Construction of Brownian Motion
نویسنده
چکیده
Brownian motion is a stochastic process (Wt, t ≥ 0) such that • W0 = 0. • (Wt2 −Wt1) and (Wt4 −Wt3) are independent, for any t1 < t2 ≤ t3 < t4. • (Wt2 −Wt1) ∼ N(0, t2 − t1). • For almost all ω, the sample function t 7→Wt(ω) is continuous. Since Wt has stationary independent increments, we already know that EWt = 0 and that the covariance function is r(t, s) = t∧ s. Thus the covariance matrix of the random vector X = (Wt1 , . . . ,Wtn) is
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