Solve the following questions appropriately.

X is a stochastic process with a discrete state space and a discrete time set. Show that if non-overlapping increments of this process are independent, then the process satisfies the Markov property. [2] (ii) Show that a white noise process in discrete time with a discrete state space does not have independent increments, but is a Markov process. [2] [Total 4]Clogs plc and Nest ple operate in a duopolistic market. Each company has fixed costs of 10.5m and variable costs of fl per unit. The product is sold for f1.50 per unit. They are considering the amount they should spend on advertising. If they each spend a low amount on advertising, ie film, the total market of 10m units will be shared equally between them. If they each spend a high amount on advertising, ie f4m, the total market of 20m units will be shared equally between them. If one firm spends a low amount and the other a high amount, the total market of 15m units will be shared in the ratio 1:4 respectively. (i) Complete the payoff table below to show the profit received by each company for each combination of strategies. Nest ple chooses High Low High Clogs ple chooses Low [4] (ii) State the maximin and maximax strategies for each firm and explain whether or not a dominant strategy exists for Clogs ple or Nest plc. [3] (iii) Comment on the likely outcome. [2] [Total 9]