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HW2_solution - solution of homework 2 Problem 1 Let Vn(x...

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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 0 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 0 α 1 [ p N ln( x + αx ) + (1 - p N ) ln( x - αx )] = max 0 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( x ) (7) The above is the strategy of gambling. Problem 2 Let V n ( x, p ) represent the maximum expected value function before the nth bet given the fortune before the nth bet is x and the probability of winning is p . So
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