5. (15%) Consider a Markov chain with the N +1 states 0, 1, ..., N and transition probabilities Pij (9) (1 – 7;)N-;, o
Posted: Wed Nov 24, 2021 9:17 am
- 5 15 Consider A Markov Chain With The N 1 States 0 1 N And Transition Probabilities Pij 9 1 7 N O I 1 (32.76 KiB) Viewed 71 times
Please help with part b.
5. (15%) Consider a Markov chain with the N +1 states 0, 1, ..., N and transition probabilities Pij (9) (1 – 7;)N-;, o<i,j<N, TT where e 1- e-2ai/N Ti = 0 <i<N, a > 0. -2a 1 - Note that 0 and N are absorbing states.
= (a). Verify that exp(-2aXt) is a martingale (or, equivalently, prove the identity E (exp(-2aX++1) |X+) = exp(-2aX+)], where Xų is the state at time t (t = 0,1, 2...). (b). Prove that the probability Pn(k) of absorption into state N starting at state k is given by 1 -2ak - e Pn(k) = - e -2aN
Posted: Wed Nov 24, 2021 9:17 am
- 5 15 Consider A Markov Chain With The N 1 States 0 1 N And Transition Probabilities Pij 9 1 7 N O I 1 (32.76 KiB) Viewed 71 times
5. (15%) Consider a Markov chain with the N +1 states 0, 1, ..., N and transition probabilities Pij (9) (1 – 7;)N-;, o<i,j<N, TT where e 1- e-2ai/N Ti = 0 <i<N, a > 0. -2a 1 - Note that 0 and N are absorbing states.
= (a). Verify that exp(-2aXt) is a martingale (or, equivalently, prove the identity E (exp(-2aX++1) |X+) = exp(-2aX+)], where Xų is the state at time t (t = 0,1, 2...). (b). Prove that the probability Pn(k) of absorption into state N starting at state k is given by 1 -2ak - e Pn(k) = - e -2aN