ma503a-09-hm4

# ma503a-09-hm4 - 5) Assume that X is a solution to the...

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MA503a (Fall 2009) HOMEWORK #4 Due Friday, October 30, 2009 1) Suppose that { B t : t 0 } is a standard Brownian motion. Show that for any constant c > 0, the scaled process W t := cB t/c , t 0 is also a standard Brownian motion. 2) Let { B t : t 0 } be a standard Brownian motion. For any t 0 and n 0, deﬁne Z n = 2 n X i =1 B it/ 2 n - B ( i - 1) t/ 2 n · 2 - t. Calculate EZ n and E [ Z n ] 2 . (Hint: you might want to use Problem 1).) 3) § 4.9 of Bj¨ork’s book: all exercises except for #4.6 and 4.7. 4) § 5.7 of Bj¨ork’s book: #5.1, 5.4, 5.5, 5.6, 5.9, 5.10.
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Unformatted text preview: 5) Assume that X is a solution to the following SDE: dX t = b ( t,X t ) dt + σ ( t,X t ) dB t , X = x, t ≥ , and u = u ( t,x ) is a classical ( C 1 , 2 ) solution to the PDE: 0 = u t + 1 2 σ 2 ( t,x ) u xx + b ( t,x ) u x + c ( t ) u + f ( t,x ); u ( T,x ) = g ( x ) . Argue that u ( t,x ) = E n e R T t c ( s ) ds g ( X T ) + Z T t e R s t c ( r ) dr f ( s,X s ) ds ﬂ ﬂ ﬂ X t = x o . 1...
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## This note was uploaded on 10/08/2010 for the course MA 503 taught by Professor Majin during the Fall '09 term at USC.

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