Let Xn be the price of a certain stock at the start of the nth day, and assume that X0

Chapter 11, Problem 10

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Let Xn be the price of a certain stock at the start of the nth day, and assume that X0, X1, X2,... follows a Markov chain with transition matrix Q. (Assume for simplicity that the stock price can never go below 0 or above a certain upper bound, and that it is always rounded to the nearest dollar.) s (a) A lazy investor only looks at the stock once a year, observing the values on days 0, 365, 2 365, 3 365,... . So the investor observes Y0, Y1,..., where Yn is the price after n years (which is 365n days; you can ignore leap years). Is Y0, Y1,... also a Markov chain? Explain why or why not; if so, what is its transition matrix? (b) The stock price is always an integer between $0 and $28. From each day to the next the stock goes up or down by $1 or $2, all with equal probabilities (except for days when the stock is at or near a boundary, i.e., at $0, $1, $27, or $28). If the stock is at $0, it goes up to $1 or $2 on the next day (after receiving government bailout money). If the stock is at $28, it goes down to $27 or $26 the next day. If the stock is at $1, it either goes up to $2 or $3, or down to $0 (with equal probabilities); similarly, if the stock is at $27 it either goes up to $28, or down to $26 or $25. Find the stationary distribution of the chain.

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