Stochastic process

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See also: Stochastic system

The bag index is an example of non-stationary type stochastic process.

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

Each variable or set of variables subject to influences or random effects constitutes a stochastic process. A stochastic process ${displaystyle X_{t}}$ can be understood as a uniparametric family of random variables indexed over time t. Stochastic processes allow us to treat dynamic processes in which there is a certain randomity. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. 2125213111

Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Stock Exchange, and the Poisson process, used by A. K. Erlang to study the number of phone calls that occur in a given period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.

The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often without a specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is often called a random field. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.

Based on their mathematical properties, stochastic processes can be grouped into several categories, including random walk, martingales, Markov chain, Lévyes process, Gaussian process, random fields, renewal process, and branching process. The study of stochastic processes uses mathematical knowledge and techniques of probability, calculus, linear algebra, set theory and topology as well as branches of mathematical analysis such as real analysis, measurement theory, analysis Fourier and functional analysis. The theory of stochastic processes is considered an important contribution to mathematics and remains an active topic of research both for theoretical reasons and for its applications.

Notion of process

Many fields use observations as a function of time (or, more rarely, of a spatial variable). In the simplest cases, these observations give rise to a well-defined curve. In reality, from the earth sciences to the humanities, observations tend to occur more or less erratically. Therefore, the interpretation of these observations is subject to some uncertainty, which may be reflected in the use of probabilities to represent them.

A random process generalizes the notion of a random variable used in probability. It is defined as a family of random variables X(t) associated to all values t ∈ T (often time).

From a statistical point of view, we consider all available observations x(t) as a realization of the process, which gives rise to certain difficulties. A first problem refers to the fact that the duration over which the process is built is generally infinite, while a realization spans a finite duration. Therefore, it is impossible to represent reality perfectly. A second, much more serious difficulty is that, unlike the random variables problem, the information available about a process is generally reduced to a single realization.

Types of processes

A distinction is usually made between discrete and continuous time processes, with discrete and continuous values.

If the set T is countable, it is called a discrete process or time series, if the set is uncountable it is called a continuous process. The difference is not fundamental: in particular, stationarity, the constancy of statistical properties as a function of time, is defined in the same way. This is not even a practical difference, since the calculations on a continuous process are made by sampling from one realization of the process. The difference is rather in the attitude towards the use of a single embodiment.

There is a somewhat more marked difference between continuous value processes and discrete value count processes. The latter replace the integrals used by the former with algebraic sums.

Examples

The following are examples within the broad group of time series:

telecommunication signals;
biomedical signals (electrocardiogram, encephalogram, etc.);
seismic signals;
the number of sunspots year after year;
the stock exchange index second to second;
the evolution of the population of one municipality year after year;
the waiting time in the tail of each of the users who are coming to a window;
the climate, a gigantic set of interrelated stochastic processes (wind speed, air humidity, etc.) that evolve in space and time;
stochastic processes of order greater than one, such as a series of order 2 time and a zero correlation with the other observations.

Special cases

Stationary Process: A process is stationary in a strict sense if the joint distribution function of any subset of variables is consistent with a shift in time. It is said that a process is stationary in a broad (or weakly stationary) sense when it is verified that:

The theoretical media is independent of time, and
Self-covariance of order s They are only affected by the span of time between the two periods and do not depend on time.

homogeneous process: independent and identically distributed random variables. They are process where the domain has some symmetry and the distributions of finite-dimensional probability have the same symmetry. A special case includes stationary processes, also called homogeneous processes in time.
Márkov Process: those discrete processes in which evolution only depends on the current state and not the previous ones.
Discreet time processes
- Bernoulli Process: they are discreet processes in which the number of events is given by a binomial distribution.
- Galton-Watson Process: It is a type of Markov process with branching.
Continuous time processes:
- Gauss Process: Continuous process in which every linear combination of variables is a normal distribution variable.
- Continuous Márkov Process
  - Gauss-Márkov Process: these are processes, at the same time, of Gauss and of Markov.
  - Feller Process: These are stochastic processes that take values on operator spaces of some functional space.
  - Lévy Process: they are homogeneous Markov processes of "continuous time" that generalize the random walk that is usually defined as "discreet time".
    - Poisson Process: it is a particular case of Lévy process where the time elapsed between jumps follows an exponential distribution and therefore the number of events at an interval is given by a distribution of Poisson.
    - Wiener Process: the increase in the variable between two moments has a gaussian distribution and, therefore, in addition to a Lévy process is a Gauss process simultaneously.
  - Double stochastic process: it is a type of stochastic model used to model certain time series, in which the parameters given by the distributions also vary randomly (hence the term of double stochastic).
    - Cox Process: It is a dually stochastic process that generalizes the Poisson process, where the intensity parameter varies randomly.
- Continuous stochastic process: it is a type of continuous time stochastic process in which the trajectories are also continuous paths.

Mathematical definition

A stochastic process can be defined equivalently in two different ways:

As a set of temporary performances and a random index that selects one of them.
As a set of random variables ${displaystyle X_{t},}$ index $t,$ , since ${displaystyle tin T,}$ , with ${displaystyle Tsubseteq mathbb {R} ,}$ .

A process is called "continuous time" if $T,$ is an interval (usually this interval is taken as ${displaystyle [0,infty)}$ or "discreet time" if $T,$ is a numberable set (only it can assume certain values, usually taken ${displaystyle Tsubseteq mathbb {N} }$ ). Random variables ${displaystyle X_{t},}$ they take values in a set called probabilistic space. Sea ${displaystyle (Omega{mathcal {B}},P)}$ a probabilistic space. In a random sample size $n$ a composite event observed $E$ formed by elemental events $omega$ :

${displaystyle E={omega _{1},omega _{2},...,omega _{n}}subset Omega ,}$ , so that ${displaystyle Ein B,}$ .

The composite event is a subset contained in the sample space and is a Boole algebra $B$ . To each event $omega$ corresponds to a value of a random variable $V$ , so that $V$ is function of $omega$ :

$<math alttext="{displaystyle V=V(omega);qquad omega in Omega,-infty <VV=V(ω ω );ω ω 한 한 Ω Ω ,- - \infty \infty .V.\infty \infty {displaystyle V=V(omega);qquad omega in Omega,-infty θinfty } <img alt="{displaystyle V=V(omega);qquad omega in Omega,-infty <V$

The domain of this function, or the field of variability of the elemental event, is the sample space, and its route, that is the one of the random variable, is the field of the actual numbers. It is called random process to value in ${displaystyle (A,{mathcal {A}})}$ of an element ${displaystyle X=(Omega{mathcal {B}},(X_{t})_{tgeq 0},P)}$ where for everything ${displaystyle tin mathbb {R}X_{t},}$ is a random variable of value in ${displaystyle (A,{mathcal {A}})}$ .

If the event is observed $omega$ in a moment $t$ Time:

<math alttext="{displaystyle V=V(omegat),qquad omega in Omegatin T,-infty <VV=V(ω ω ,t),ω ω 한 한 Ω Ω ,t한 한 T,- - \infty \infty .V.\infty \infty {displaystyle V=V(omegat),qquad omega in Omegatin T,-infty θinfty } <img alt="{displaystyle V=V(omegat),qquad omega in Omegatin T,-infty <V

$V$ thus defines a stochastic process.

Yeah. ${displaystyle ({mathcal {B}}_{t})_{t},}$ It's a filtration, it is called random process adapted to the value in ${displaystyle (A,{mathcal {A}})}$ of an element ${displaystyle X=(omega{mathcal {B}},{mathcal {B}}_{t},(X_{t})_{t},P)}$ Where ${displaystyle X_{t},}$ is a random variable ${displaystyle {mathcal {B}}_{t}}$ - Measurable of value in ${displaystyle (A,{mathcal {A}})}$ . Function ${displaystyle mathbb {R} rightarrow A: tmapsto X_{t}(omega)}$ is called the path associated with the event $omega ,$ .

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