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HO6_09

HO6_09 - Math 236 Stochastic Dierential Equations Final...

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Math 236, Stochastic Differential Equations Final problem set 6 March 12, 2009 These problems are due on Friday March 20. Please put them in my mailbox outside the mathematics department, slip them under the door of my office or give them to me in class. Problem 1 Consider the controlled diffusion process X ( t ) that satisfies the Ito equation dX ( t ) = b ( X ( t ) , U ( t )) dt + σ ( X ( t ) , U ( t )) dB ( t ) with X (0) = x . Here U ( t ) ∈ U is the control process, a non-anticipating function with values in the set U . We assume that the coefficients b ( x, u ) and σ ( x, u ) satisfy the Ito conditions as functions of x , uniformly in u ∈ U . The HJB equation for the value function V ( t, x ) = inf U E t,x { g ( X ( T )) } , t T has the form V t ( t, x ) + inf u {L u V ( t, x ) } = 0 , t < T with terminal conditions V ( T, x ) = g ( x ). Here L u is the generator of the controlled diffusion L u = 1 2 σ 2 ( x, u ) 2 ∂x 2 + b ( x, u ) ∂x We assume here that the HJB equation has a classical solution and denote the unique minimal u by u * = u * ( t, x ). The optimal, Markovian, control is U * ( t ) = u * ( t, X * ( t )) where X * ( t ) is the

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