Construction and Properties of Brownian Motion
نویسنده
چکیده
Brownian motion refers to the mathematical models that are used to describe the random movement of particles immersed in a fluid. This random movement was discovered by botanist Robert Brown, in 1827, while studying pollen grains in water under the microscope. One of the first to describe Brownian motion was Thorvald N. Thiele, in 1880, in a paper on the method of least squares. In 1900, Louis Bachelier, in his PhD thesis applied Brownian motion to the stock and option market fluctuations. In 1905, Albert Einstein, unaware of the earlier work on the subject, discussed Brownian motion in his paper on the molecular kinetic theory of heat. Brownian motion has been used to model thermal noise in electrical circuits, limiting behavior in queueing systems, and random fluctuations in many physical, biological, and economic systems. It is widely used in part because it is among the simplest stochastic processes on a continuous domain. It is also the limit of other stochastic processes. However, it should be noted that it is often convenience rather than accuracy that motivates its use. This paper will give an existence proof of Brownian motion, and then outline some properties of Brownian motion. We begin with a brief review of the normal random variable. For a fixed x ∈ R and t > 0, suppose that Yt,x is an R-valued normal random variable, i.e. the distribution measure μYt,x has a density (where dy = dy1 · · · dyd) dμYt,x = p(t, x, y) dy := (2πt) −d/2 exp 3 −12 ky − xk 2 /t ́ dy. (1)
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