As you know, stock market prices fluctuate in a way that looks very much random. The following model has been used as a model of the price of a share stock. Let X n be the price at day n. Define Y n =...

As you know, stock market prices fluctuate in a way that looks very much random. The following model has been used as a model of the price of a share stock. Let X_n
be the price at day n. Define

Y_n
= log X_n
with Y₀
= 0 and suppose that:

Where a = 1+ p, and p is the mean growth per day. Usually p is very small; for example, if the annual growth is 10%, then
  if we count 250 business days per year. Suppose that en are independent normal variables with mean zero and variance (K loga)².

(a) Compute the expectation,
  and the covariance function,
  Is
  a stationary process?

(b) It is reasonable to believe that K = 30. Compute the probability that the share price rises more than 3.8% (use p = 0.00038) from one day to another. What is the expected number of events of this type during one year (250 days)?