tu12 - dr t = a ( b-r t ) dt + dB t , where a and b are...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: dr t = a ( b-r t ) dt + dB t , where a and b are positive constants. (a) A stochastic process { U t , t } is dened by U t = e at r t . i. Derive an equation for the stochastic dierential dU t . ii. Solve the equation to nd U t . iii. Show that the spot rate satises r t = b + ( r-b ) e-at + Z t e a ( s-t ) dB s . 5. Dene ( n,t ) = E [ B n t ] for positive integer n and t 0. Use Itos formula to nd the recursive equation satised by ( n,t ).- End-1...
View Full Document

Ask a homework question - tutors are online