Random Walks and Brownian Motion
نویسنده
چکیده
Tags for today's lecture: Donsker's invariance priciple, Stochastic integration, Itô's formula In this lecture we show an application of Donsker's invariance principle and then proceed to the construction of Itô's stochastic integral. We recall the definitions and give a simple example of an application of the invariance principle. Consider a random walk S n = Σ n i=1 x i with E(x) = 0, E(x 2) = 1. Let S(t) be its linear interpolation and define S * n (t) = S(nt) √ n t ∈ [0, 1]
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Constructing a sequence of random walks strongly converging to Brownian motion
It is one of the most basic facts in probability theory that random walks, after proper rescaling, converge to Brownian motion. However, Donsker’s classical theorem [Don51] only states a convergence in law. Various results of almost sure convergence exist (see e.g. [KMT75, KMT76] and the references therein) but involves rather intricate relations between the converging sequence of random walks ...
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