2 if there is no arbitrage then there must be an

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Unformatted text preview: Theorem 1.27. 5.17. Expectations of hitting times. Consider a Markov chain state space S . Let A ⇢ S and suppose that C = S A is a finite set. Let VA = min{n 0 : Xn 2 A} be the time of the first visit to A. Suppose that g (x) = 0 for x 2 A, while for x 2 C we have X g (x) = 1 + p(x, y )g (y ) y (a) Show that g (XVA ^n ) + (VA ^ n) is a martingale. (b) Conclude that if Px (VA < 1) > 0 for all x 2 C then g (x) = Ex VA , giving a proof of Theorem 1.28. 5.18. Lyapunov functions. Let Xn be an irreducible Markov chain with state space {0, 1, 2, . . .} and let 0 be a function with limx!1 (x) = 1, and Ex (X1 ) (x) when x K . Then Xn is recurrent. This abstract result is often useful for proving recurrence in many chains that come up in applications and in many cases it is enough to consider (x) = x. 5.19. GI/G/1 queue. Let ⇠1 , ⇠2 , . . . be independent with distribution F and Let ⌘1 , ⌘2 , . . . be independent with distribution G. Define a Markov chain by Xn+1 = (Xn + ⇠n ⌘n+1 )+ where y + = max{y, 0}. Here Xn is the workload in the queue at the time of arrival of the nth customer, not counting the service time of the nth customer, ⌘n . The amount of wo...
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