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MATH 5050: Homework 1 (Due Monday, Feb. 1, at the beginning of lecture) Problem 1 Consider a simple random walk with absorbing boundaries on {0, 1, 2, . . . , 10}. From a site k that is not 0 nor 10 the walk goes equally likely to k + 1 or k − 1. 0 and 10 are absorbing states. The payoff function is [0, 2, 4, 3, 10, 0, 6, 4, 3, 3, 0]. a) Find the optimal strategy and optimal expected gain. b) Find the optimal strategy and optimal expected gain if there is a discount factor of 0.9. c) Find the optimal strategy and optimal expected gain if there each move costs 2. d) Use a computer simulation to verify your answers. Problem 2 Consider the Markov chain on state space {1, 2, . . . , 10} with transition matrix 1 0 0 0 0 0 0 0 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0 0 1 0 0 0 0 0 0 0 0.4 0.1 0 0 0 0.5 0 0 0 0 0.3 0.1 0 0 0.1 0.2 0 0.1 0.1 0.1 . P = 0.2 0.1 0.1 0 0.1 0.4 0 0 0.1 0 0.1 0.2 0.1 0 0.2 0.2 0 0 0.1 0.1 0 0 0 0 0 0 0 1 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0 0.2 0.1 0 0.1 0.1 0.1 The pay off function is [2, 4, 3, 10, 0, 6, 4, 3, 3, 0]. Solve the following questions numerically. a) Find (approximately) the optimal strategy and optimal expected gain. b) Find (approximately) the optimal strategy and optimal expected gain if there is a discount factor of 0.6. c) Find (approximately) the optimal strategy and optimal expected gain if the cost of each move out of the different states is given by [∞, 2, ∞, 1, 0, 0, 1, ∞, 3, 2]. d) Can you find (approximately) the optimal strategy and optimal expected gain if the cost of each move out of the different states is given by [∞, 2, ∞, 1, 0, 0, 1, ∞, 3, 2] AND the pay off is discounted by a factor of 0.6 at each move? 1