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Unformatted text preview: solution of homework 2 Problem 1 Let V n ( x ) represent the maximum expected value function before the nth bet given the fortune before the nth bet is x . So we have the following equations: V n ( x ) = max ≤ α ≤ 1 [ p n V n +1 ( x + αx ) + (1 p n ) V n +1 ( x αx )] V N +1 ( x ) = ln( x ) (1) Based on the above equations, we could get: V N ( x ) = max ≤ α ≤ 1 [ p N ln( x + αx ) + (1 p N )ln( x αx )] = max ≤ α ≤ 1 [ p N ln(1 + α ) + (1 p N )ln(1 α )] + ln( x ) The optimal value for α is α * N = (2 p N 1) + and V N ( x ) * = C * N + ln( x ). C * N = p N ln(1 + α * N ) + (1 p N )ln(1 α * N ) By induction, if V n ( x ) = C + ln( x ), C is constant. Then through equation 1, we could get: α * n 1 = (2 p n 1 1) + (2) C * n 1 = p n 1 ln(1 + α * n 1 ) + (1 p n 1 )ln(1 α * n 1 ) (3) V n 1 ( x ) = C * n 1 + C + ln( x ) = C * n 1 + V n ( x ) (4) Therefore, for any 1 ≤ n ≤ N , we have: α * n = (2 p n 1) + (5) C * n = p n ln(1 + α * n ) + (1 p n )ln(1 α * n ) (6) V n ( x ) = N X i = n C * i + ln(...
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This note was uploaded on 03/16/2012 for the course IEOR 466 taught by Professor Richard during the Spring '12 term at Columbia.
 Spring '12
 Richard

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