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In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, and physics. The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and unknown forces in control theory. The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model. == Characterisations of the Wiener process == The Wiener process ''Wt'' is characterised by the following properties:〔Durrett 1996, Sect. 7.1〕 #''W0'' = 0 a.s. #''W'' has independent increments: ''Wt+u - Wt'' is independent of ''σ(Ws : s ≤ t)'' for ''u ≥ 0'' #''W'' has Gaussian increments: ''Wt+u - Wt'' is normally distributed with mean ''0'' and variance ''u'', ''Wt+u−Wt ~ N(0, u)'' #''W'' has continuous paths: With probability ''1'', ''Wt'' is continuous in ''t''. The independent increments means that if 0 ≤ ''s''1 < ''t''1 ≤ ''s''2 < ''t''2 then ''W''''t''1−''W''''s''1 and ''W''''t''2−''W''''s''2 are independent random variables, and the similar condition holds for ''n'' increments. An alternative characterisation of the Wiener process is the so-called ''Lévy characterisation'' that says that the Wiener process is an almost surely continuous martingale with ''W''0 = 0 and quadratic variation (''W''''t'' ) = ''t'' (which means that ''W''''t''2−''t'' is also a martingale). A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ''N''(0, 1) random variables. This representation can be obtained using the Karhunen–Loève theorem. Another characterisation of a Wiener process is the Definite integral (from zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process. The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant, meaning that : is a Wiener process for any nonzero constant α. The Wiener measure is the probability law on the space of continuous functions ''g'', with ''g''(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wiener process」の詳細全文を読む スポンサード リンク
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