2. (a) (5pts) A stochastic process {Y{}t is a martingale if, for s < t, E(Y7|Y4,0 < u

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2 A 5pts A Stochastic Process Y T Is A Martingale If For S T E Y7 Y4 0 U S Ys T Show That If B 1 (89.25 KiB) Viewed 17 times

2. (a) (5pts) A stochastic process {Y{}t is a martingale if, for s < t, E(Y7|Y4,0 < u <s) = Ys. -t = : Show that, if {B(t)} is the standard Brownian motion, Yé = B(t)2 – sa martingale.. (b) (10pts) Use the above result to show find E(T) where T = inf{t> 0 : B(t) = a or —b}, a, b > 0. State the theorems you need to use. (c) (10pts) For a standard BM, let Ta inf{t > 0: B(t) = a}. Show that E(Ta) (d) (10pts) Let {X(t)} be a BM with a drift, with drift parameter M, variance parameter o2, and X(0) = 0. Assume u > 0 and consider a > 0. Use the fact that B(t) is a martingale to find E(Ta) where = = 0. a - Ta = inf{t > 0: X(t) = a} = inf{t > 0: B(t) = : = ut -}. о