hwk10 - E3106 Solutions to Homework 10 Columbia University...

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E3106, Solutions to Homework 10 Columbia University Exercise 10.6 . Since Brownian motion has independent and stationary incre- ments, the probability that you recover your purchase price is the probability that a Brownian motion goes up c by time t . Hence the desired probability is 1 P μ max 0 s t B ( s ) c =1 P ( T c t ) =1 2 2 π Z c/ t e y 2 / 2 dy =1 2 Φ μ c t where the last equality follows from Equation (10.7) in the textbook. Exercise 10.7 . By conditioning on B ( t 1 ), we have P μ max t 1 s t 2 B ( s ) >x = Z −∞ P μ max t 1 s t 2 B ( s ) >x | B ( t 1 )= y f t 1 ( y ) dy If y>x ,itisobv iousthat P μ max t 1 s t 2 B ( s ) >x | B ( t 1 )= y =1 (1) If y x , it follows from the independent and stationary increments of Brownian motion that P μ max t 1 s t 2 B ( s ) >x | B ( t 1 )= y = P μ max t 1 s t 2 ( B ( s ) B ( t 1 )) >x y | B ( t 1 )= y = P μ max t 1 s t 2 ( B ( s ) B ( t 1 )) >x y (by independent increments) =
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hwk10 - E3106 Solutions to Homework 10 Columbia University...

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