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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 continuoustime 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 α ( s ) is the rate at which shares are purchased. The state equation is: dw = αds for t < s < T, w ( t ) = w dp = θαds + σdz for t < s < T, p ( t ) = p where dz is Brownian motion and θ , σ are fixed constants. The goal is to minimize, among (nonanticipating) controls α ( s ), the expected cost E ( Z T t [ p ( s ) α ( s ) + θα 2 ( s )] ds + [ p ( T ) w ( T ) + θw 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 ) + σ 2 2 u pp = 0 for t < T , with Hamiltonian H ( u w ,u p ,p ) = 1 4 θ ( p + θu p u w ) 2 ....
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 Fall '11
 Bayou
 Finance, Normal Distribution, Brownian Motion, Trigraph, Ji

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