610hw2 - FE 610 Homework 2 Lena Yemelyanov Due: Wednesday,...

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FE 610 Homework 2 Lena Yemelyanov Due: Wednesday, October 21, 2009 The Full Assignment is from PDF available online Question 1 : Problem statement : Let W t be a Wiener process and t denote denote the time, are the following stochastic processes martingales? If not, how to change the formula to make it a martingale? Draw the figure of them to testify your conclusion. (1) X t W t 2 t (2) X t W t 2 Solution : A Wiener process is illustrated below, it is a combination of normally distributed random variables, so that the mean is 0. 1
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(1) X t W t 2 t is illustrated below in Red, the graph in Blue is the conversion of X t to Wiener process 2
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E M t T E M t X t W t 2 t E X t T E W t T 2 t T  E W t T 2 E t T W t 2 t T W t 2 t 2 T X t T X t Hence X t is not a martingale. From the above it is clear that in order to transform X t into a martingale we need to subtract 2 t Thus M t X t 2 t is a martingale. (2)
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This note was uploaded on 02/27/2010 for the course FE 610 taught by Professor Prasad during the Spring '10 term at Stevens.

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610hw2 - FE 610 Homework 2 Lena Yemelyanov Due: Wednesday,...

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