A company’s cash position, measured in millions of dollars, and denoted by (Xt) follows an arithmetic Brownian motion with dynamics dXₜ = αdt + σdWt along with the drift rate of α = 1.2 per annum and variance rate of σ2 = 1.92 per annum. The initial cash position is X₀ = 2.
(a) What are the probability distributions of the cash position at the end of 1 month, 1 year and 3 years?
(b) What are the probabilities of a negative cash position at the end of 1 month, 1 year and 3 years?