Applied Stochastic Processes for Statistics

Applied Stochastic Processes is a collection of papers on stochastic processes, stochastic equations, and their applications in a variety of scientific fields. One paper discusses stochastic systems, which can be large dynamical multi-input, multi-output systems with randomness in the system itself.
A stochastic or random process is a mathematical object defined as a family of random variables in probability theory and related fields. Stochastic processes are commonly used as mathematical models for systems and phenomena that appear to vary at random. Examples include bacterial population growth, electrical current fluctuation caused by thermal noise, and gas molecule movement. Many disciplines use stochastic processes, including biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography, and telecommunications. Furthermore, the widespread use of stochastic processes in finance has been motivated by seemingly random changes in financial markets.

KEY FEATURE

Markov chains in discrete and continuous time, renewal and regenerative processes, Poisson processes, and Brownian motion are all commonly used in applications.

A stochastic process is one in which there are observations at specific times and the outcome, that is, the observed value at each time, is a random variable.

The increment of a stochastic process is the amount that it changes between two index values, which are frequently interpreted as two points in time.

A stochastic process can have many outcomes due to its randomness, and a single outcome of a stochastic process is known as, among other things, a sample function or realisation.

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    Applications and the study of phenomena have led to the development of new stochastic processes. The Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse and the Poisson process, used by A. K. Erlang to study the number of phone calls made in a given period of time, are two examples of stochastic processes. These two stochastic processes are regarded as the most important and central in stochastic process theory and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in various settings and countries.

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