Stochastic Processes (Week 6)

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Camilla McGee
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1 Stochastic Processes (Week 6) October 30th, Discrete-time Finite Markov Chains 2 Countable Markov Chains 3 Continuous-Time Markov Chains 3.1 Poisson Process 3.2 Finite State Space Kolmogrov s backward and forward equation Kolmogorov s forward and backward equation, Embedded Markov chain Large time behavior Definition 3.1. A continuous-time Markov chain is irreducible if all states communicate, i.e, for each x, y S, there exists a sequence of z 1, z 2,, z j S with α(x, z 1 ), α(z 1, z 2 ),, α(z j 1, z j ), α(z j, y) all strictly positive. About periodicity, it does not occur to continuous-time Markov chain because Lemma 3.1. For any irreducible continuous-time Markov chain, P t has strictly positive entries for all t > 0. Proof. (i). if x = y, it is trivial to check that for any integer n, P t (x, x) Pt/n n (x, x), if P t(x, x) = 0, then P t/n (x, x) = 0 for any integer n, it contradicts with P 0 (x, x) = 1; 1

2 (ii). if x y, by definition, there exists k 1, k 2,, k m such that P t/2 n(x, k 1 )P t/2 n(k 1, k 2 ) P t/2 n(k m 1, k m )P t/2 n(k m, y) lim = n + (t/2 n ) m+1 = α(x, k 1 )α(k 1, k 2 ) α(k m 1, k m )α(k m, y) > 0. That is, P t/2 n(x, k 1 )P t/2 n(k 1, k 2 ) P t/2 n(k m 1, k m )P t/2 n(k m, y) > 0 for sufficiently large n. Therefore, P t (x, y) P t/2 n(x, k 1 )P t/2 n(k 1, k 2 ) P t/2 n(k m, y)p t (m+1)/2 n t(y, y) > 0. Remark: From the proof we can conclude that, an equivalent definition for irreducibility is, a continuous-time Markov chain is irreducible if for every x, y S, P t (x, y) > 0 for some t. Corollary 3.1. A continuous time Markov chain is reducible if and only if its embedded chain is irreducible. Denote H x as the holding time in state x, i.e, H x = inf{t > 0 : X t x, X 0 = x}, and T x,x = inf{t H x : X t = x, X 0 = x}, the amount of time until the Markov chain re-visits state x after the first change of state if X 0 = x. Definition 3.2. State x is called recurrent if with probability 1, the Markov chain will return to state x within a finite interval of time, i,e, P (T x,x < ) = 1. Otherwise, it is called transient. Remark: A state x for a continuous time Markov chain is recurrent/transient if and only if it is recurrent/transient for the embedded discrete-time chain. Consequently, an irreducible Markov continuous time Markov chain is recurrent. Moreover, if it is recurrent, the total amount of time that X t stays at x, 0 I(X s = x)ds is infinite with probability 1. Proof. Note that X 0 = Y 0 = x, define τ x = inf{n 1 : Y n = x}, then T x,x = T 1 + T T τx. T x,x = if and only if τ x =. I(X 0 s = x)ds = Nx H x,k where H x,k s are independent exponential distributed random variables with rate α(x) and N x = lim I(Y n = x). n=0 k=1 2

3 Definition 3.3. For continuous time Markov chain, is said to be an invariant probability distribution if P t = for all t > 0. Lemma 3.2. A nonnegative vector with 1 = 1 is an invariant probability distribution if and only if A = 0. Proof. If P t = for all t > 0, then 0 = d(p t)(y) = π(x) dp t(x, y) = π(x) p t (x, z)a z,y z S = π(x)p t (x, z)a z,y = π(z)a z,y = (A)(y). z S z S Conversely, if A = 0, then ( ) d π(x)p t (x, y) = π(x) dp t(x, y) = π(x) A xz P t (z, y) z S = π(x)a xz P t (z, y) = (A)(z)P t (z, y) = 0. z S z S P t is constant and P t = P 0 =. Note that for an irreducible continuous-time Markov chain X t with finite state space S, the embedded discrete time Markov chain Y n is irreducible, recurrent with finite state space S, hence there exists a unique positive invariant probability vector. Denote the one-step transition matrix of Y n is P. Since A x,y = α(x)p (x, y) for x y, and α(x) for x = y, Through direct calculation, for any η = (η(x)), we have η τ A = 0 η(x)a x,y = 0 y S x y η(x)α(x)p (x, y) = η(y)α(y) P = where = (π(x)), π(x) = η(x)α(x). i.e, this is proportional to the unique invariant probability vector of Y n. Hence, there exists a unique positive probability vector η satisfying η τ A = 0 and η(x) = ( ) 1 π(x) π(x) α(x) α(x). Another method of verification is to follow exercise 3.4 in the textbook: 3

4 Theorem 3.1. For an irreducible continuous-time Markov chain with finite s- tate space S, there is a unique probability vector satisfying A = 0; all the eigenvalues of A have negative real part. Proof. Note that A is the infinitesimal generator for an irreducible continuoustime Markov chain, it has such properties: the row sums equal to 0; diagonal elements are nonnegative; off-diagonal elements are nonnegative. Let a be some positive number greater than the absolute values of all the entries of A, then P = 1 a A + I is the transition matrix for a discrete-time, irreducible, aperiodic Markov Chain: (i). P xy = 1 a A xy + I(x = y) 0; (ii). y P xy = 1 A xy + 1 = 1; a y (iii). P xx = 1 a A xx + 1 > 0; (iv). by definition, for each x, y S, there exists j distinct z 1, z 2,, z j with A xz1, A z1 z 2,, A zj 1 z j, A zj y are strictly positive. S P j+1 xy i.e, x y. P xz1 P z1 z 2 P zj 1 z j P zj y A xz 1 A z1 z 2 A zj 1 z j A zj y a j+1 > 0, For an irreducible, aperiodic Markov chain, by Perron-Frobenius Theorem, P has a unique left eigenvector with eigenvalue 1 and that is a probability vector, all the other eigenvalues of P have absolute values strictly less than 1. Theorem 3.2. For an irreducible continuous-time Markov chain with finite state space S, lim P t =, where A = 0. t + Proof. Fixed t > 0, Q = P t can be considered as the one-step transition probability matrix for an irreducible, aperiodic discrete-time Markov chain with finite state space S, lim Q l = where is the invariant probability distribution l of the Markov chain, it is independent of choice of t > 0. Moreover, A dp t 0 = lim t = lim P ta =. t A 4

5 3.2.3 Exit distributions and hitting times Suppose X t is a continuous time irreducible Markov chain on finite state space S. I. Define T = inf{t 0 : X t x}, i.e, the time of the first exist from x then T is exponential with parameter α(x) if assume X 0 = x, therefore, E(T X 0 = x) = 1/α(x). II. For a fixed state z S, define Y = inf{t 0 : X t = z}, i.e, the time of the first visit to z Define b(x) = E(Y X 0 = x),clearly, b(z) = 0. Denote b = (b(x)) x z. Theorem 3.3. Let Ã be the matrix obtained from A by deleting the row and the column associated to the state z, then b = Ã 1 1. Lemma 3.3. The Ã in Theorem 3.3 is invertible. Proof. We observe that for Ã, the row sums are all nonpositive and at least one of these row sums is strictly negative. Otherwise, α(z, y) = 0 for all y z, contradicts with the irreducibility assumption. From Theorem 3.1, we know that ( πa ) = 0, 1 T π = 1 has a unique positive A T solution. Suppose S = n, then rank = n, and rank(a) = n 1, hence {π : πa = 0} has dimension 1. Denote the adjoint matrix of A by A, then A A = 0, each row of A is a solution to πa = 0 and all entries are nonzero, therefore, Ã is invertible. Proof of Theorem 3.3. By definition, for x z, b(x) = E(Y X 0 = x) = E(T X 0 = x) + = 1 α(x) + i.e, α(x)b(x) = 1 + y S,y x = 1 + y S,y x y S,y x,z 1 T α(x, y) α(x) b(y), α(x, y)b(y) α(x, y)b(y), 5 y S,y x P (X T = y X 0 = x)e(y X 0 = y)

6 which implies 1 + y z A x,y b(y) = 0, i.e, 0 = 1 + Ã b, b = Ã 1 1. Example 3.1. Consider a Markov chain with four states {0, 1, 2, 3}, and infinitesimal generator, A = let z = 3, compute b = (b(x)) x 3 where b(x) = E(Y X 0 = x), Y = inf{t 0 : X t = 3}. Exercises: Ã = , b = Ã 1 1 = (8/3, 5/3, 4/3) Assume = (π(x)) is the invariant probability distribution for an irreducible continuous time Markov chain, let V x (t) = t I(X 0 s = x)ds, i.e, the time spent in state x up to time t, show that V x (t) lim t t = π(x) almost surely, i.e, π(x) is the proportion of time spent in state x over long periods of time. 2. (Detailed balance condition) Please show that (1) For a discrete-time Markov chain with one-step transition probability matrix P and state space S, if a nonnegative vector = (π(x)) satisfies π(x)p(x, y) = π(y)p(y, x) for all x y and π(x) = 1, then is an invariant probability distribution. (2) For a continuous-time Markov chain with infinitesimal generator A and state space S, if a nonnegative vector = (π(x)) satisfies π(x)a x,y = π(y)a y,x for all x y and π(x) = 1, then is an invariant probability distribution. 6

MATH 56A: STOCHASTIC PROCESSES CHAPTER 2

MATH 56A: STOCHASTIC PROCESSES CHAPTER 2 2. Countable Markov Chains I started Chapter 2 which talks about Markov chains with a countably infinite number of states. I did my favorite example which is on