HO5_09 - Math 236 Stochastic Differential Equations Problem...

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Unformatted text preview: Math 236, Stochastic Differential Equations Problem set 5 March 7, 2009 These problems are due on Friday March 13. 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 discrete, Euler approximations X n and Y n to the solution of the SDE’s dX ( t ) = σ ( X ( t )) dB ( t ) , dY ( t ) = b ( Y ( t )) dt + σ ( Y ( t )) dB ( t ) , X (0) = Y (0) = x which are defined by X n +1- X n = σ ( X n )Δ B n +1 , Y n +1- Y n = b ( Y n )Δ t + σ ( Y n )Δ B n +1 Here t = n Δ t is fixed and as Δ t → P { max ≤ k ≤ n | X ( k Δ t )- X k | > δ } → for any δ > 0, along with the same type of convergence for Y n to Y ( t ). The coefficients b and σ satisfy the usual Lipschits and linear growth conditions and Δ B n +1 = B (( n +1)Δ t )- B ( n Δ t ), where B ( t ) is a standard Brownian motion. Let p * ( k,x k ,x k +1 ) be the transition probability density of Y k +1 given Y k = x k , which is a Gaussian with mean x k + b ( x k )Δ t and variance σ 2 ( x k )Δ t . Similarly, let p ( k,x k ,x k +1 ) be the transition probability density of...
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HO5_09 - Math 236 Stochastic Differential Equations Problem...

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