37. Let X be an irreducible continuous-time Markov chain on the state space S with transition prob-abilities Pjk (t) and unique stationary distribution, and write P(X (t) = j) = aj(t). If c(x) is aconcave function, show that d (t) = Σjes ¹jc(a¡(t)/Ã¡) increases to c(1) as t → ∞.

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Answered: 37. Let X be an irreducible…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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[stochastic process1]
• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain starts in state x, Pμ(A) the probability when the initial state is random with distribution μ. Ty = min{n ≥ 1: X₂ = y} is the first time after 0 that the chain visits state y. Px,y = Px(Ty < ∞) . Ny is the number of visits to state y after time 0. 4. Let Sn, n20 be a random walk on Z with step distribution 1-p 2 P 1) = 1/², P(X₁ = 1) = 2/₁ 2' 2' P(X₂ = 0) P(X₂ = − 1) = for some 0 < p < 1, p ‡ / . We may denote q=1 - p. That is, the increments (X₂)₁ are i.i.d. and S₂ = X₁ + … + Xn for n ≥ 1 and S = 0. Compute E[S2+1 Sn] for n ≥ 1. (b) Show that Mn = (q/p) Sn defines a martingale (with respect to (Xk)k-1). (c) Does the limit limn→∞ Mn exist almost surely? If yes, give a justification. If your answer is no, explain why. (d) Let T be the first time that S is equal to either −3 or 3. Compute P(ST = 3). Hint: You may use, without proof, the fact P(T<∞) = 1 and that the Optional Stopping Theorem…
The sequence (Xn)n>o is a Markov chain with transition matrix 0 을 0 0 0 3 1 0 0 0 4 4 0 } 0 0 0 1 2 1 1 0 0 3 3 0 } 0 } 0 0 0 0 0 0 0 (a) Draw the transition diagram for (Xn). (b) Determine the communicating classes of the chain, stating whether each class is recurrent or transient. (c) Determine the period of each communicating class.
a)
2. Consider a Markov chain {Xn}n≥o having the following transition diagram: 1/2 1 2 5 1/4 3 4 1/2 1/4 6 7 1/2 For this chain, there are two recurrent classes R₁ = {6,7} and R₂ = {1,2,5}, and one transient class R3 = {3,4}. (a) Find the period of state 3. (b) Find f33 and f22. (c) Starting at state 3, find the probability that the chain is absorbed into R₁. (d) Starting at state 3, find the mean absorbation time, i.e., the expected number of steps that the chain is absorbed into R₁ or R₂. Note: there are missing transition probabilities for this chain, but no impact for your solution.
E5.2. Consider a Markov chain with state space S = {1,2,3, 4, 5, 6} and transition matrix 1 0 0 0 0 0 0 0 0 P = 0 0 0 1 0 0 0 0 0 0 0 1 0 (a) Draw the transition graph of this chain. (b) Show that state 1 is absorbing and conclude that it is positive re- current. (c) Show that states 2 and 3 are transient. (d) Verify that states 4, 5 and 6 are positive recurrent.
Please solve for (c)
2. Consider the Markov jump process with generator matrix 1 0 0 −1 1 1 0 −1 0 0 2 -4 2 1007 0 0 0 0000 0 0 0 1 -2 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 and let (P(t)) be the associated transition semigroup. Find lim→∞ P(t). [10] Hint: find the communicating classes and their hitting probabilities.

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37. Let X be an irreducible continuous-time Markov chain on the state space S with transition prob- abilities Pjk (t) and unique stationary distribution, and write P(X (t) = j) = a; (t). If c(x) is a concave function, show that d (t) = Σjes ¹jc(a¡(t)/¹;) increases to c(1) as t → ∞.