, the joint distribution of X(t1) and X(t2) is the same as the joint distribution of X(t1+ Δ ) and X(t2+ Δ ). For example, if you have a stationary process X( t ), then

In practice, it is desirable for a random process X( t )to be stationary. In particular, if a process is stationary, its analysis is usually simpler, since the probabilistic properties do not change with time.

### Types of stochastic processes

A system whose intrinsic behavior is nondeterministic is called stochastic (from the Latin stochasticus, which in turn comes from the Greek στοχαστικός stochastikós “skilled in conjecture”). A stochastic process is one whose behavior is nondeterministic, insofar as the subsequent state of the system is determined both by the predictable actions of the process and by random elements. However, according to Mark Kac and E. Nelson, any temporal development (whether deterministic or essentially probabilistic) that can be analyzed in terms of probability deserves to be called a stochastic process. Specifically: the term stochastic applies to processes, algorithms and models in which there is a changing sequence of probabilistically analyzable events as time passes.

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The use of the term stochastic to refer to something based on probability theory can be traced back in time to Ladislaus Bortkiewicz, who gave it the meaning of “making guesses”, and who carries the Greek term from the ancient philosophers, from the title of Ars Conjectandi that Jakob Bernoulli gave to his work on probability theory. The term stochastic is also used to refer to something based on the theory of probability. The term stochastic is also used to refer to something based on the theory of probability.