Stochastic

# If the stock price is 120 you will choose to buy the

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Unformatted text preview: 0. show that X = limn!1 Xn has a beta distribution (g + r 1)! g x (g 1)!(r 1)! 1 (1 x)r 1 5.6. An unfair fair game. Deﬁne random variables recursively by Y0 = 1 and for n 1, Yn is chosen uniformly on (0, Yn 1 ). If we let U1 , U2 , . . . be uniform on (0, 1), then we can write this sequence as Yn = Un Un 1 · · · U0 . (a) Use Example 5.5 to conclude that Mn = 2n Yn is a martingale. (b) Use the fact that log Yn = log U1 + · · · + log Un to show that (1/n) log Xn ! 1. (c) Use (b) to conclude Mn ! 0, i.e., in this “fair” game our fortune always converges to 0 as time tends to 1. 176 CHAPTER 5. MARTINGALES 5.7. General birth and death chains. The state space is {0, 1, 2, . . .} and the transition probability has p(x, x + 1) = px p(x, x 1) = qx p(x, x) = 1 px qx for x > 0 for x 0 while the other p(x, y ) = 0. Let Vy = min{n 0 : Xn = y } be the time of the Pz Qy 1 ﬁrst visit to y and let hN (x) = Px (VN < V0 ). Let (z ) = y=1 x=1 qx /px . Show that (x) (a) Px (Vb < Va ) = (b) (a) From this it follows that 0 is recurrent if and o...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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