Finm345A09Homework2Corr - FINM345/STAT390 Stochastic...

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FINM345/STAT390 Stochastic Calculus Hanson Autumn 2009 Lecture 2 Homework: Stochastic Jump and Diffusion Processes (Due by Lecture 3 in Chalk FINM345 Digital Dropbox) You must show your work, code and/or worksheet for full credit. There are 10 points per question if correct answer. Corrections are in Red , 10/06/2009. 1. (a) Show that when 0 s t that E ± W 3 ( t ) ² ² W ( s ) ³ = W 3 ( s ) + 3( t - s ) W ( s ) , justifying every step with a reason, such as a property of the process or a property of conditional expectations. (b) Use this result to derive the primary martingale property for Markov processes: E ± W 3 ( t ) - 3 tW ( t ) ² ² W ( s ) ³ = W 3 ( s ) - 3 sW ( s ) . { Remark: The general technique is to seek the expectation of m th power in the separable form, E h M ( m ) W ( W ( t ) ,t ) ² ² ² W ( s ) i = M ( m ) W ( W ( s ) ,s ) , where M ( m ) W ( W ( t ) ,t ) = W m ( t ) + m - 1 X k =0 α k ( t ) W k ( t ) , satisfied for the sequence of functions { α 0 ( t ) ,...,α m - 1 ( t ) } , that can be recursively solved using the separable form α k ( t ) in the order k = 0 : m - 1 ; or just use the binomial theorem. Obviously, m = 3 here. } 2. (a) Verify that when 0 s t and constant jump rate λ 0 > 0 that E ± P 2 ( t ) ² ² P ( s ) ³ = P 2 ( s )+2 λ 0 ( t - s ) P ( s )+ λ
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This note was uploaded on 04/19/2010 for the course FIN 390 taught by Professor Hansen during the Fall '09 term at University of Illinois, Urbana Champaign.

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Finm345A09Homework2Corr - FINM345/STAT390 Stochastic...

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