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hw4 (2)

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

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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 ρ . The dynamical law is thus dy/ds = f ( y ( s ) , α ( s )) , y ( t ) = x, and the optimal utility discounted to time 0 is u ( x, t ) = max α A ( Z T t e - ρs h ( y ( s ) , α ( s )) ds + e - ρT 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 ρt u ( x, t ) = max α A ( Z T t e - ρ ( s - t ) h ( y ( s ) , α ( s )) ds + e - ρ ( T - t ) g ( y ( T )) ) . (a) Show (by a heuristic argument similar to those in the Section 4 notes) that v satisfies v t - ρv + 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 α A Z t e - ρ ( s - t ) h ( y ( s ) , α ( 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 - ρ ¯ v + H ( x, ¯ v ) = 0 . 2) Recall Example 1 of the Section 4 notes: the state equation is dy/ds = ry - α with y ( t ) = x , and the value function is u ( x, t ) = max α 0 Z τ t e - ρs h ( α ( s )) ds 1

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with h ( a ) = a γ for some 0 < γ <
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hw4 (2) - PDE for Finance Spring 2011 Homework 4...

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