hmw3 - formula when necessary. b Assume now that | a ( x...

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Numerical Methods for SDEs, Fall 2006. Course Instructor: Ra´ul Tempone. Homework Set 3, due Thursday Sept 21. Last revised, Aug 27, 2006. Exercise 1 Formulate and motivate a forward Euler method for approximation of the Stratonovich SDE dX ( t ) = a ( t,X ( t )) dt + b ( t,X ( t )) dW ( t ) . Exercise 2 Consider the deterministic differential equation dZ ( t ) = a ( Z ( t )) dt, Z (0) = x 0 , 0 t T, and a perturbation of it, the Ito stochastic differential equation dX ( t ) = a ( X ( t )) dt + bdW ( t ) , X (0) = x 0 , 0 t T, where a is a smooth function and b > 0 is a positive constant. The aim of this excercise is to compare the solution of both equations. Define then the difference e ( t ) = X ( t ) - Z ( t ) . a Consider a ( x ) = ax (linear case) and compute E ( e ( t )) , and var ( e ( t )) . Hint: Use Ito’s
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Unformatted text preview: formula when necessary. b Assume now that | a ( x )-a ( y ) | C a | x-y | with a positive constant C a . Find bounds for the expectation E ( | e ( t ) | 2 ) use it to bound the variance var ( e ( t )) . Discuss what happens as b . c Implement a uniform time step forward Euler discretization of the above equations taking a ( x ) = cos( x ) , b = 0 . 1 and T = 6 . Plot the sample estimator for var ( e ( t )) vs. time, and compare it with the bound obtained in part (b). Use M = 10 3 sample paths and dierent number of time steps: N = 10 , 20 , 40 . 1...
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