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Homework Assignment #3 1: Let B(t), Ft , P be an RN -valued Brownian motion. (i) If ζ is a bounded stopping time relative to {Ft : t ≥ 0}, show that B(t + ζ) − B(ζ), Ft+ζ , P is a Brownian motion that is independent of Fζ . This can be done as an application of Doob’s stopping time theorem to the martingale ei(ξ,B(t))RN + |ξ|2 2 t , Ft , P . (ii) Let ζ be a stopping time, and set B̌(t) = B(t ∧ ζ) − B(t) − B(t ∧ ζ) = 2B(t ∧ ζ) − B(t). In other words, B̌( · ) is the path obtained by reflecting B( · ) at time ζ. Show that B̌(t), Ft , P is a Brownian motion. R (iii) Assume that N = 1, and, for R > 0, define ζR = {t ≥ 0 : B(t) ≥ R} and B̌R (t) = 2B(t ∧ ζR ) − B(t), and observe that ζR = ζ̌R ≡ inf{t ≥ 0 : B̌R (t) ≥ R}, Obviously, P B(t) > a & ζR < t = P B(t) > a) if a > R. Show that if a ≤ R, then P B(t) < a & ζR < t = P B̌R (t) < a & ζ̌R < t = P B(t) > 2R − a , and from this conclude that P(ζR < t) = P(ζR ≤ t) = 2P B(t) > R and P B(t) ≤ a & ζR > t = P B(t) < a − P B(t) > 2R − a for a < R. (iv) Continuing in the setting of (iii), show that P(ζR < ∞) = 1 and that 1 R2 dP(ζR ≤ t) = 1(0,∞) (t)(2πt3 )− 2 e− 2t dt. Next, use Doob’s stopping time to show that λ2 EP e− 2 ζR = e−λR for λ > 0, and use this to recover the result Z ∞ 3 λ2 t R2 t 1 1 t− 2 e− 2 − 2 dt = (2π) 2 dt = (2π) 2 e−λR 0 proved in the first homework assignment.