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4703-10-Fall-h8

# 4703-10-Fall-h8 - IEOR 4703 Homework 8 Given a stochastic...

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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 0 , where { B ( t ) : t 0 } 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 0 def = X 0 ) X h 1 = X 0 + a ( X 0 ) h + b ( X 0 ) 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, 0 i N. (1) In what follows, we shall use the piecewise constant approach for simplicity. For example, if T = 1, and we choose h = 1 / 100, then N = 100 (we have partitioned the interval [0 , 100] into 100 subintervals of length 1 / 100) and we have approximated the path { X ( t ) : 0 t 1 }

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4703-10-Fall-h8 - IEOR 4703 Homework 8 Given a stochastic...

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