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Unformatted text preview: PDE for Finance, Spring 2011 – Homework 1 Distributed 1/24/2011, due 2/14/2011. HW must be turned in by the due date to get credit, unless an extension has been granted. 1) Consider the lognormal random walk dy = μydy + σydw starting at y (0) = x . Assume μ 6 = 1 2 σ 2 . The Section 1 notes examine the mean exit time from an interval [ a, b ] where 0 < a < x < b . There we used the PDE for the mean exit time μxu x + 1 2 σ 2 x 2 u xx =- 1 for a < x < b (1) with boundary conditions u ( a ) = u ( b ) = 0 to derive an explicit formula for u . (a) Show that the general solution of (1), without taking any boundary conditions into account, is u = 1 1 2 σ 2- μ log x + c 1 + c 2 x γ with γ = 1- 2 μ/σ 2 . Here c 1 and c 2 are arbitrary constants. [The formula given in the notes for the mean exit time is easy to deduce from this fact, by using the boundary conditions to solve for c 1 and c 2 ; however you need not do this calculation as part of your homework.] (b) Argue as in the notes to show that the mean exit time from the interval (...
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- Fall '11
- Finance, Probability theory, mean exit time