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Problem 4 (Strong Laws of Large Numbers) (8+8=16 points) Elon is a hedge fund manager who is interested in investing Dogecoin, which is a
Problem 4 (Strong Laws of Large Numbers) (8+8=16 points) Elon is a hedge fund manager who is interested in investing Dogecoin", which is a highly volatile cryptocur- rency. Suppose the price of Dogecoin either rises by 50% or drops by 30% with equal probabilities every day, and the outcomes in different days are independent. (a) Suppose Elon always invests half of his current fortune into Dogecoin in each day. Let V denote the total fortune after n days, starting from an initial fortune of V = 1000. What would happen to V as n ? (Hint: Derive the ratio Vn+1/Vn and find out a way to leverage SLLN) (b) On the second thought, Elon decides to take a more general approach. In each day, Elon would invest a fraction a [0,1] of his current fortune into Dogecoin (and in Problem 4(a), Elon chooses a = 0.5). We use the same definition of V and have Vo: 1000. Please find the function g(a) such that as n , = log Vn g(a) n almost surely. (Hint: Try to express log(Vn/Vo) as the sum of a sequence of i.i.d. random variables)
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Making Hard Decisions with decision tools
Authors: Robert Clemen, Terence Reilly
3rd edition
538797576, 978-0538797573
