"Frontmatter". In: Analysis of Financial Time Series

224 CONTINUOUS-TIME MODELSw-0.4 0.0 0.2 0.4 0.60.0 0.2 0.4 0.6 0.8 1.0timew-1.5 -1.0 -0.5 0.0 0.50.0 0.2 0.4 0.6 0.8 1.0timew-0.6 -0.2 0.2w-0.4 0.0 0.2 0.40.0 0.2 0.4 0.6 0.8 1.0time0.0 0.2 0.4 0.6 0.8 1.0timeFigure 6.1. Four simulated Wiener processes.Figure 6.1 shows four simulated Wiener processes on the unit time interval [0, 1].They are obtained by using a simple version of the Donsker’s Theorem in the statisticalliterature with n = 3000; see Donsker (1951) or Billingsley (1968).Donsker’s TheoremAssume that {z i }i=1 n is a sequence of independent standard normal random variates.For any t ∈[0, 1],let[nt] be the integer part of nt.Define w n,t = √ 1 ∑ [nt]n i=1 z i.Thenw n,t converges in distribution to a Wiener process w t on [0, 1] as n goes to infinity.The four plots start with w 0 = 0, but drift apart as time increases, illustrating thatthe variance of a Wiener process increases with time. A simple time transformationfrom [0, 1) to [0, ∞) can be used to obtain simulated Wiener processes for t ∈[0, ∞).Remark: Aformaldefinition of a Brownian motion w t on a probability space(, F, P) is that it is a real-valued, continuous stochastic process for t ≥ 0 withindependent and stationary increments. In other words, w t satisfies1. continuity: the map from t to w t is continuous almost surely with respect tothe probability measure P;2. independent increments: if s ≤ t, w t − w s is independent of w v for all v ≤ s;