Transcription of A Rigorous Introduction to Brownian Motion - …
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A Rigorous Introduction to Brownian Motion - …
A Rigorous Introduction to Brownian MotionAndy DahlAugust 19, 2010 AbstractIn this paper we develop the basic properties of Brownian Motion thengo on to answer a few questions regarding its zero set and its local The Basics12 The Relevant Measure Theory53 Markov Properties of Brownian motion64 Further Properties of Brownian motion91 The BasicsThe concept of a Brownian Motion was discovered when Einstein observedparticles oscillating in liquid. Since fluid dynamics are so chaotic and rapidat the molecular level, this process can be modeled best by assuming theparticles move randomly and independently of their past Motion . We canalso think of Brownian Motion as the limit of a random walk as its timeand space increments shrink to 0. In addition to its physical importance, Brownian Motion is a central concept in stochastic calculus which can beused in finance and economics to model stock prices and interest Brownian Motion DefinedSince we are trying to capture physical intuition, we define a Brownian mo-tion by the properties we want it to have and worry about proving the exis-tence of and explicitly constructing such a process stochastic process{B(t) :t 0}is called ad-dimensionalBrownian motionstarting atx Rdif it has the followingfour properties: Start at x.
0. But we can also look at the process at some time sat which the set fX tj0 t sgis known, and the probability of events occuring past swill depend on this information. De nition 6. A ltration on a probability space (;F;P) is a family fF tjt 0gof ˙-algebras such that F s ˆF t ˆFfor all s<t. A stochastic process fX tjt 0gis adapted to the ...
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