Stochastic

# 2 if there is no arbitrage then there must be an

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

Ask a homework question - tutors are online