assignment4 - 3. Determine the probability law (including...

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Mathematical Finance Assignment 4 - to be handed in on or before November 05, 2009. Grading scheme: 20 points per question. Let ( B t ) t R + a standard Brownian motion. 1. Let c > 0. Among the following processes, tell which is a standard Brownian motion and which is not. Justify your answer. (a) ( B c + t - B c ) t R + . (b) ( cB t/c 2 ) t R + . (c) ( B ct 2 ) t R + . 2. Compute the stochastic integrals Z T 0 2 dB t and Z T 0 (2 × 1 [0 ,T/ 2] ( t ) + 1 ( T/ 2 ,T ] ( t )) dB t and determine their probability laws (including mean and variance).
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Unformatted text preview: 3. Determine the probability law (including mean and variance) of the stochastic integral Z 2 π sin( t ) dB t . 4. Compute E [ B t B s ] in terms of s,t ≥ 0. 5. Let T > 0. Show that if f is a differentiable function with f (0) = f ( T ) = 0 we have Z T f ( t ) dB t =-Z T f ( t ) B t dt. Hint: Apply Itˆo’s calculus to t 7→ f ( t ) B t ....
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This note was uploaded on 11/02/2009 for the course MATH 151 taught by Professor Pyke during the Spring '08 term at Simon Fraser.

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