Math 489/889 Stochastic Processes and Advanced Mathematical Finance Homework 10 Steve Dunbar Due Wednesday, December 8, 2010
<Text-field style="Heading 1" layout="Heading 1">Problem 1</Text-field> <Text-field style="Text" layout="Heading 1"> Simulate the solution of the stochastic differential equation dX(t) = X(t)dt + 2*X(t)*dW on the interval [0, 1] with initial condition X (0) = 1 and step size dt = 1/10. Put your calculations in a table as done in the lesson. Include a record of your coin flips. Note that below I using a freshly generated sum of 10 coin flips in each time step of 1/10. 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<Text-field style="Heading 1" layout="Heading 1">Problem 2</Text-field> Find the solution of the stochastic differential equation dX(t) = X(t) dt + 2 X(t) dW Solution: Guess a solution of the form 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= are parameters to be determined. Then by Ito's formula dX = (a + b^2/2) X dt + b X dW so matching coefficients 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 and b = 2, so a = -1. The solution 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.
<Text-field style="Heading 1" layout="Heading 1">Problem 3</Text-field> Find the mode (the value of the independent variable with the highest probability) of the lognormal probability density function. (Use parameters \316\274 and \317\203.) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVA== print(); # input placeholder 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 Note very carefully that Maple uses different parameters than I do! I define the lognormal in terms of the parameters \316\274 and \317\203^2, but Maple uses the parameters \316\274 and \317\203 to define the same distribution. So be careful to get the distribution you really want! 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJkkjbXVHRiciIiIqJClJJnNpZ21hR0YnIiIjRishIiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=