Brownian bridge of the Brownian bridge based on the Brownian motion is, for "t" ∈ [0, T] Conversely, for "t" ∈ [0, ∞] The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as where formula_9 are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem). A Brownian bridge is the result of Donsker's theorem in the area of empirical processes. It is also used in the Kolmogorov–Smirnov test in the area of statistical inference. A standard Wiener process satisfies "W"(0) = 0 and is therefore "tied down" to the origin, but other points are not
Brownian motion was able to rule out that the motion was life-related, although its origin was yet to be explained. The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian motion model of the stock market is often cited, but Benoit Mandelbrot rejected its applicability to stock price movements in part because these are