15 Facts About Stochastic processes

Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.

Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications.

The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.

The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.

Related searches

One of the simplest stochastic processes is the Bernoulli process, which is a sequence of independent and identically distributed random variables, where each random variable takes either the value one or zero, say one with probability and zero with probability.

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent.

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process.

Martingales can be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process resulting in a martingale called the compensated Poisson process.

Levy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.

The main defining characteristics of these Stochastic processes are their stationarity and independence properties, so they were known as Stochastic processes with stationary and independent increments.

Levy process can be defined such that its state space is some abstract mathematical space, such as a Banach space, but the Stochastic processes are often defined so that they take values in Euclidean space.

In 1953 Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.

Theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.