The price of a stock moves day by day within the values {$1, $2, $3, $4}. Let Xn, denote the price of the stock on day in. If the price of the stock at day n is Xn =j with j=2 or j = 3, then, the price of the stock the next day is Xn+1 = j-1 with probability 1/3 and Xn+1 = $(j + 1) with probability 2/3. If X„ = $1, then, Xn+1=$2 with probability one. Finally, if X„ = $4, then, Xn+1= $3 with probability 1.
(a) Develop this as a DTMC. Also, write the one-step transition probability matrix.
(b) Classify all states (recurrent/transient). Also, determine periodicity of each state.
(c) Determine the long-run fraction of time stock price is $4? How about $1?
(d) Assume some time in future the stock price becomes $1. Determine the expected number of days until the stock price returns to $1 again?