8.9 Consider a correlated random walk with stay. That is, a walker can move to the right, to the left, or not move at all. Given that the move in the current step is to the right, then in the next step it will move to the right again with probability

a, to the left with probability b and remain in the current position with probability 1 2 a 2

b. Given that the walker did not move in the current step, then in the next step it will move to the right with probability

c, to the left with probability

d, and not move again with probability 1 2 c 2

d. Finally, given that the move in the current step is to the left, then in the next step it will move to the right with probability g, to the left again with probability h, and remain in the current position with probability 1 2 g 2 h. Let the process be represented by the bivariate process fXn; Yng, where Xn is the location of the walker after n steps and Yn is the nature of the nth step (i.e., right, left, or no move). Let π1 be the limiting probability that the process is in “right” state, π0 the limiting probability that it is in the “no move” state, and π21 the limiting probability that it is in the “left” state, where π1 1 π0 1 π21 5 1. Let Π 5 fπ1; π0; π21g.

a. Find the values of π1; π0, and π21.

b. Obtain the transition probability matrix of the process. 8.10 Consider a CTRW fXðtÞjt $ 0g in which the jump size, Θ, is normally distributed with mean μ and variance σ2, and the waiting time, T, is exponentially distributed with mean 1=λ, where Θ and T are independent. Obtain the master equation, Pðx; tÞ, which is the probability that the p the company sells enough treasury bills to bring the cash level up to x, where 0 , x , K. Assume that in any given period the probability that the cash level increases by $1 is p, and the probability that it decreases by $1 is q 5 1 2 p. We define an intervention cycle as the period from the point when the cash level is x until the point when it is either 0 or K. Let T denote the time at which the available cash first reaches 0 or K, given that it starts at level x.

a. What is the expected value of T?

b. What is the mean number of visits to level m up to time T, where 0 , m , K? 8.9 Consider a correlated random walk with stay. That is, a walker can move to the right, to the left, or not move at all. Given that the move in the current step is to the right, then in the next step it will move to the right again with probability

a, to the left with probability b and remain in the current position with probability 1 2 a 2

b. Given that the walker did not move in the current step, then in the next step it will move to the right with probability

c, to the left with probability

d, and not move again with probability 1 2 c 2

d. Finally, given that the move in the current step is to the left, then in the next step it will move to the right with probability g, to the left again with probability h, and remain in the current position with probability 1 2 g 2 h. Let the process be represented by the bivariate process fXn; Yng, where Xn is the location of the walker after n steps and Yn is the nature of the nth step (i.e., right, left, or no move). Let π1 be the limiting probability that the process is in “right” state, π0 the limiting probability that it is in the “no move” state, and π21 the limiting probability that it is in the “left” state, where π1 1 π0 1 π21 5 1. Let Π 5 fπ1; π0; π21g.

a. Find the values of π1; π0, and π21.

b. Obtain the transition probability matrix of the process.