Stochastic process

In probability theory and related fields a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and real-world phenomena have repeatedly motivated mathematicians to propose new stochastic processes. Two classic examples are the Wiener process (also called the Brownian motion process) and the Poisson process. Louis Bachelier used the Wiener process to model price changes on the Paris Bourse, while A. K. Erlang used the Poisson process to model the number of phone calls occurring in a given period of time. These two processes are widely treated as central to the theory of stochastic processes, and they were invented 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 with no 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...