# tu12 - dr t = a b-r t dt σdB t where a and b are positive...

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1. Ross: Chapter 10, Q25. Compute the mean and the variance of (a) R 1 0 tdB ( t ) (b) R 1 0 t 2 dB ( t ) 2. Given f ( t ) = 3 1 t < 2 4 2 t < 3 Find the distribution of R 2 . 5 1 . 5 f ( t ) dB t . 3. Let X t be the stochastic process deﬁned by X t = X 0 exp( μt + σB t ) , where μ and σ are ﬁxed constants, X 0 > 0, and B t is a standard Brownian motion. (a) Write down the SDE satisﬁed by the stochastic process Y t = log ( X t ) , where log is the natural logarithm. (b) Write down the SDE satisﬁed by X t . 4. [September 2001 Institute Examination] Suppose { B t , t 0 } is a stan- dard Brownian motion. The spot rate of interest, r t , is governed by the stochastic diﬀerential equation
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Unformatted text preview: dr t = a ( b-r t ) dt + σdB t , where a and b are positive constants. (a) A stochastic process { U t , t ≥ } is deﬁned by U t = e at r t . i. Derive an equation for the stochastic diﬀerential dU t . ii. Solve the equation to ﬁnd U t . iii. Show that the spot rate satisﬁes r t = b + ( r-b ) e-at + σ Z t e a ( s-t ) dB s . 5. Deﬁne μ ( n,t ) = E [ B n t ] for positive integer n and t ≥ 0. Use Itˆo’s formula to ﬁnd the recursive equation satisﬁed by μ ( n,t ).- End-1...
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## This note was uploaded on 06/12/2011 for the course ASB 2003 taught by Professor Kim during the Three '11 term at University of New South Wales.

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