1. Let B(t) he a standard Brownian Motion (BM). Sketch a proof that tB(t- I) is a BM. Hint: use similar way as in printing that c2 B(ei) is a BM, c-const. 2. Let k2-",k = 0,1...,2" be a partition of...

1. Let B(t) he a standard Brownian Motion (BM). Sketch a proof that tB(t- I) is a BM. Hint: use similar way as in printing that c2 B(ei) is a BM, c-const.
2. Let k2-",k = 0,1...,2" be a partition of [0,1], define Ak = 1B (k2-4) - B((k - 1)2-n)r and Sm = Er(Lik) - 2-"). Show that. S„, is a martingale 3. Using the maximal inequality for martingales show that P(maxo n2-"/2)

7. Let 7,, be the first moment when B(•) crosses a positive level a. Argue that P(ra)