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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 = at x = 0 is w ( x,t ) = Z t G y ( x, ,t- s ) ( 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...
View Full Document

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online