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Unformatted text preview: PDE for Finance, Spring 2011 Homework 6 Distributed 4/18/11, due 5/2/11 . No extensions! 1) This problem develops a continuous-time analogue of the simple Bertsimas & Lo model of Optimal control of execution costs presented in the Section 7 notes. The state is ( w,p ), where w is the number of shares yet to be purchased and p is the current price per share. The control ( s ) is the rate at which shares are purchased. The state equation is: dw =- ds for t < s < T, w ( t ) = w dp = ds + dz for t < s < T, p ( t ) = p where dz is Brownian motion and , are fixed constants. The goal is to minimize, among (nonanticipating) controls ( s ), the expected cost E ( Z T t [ p ( s ) ( s ) + 2 ( s )] ds + [ p ( T ) w ( T ) + w 2 ( T )] ) . The optimal expected cost is the value function u ( w ,p ,t ). (a) Show that the HJB equation for u is u t + H ( u w ,u p ,p ) + 2 2 u pp = 0 for t < T , with Hamiltonian H ( u w ,u p ,p ) =- 1 4 ( p + u p- u w ) 2 ....
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- Fall '11