Brownian motion are independent formula_45 random variables. This representation can be obtained using the Karhunen–Loève theorem. 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. The time evolution of the position of the Brownian particle
Brownian model of financial markets for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models. Consider a financial market consisting of formula_1 financial assets, where one of these assets, called a "bond" or "money market", is risk free while the remaining formula_2 assets, called "stocks", are risky. A "financial market" is defined as formula_3 that satisfies the following: Let formula_17 be a probability space, and a formula_18 be D-dimensional Brownian motion stochastic process, with the natural filtration: If formula_20 are the measure 0 (i.e.