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Brownian Motion and Related Processes 419 8.3.6 8.3.7 8.3.8 8.3.9 The answer is E[A(1)|A(0) = x] = x. Show that E[A(t)|A(0) = x] = x for all t > 0. Let M = max{A(t); t 0} be the largest value assumed by an absorbed Brown- ian motion A(t). Show that Pr{M > z|A(0) = x} = x/z for 0 < x < z. Let t 0 = 0 < t 1 < t 2 < · · · be time points, and define X n = A(t n ), where A(t) is absorbed Brownian motion starting from A(0) = x. Show that {X n } is a nonneg- ative martingale. Compare the maximal inequality (2.53) in Chapter 2 with the result in Problem 8.3.6. Show that the transition densities for both reflected Brownian motion and absorbed Brownian motion satisfy the diffusion equation (8.3) in the region 0 < x < . Let F(t) be a cumulative distribution function and B 0 (u) a Brownian bridge. (a) Determine the covariance function for B 0 (F(t)). (b) Use the central limit principle for random functions to argue that the empir- ical distribution functions for random variables obeying F(t) might be approximated by the process in (a). 8.4 Brownian Motion with Drift Let {B(t); t 0} be a standard Brownian motion process, and let µ and > 0 be fixed.