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Unformatted text preview: IEOR 4703: Homework 8 Given a stochastic differential equation (SDE) for a diffusion, dX ( t ) = a ( X ( t )) dt + b ( X ( t )) dB ( t ) , X (0) = X , where { B ( t ) : t } denotes a standard BM, the Euler method for approximating the sample paths of X = { X ( t ) : 0 t T } is given by choosing a small h > 0, setting N = b T/h c , the integer part of T/h , then generating N iid N (0 , 1) rvs, Z 1 ,...,Z N , and using the recursion (with X h def = X ) X h 1 = X + a ( X ) h + b ( X ) hZ 1 X h 2 = X h 1 + a ( X h 1 ) h + b ( X h 1 ) hZ 2 . . . X h N = X h N 1 + a ( X h N 1 ) h + b ( X h N 1 ) hZ N . This approximation yields N values, { X h i : 1 i N } , which approximate X ( t ) only at the time points t { h, 2 h,...,Nh } , that is X ( ih ) X h i . To get an approximation over the entire time interval [0 ,T ] (denoted by { X h ( t ) : 0 t T } ) we can either linearly interpolate between time points (yielding continuous sample paths), or use the piecewise constant approach via X h ( t ) def = X h i , ih t < ( i + 1) h, i N. (1) In what follows, we shall use the piecewise constant approach for simplicity....
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This note was uploaded on 03/14/2012 for the course IEOR 4703 taught by Professor Sigman during the Spring '07 term at Columbia.
 Spring '07
 sigman

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