![]() ![]() These depend on the settings such as the level of mathematical rigour. ![]() In the stochastic processes literature there are different definitions of the Wiener process. In future posts I will cover the history and generalizations of this stochastic process. I will also describe some of its key properties and importance. In this post I will give a definition of the standard Wiener process. I have written that and the current post with the same structure and style, reflecting and emphasizing the similarities between these two fundamental stochastic process. The other important stochastic process is the Poisson (stochastic) process, which I cover in another post. The Wiener process is arguably the most important stochastic process. I will use the terms Wiener process or Brownian (motion) process to differentiate the stochastic process from the physical phenomenon known as Brownian movement or Brownian process. But the physical process is not true a Wiener process, which can be treated as an idealized model. The Wiener process is named after Norbert Wiener, but it is called the Brownian motion process or often just Brownian motion due to its historical connection as a model for Brownian movement in liquids, a physical phenomenon observed by Robert Brown. It plays a key role different probability fields, particularly those focused on stochastic processes such as stochastic calculus and the theories of Markov processes, martingales, Gaussian processes, and Levy processes. This continuous-time stochastic process is a highly studied and used object. The Wiener process can be considered a continuous version of the simple random walk. In a previous post I gave the definition of a stochastic process (also called a random process) with some examples of this important random object, including random walks. One of the most important stochastic processes is the Wiener process or Brownian (motion) process. ![]()
0 Comments
Leave a Reply. |