Additional Problems on Stochastic Processes

From ECON 520

Here are two additional problems (from previous year homeworks) on stochastic processes.

Problem 1:

Suppose that there is a classroom containing 3 white boards. Initially (at time t = 1) the boards are blank. In each time period, a professor either: (a) fills an extra board with equations (with probability .25); (b) erases all of the boards (with probability .25); or (c) talks and leaves the same number of boards filled with equations. If all of the boards are already filled with equations, then (a) is not an option, and the professor either erases all of the boards (with probability .5) or leaves the boards as they are (with probability .5). If all of the boards are currently empty, then (b) and (c) both result in the boards being empty in the next time period.

Let X_t denote the number of boards filled with equations at time t. Write down the Markov chain transition matrix. Do you expect the chain to converge to a stationary distribution? If so what is the stationary distribution? (Give as precise a characterization as possible.)

Problem 2:

Consider the random walk

where the ε_t are independent N(0,1). Suppose that the initial value Y₁ is distributed as N(1,4).

What are E[Y_t] and V[Y_t], as functions of time t?

Solutions to Additional Problems