hw4 (2) - PDE for Finance, Spring 2011 Homework 4...

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Unformatted text preview: PDE for Finance, Spring 2011 Homework 4 Distributed 3/21/11, due 4/4/11 . These problems concern deterministic optimal control (Section 4 material). Warning: some of the problems here are a bit laborious (though they are not necessarily difficult). 1) Consider the finite-horizon utility maximization problem with discount rate . The dynamical law is thus dy/ds = f ( y ( s ) , ( s )) , y ( t ) = x, and the optimal utility discounted to time 0 is u ( x,t ) = max A ( Z T t e- s h ( y ( s ) , ( s )) ds + e- T g ( y ( T )) ) . It is often more convenient to consider, instead of u , the optimal utility discounted to time t ; this is v ( x,t ) = e t u ( x,t ) = max A ( Z T t e- ( s- t ) h ( y ( s ) , ( s )) ds + e- ( T- t ) g ( y ( T )) ) . (a) Show (by a heuristic argument similar to those in the Section 4 notes) that v satisfies v t- v + H ( x, v ) = 0 with Hamiltonian H ( x,p ) = max a A { f ( x,a ) p + h ( x,a ) } and final-time data v ( x,T ) = g ( x ) . (Notice that the PDE for v is autonomous, i.e. there is no explicit dependence on time.) (b) Now consider the analogous infinite-horizon problem, with the same equation of state, and value function v ( x,t ) = max A Z t e- ( s- t ) h ( y ( s ) , ( s )) ds. Show (by an elementary comparison argument) that v is independent of t , i.e. v = v ( x ) is a function of x alone. Conclude using part (a) that if v is finite, it solves the stationary PDE- v + H ( x, v ) = 0 ....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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hw4 (2) - PDE for Finance, Spring 2011 Homework 4...

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