Question:
Starting at value 0, the fortune of an investor increases per week by $ 200 with probability 3/8, remains constant with probability 3/8 and decreases by $ 200 with probability 2/8. The weekly increments of the investor's fortune are assumed to be independent. The investor stops the 'game' as soon as he has made a total fortune of $ 2000 or a loss of $1000, whichever occurs first. By using suitable martingales and applying the optional stopping theorem, determine
(1) the probability P2000 that the investor finishes the 'game' with a profit of $2000,
(2) the probability P- 1000 That the investor finishes the 'game' with a loss of $1000
(3) the mean duration of the 'game'