hw2 (2)

hw2 (2) - PDE for Finance Spring 2011 – Homework 2...

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Unformatted text preview: PDE for Finance, Spring 2011 – Homework 2 Distributed 2/14/11, due 2/28/11 . 1) Consider the linear heat equation u t- u xx = 0 in one space dimension, with discontinuous initial data u ( x, 0) = ( 0 if x < 1 if x > 0. (a) Show by evaluating the solution formula that u ( x,t ) = N x √ 2 t (1) where N is the cumulative normal distribution N ( z ) = 1 √ 2 π Z z-∞ e- s 2 / 2 ds. (b) Explore the solution by answering the following: what is max x u x ( x,t ) as a func- tion of time? Where is it achieved? What is min x u x ( x,t )? For which x is u x > (1 / 10)max x u x ? Sketch the graph of u x as a function of x at a given time t > 0. (c) Show that v ( x,t ) = R x-∞ u ( z,t ) dz solves v t- v xx = 0 with v ( x, 0) = max { x, } . Deduce the qualitative behavior of v ( x,t ) as a function of x for given t : how rapidly does v tend to 0 as x → -∞ ? What is the behavior of v as x → ∞ ? What is the value of v (0 ,t )? Sketch the graph of v ( x,t ) as a function of x for given t > 0. 2) We showed, in the Section 2 notes, that the solution of w t = w xx for t > 0 and x > 0, with w = 0 at t = 0 and w = φ at x = 0 is w ( x,t ) = Z t ∂G ∂y ( x, ,t- s ) φ ( s ) ds (2) where G ( x,y,s ) is the probability that a random walker, starting at x at time 0, reaches y at time...
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hw2 (2) - PDE for Finance Spring 2011 – Homework 2...

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