Stochastic Process in Probability and Random Variables

A stochastic process, also known as a random process, is a collection of random variables that are indexed by some mathematical set. Each probability and random process are uniquely associated with an element in the set. The index set is the set used to index the random variables. The index set was traditionally a subset of the real line, such as the natural numbers, which provided the index set with time interpretation.

Stochastic Process Meaning is one that has a system for which there are observations at certain times, and that the outcome, that is, the observed value at each time is a random variable.

Each random variable in the collection of the values is taken from the same mathematical space, known as the state space. This state-space could be the integers, the real line, or η-dimensional Euclidean space, for example. A stochastic process's increment is the amount that a stochastic process changes between two index values, which are frequently interpreted as two points in time. Because of its randomness, a stochastic process can have many outcomes, and a single outcome of a stochastic process is known as, among other things, a sample function or realization.

Classification 

A stochastic process can be classified in a variety of ways, such as by its state space, index set, or the dependence among random variables and stochastic processes are classified in a single way, the cardinality of the index set and the state space.

When expressed in terms of time, a stochastic process is said to be in discrete-time if its index set contains a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers. Time is said to be continuous if the index set is some interval of the real line. Discrete-time stochastic processes and continuous-time stochastic processes are the two types of stochastic processes. The continuous-time stochastic processes require more advanced mathematical techniques and knowledge, particularly because the index set is uncountable, discrete-time stochastic processes are considered easier to study. If the index set consists of integers or a subset of them, the stochastic process is also known as a random sequence.

If the state space is made up of integers or natural numbers, the stochastic process is known as a discrete or integer-valued stochastic process. If the state space is the real line, the stochastic process is known as a real-valued stochastic process or a process with continuous state space. If the state space is η-dimensional Euclidean space, the stochastic process is known as a η-dimensional vector process or η-vector process.

Examples

You can study all the theory of probability and random processes mentioned below in the brief, by referring to the book Essentials of stochastic processes.

Types of Stochastic Processes

The probability of any event depends upon various external factors. The mathematical interpretation of these factors and using it to calculate the possibility of such an event is studied under the chapter of Probability in Mathematics. According to probability theory to find a definite number for the occurrence of any event all the random variables are counted. These random variables are put together in a set then it is called a stochastic process. For mathematical models used for understanding any phenomenon or system that results from a very random behavior, Stochastic processes are used. Such phenomena can occur anywhere anytime in this constantly active and changing world. 

To make the learning of the Stochastic process easier it has been classified into various categories. If the sample space consists of a finite set of numbers or a countable number of elements such as integers or the natural numbers or any real values then it remains in a discrete time.  For an uncountable Index set, the process gets more complex. It will be taught in higher classes. In this article, we will deal with discrete-time stochastic processes.

Various types of processes that constitute the Stochastic processes are as follows :

Bernoulli Process

The Bernoulli process is one of the simplest stochastic processes. It is a sequence of independent and identically distributed (iid) random variables, where each random variable has a probability of one or zero, say one with probability P and zero with probability 1-P. This process is analogous to repeatedly flipping a coin, where the probability of getting a head is P and its value is one, and the probability of getting a tail is zero. In other words, a Bernoulli process is a series of iid Bernoulli random variables, with each coin flip representing a Bernoulli trial.

Random Walk

Random walks are stochastic processes that are typically defined as sums of iid random variables or random vectors in Euclidean space, implying that they are discrete-time processes. However, some people use the term to refer to processes that change in real-time, such as the Wiener process used in finance, which has caused some confusion and led to criticism. Other types of random walks are defined so that their state spaces can be other mathematical objects, such as lattices and groups, and they are widely studied and used in a variety of disciplines.

The simple random walk is a classic example of a random walk. It is a stochastic process in discrete time with integers as the state space and is based on a Bernoulli process, with each Bernoulli variable taking either a positive or negative value. In other words, the simple random walk occurs on integers, and its value increases by one with probability or decreases by one with probability 1-p, so the index set of this random walk is natural numbers, while its state space is integers. If p=0.5, This random walk is referred to as an asymmetric random walk.

Wiener Process

The Wiener process is a stationary stochastic process with independently distributed increments that are usually distributed depending on their size. The Wiener process is named after Norbert Wiener, who demonstrated its mathematical existence, but it is also known as the Brownian motion process or simply Brownian motion due to its historical significance as a model for Brownian movement in liquids

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The Wiener process, which plays a central role in probability theory, is frequently regarded as the most important and studied stochastic process, with connections to other stochastic processes. It has a continuous index set and states space because its index set and state spaces are non-negative numbers and real numbers, respectively. However, the process can be defined more broadly so that its state space is  -dimensional Euclidean space. The resulting Wiener or Brownian motion process is said to have zero drift if the mean of any increment is zero. If the mean of the increment between any two points in time equals the time difference multiplied by some constant μ, that is a real number, the resulting stochastic process is said to have drift μ.

Almost certainly, a Wiener process sample path is continuous everywhere but differentiable nowhere. It can be thought of as a continuous variation on the simple random walk. Donsker's theorem or invariance principle, also known as the functional central limit theorem, is concerned with the mathematical limit of other stochastic processes, such as certain random walks rescaled.

The Wiener process belongs to several important families of stochastic processes, including the Markov, Lévy, and Gaussian families. The process has a wide range of applications and is the primary stochastic process in stochastic calculus. It is crucial in quantitative finance, where it is used in models such as the Black–Scholes–Merton. The process is also used as a mathematical model for various random phenomena in a variety of fields, including the majority of natural sciences and some branches of social sciences.

Poisson Process

The Poisson process is a stochastic process with various forms and definitions. It is a counting process, which is a stochastic process that represents the random number of points or events up to a certain time. The number of process points located in the interval from zero to some given time is a Poisson random variable that is dependent on that time and some parameter. This process's state space is made up of natural numbers, and its index set is made up of non-negative numbers. This process is also known as the Poisson counting process because it can be interpreted as a counting process.

A homogeneous Poisson process is one in which a Poisson process is defined by a single positive constant. The homogeneous Poisson process belongs to the same class of stochastic processes as the Markov and Lévy processes.

There are several ways to define and generalize the homogeneous Poisson process. This stochastic process is also known as the Poisson stationary process because its index set is the real line. If the Poisson process's parameter constant is replaced with a nonnegative integrable function of t. The resulting process is known as an inhomogeneous or nonhomogeneous Poisson process because the average density of the process's points is no longer constant. The Poisson process, which is a fundamental process in queueing theory, is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.

Do You Know?

What is a stochastic variational inference? A scalable algorithm for approximating posterior distributions is stochastic variational inference. This technique was developed for a large class of probabilistic models and demonstrated with two probabilistic topic models, latent Dirichlet allocation and hierarchical Dirichlet process. We can analyze several large collections of documents using stochastic variational inference: 300K articles from Nature, 1.8M articles from The New York Times, and 3.8M articles from Wikipedia. The stochastic inference is capable of handling large data sets and outperforms traditional variational inference, which can only handle a smaller subset. (We also show that the Bayesian nonparametric topic model outperforms its parametric counterpart.) Stochastic variational inference lets us apply complex Bayesian models to massive data sets.