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Change of Variable

  1. Feb 10, 2007 #1

    I have to change the variable in the following eqn, although I know the answer I cannot see how the answer was arrived at:

    We start with a p.d.e in F(S,t)

    -dF = -rF + (r- delta)SdF+0.5(theta^2)(S^2)d^2F
    dt dS dS^2

    We then want to change to X=1/S F(S,t)=Sf(X,t)

    The answer is:

    -df = -deltaf + (delta- r)Sdf+0.5(theta^2)(S^2)d^2f
    dt dS dS^2

    Now the way I see it:

    -dF = -Sdf part 1
    dt dt

    -rF = -r.S.f not sure how this becomes delta? part 2

    (r-delta)SdF=(r-delta)S^2df. dX=(delta-r)S^2df . 1
    dS dX dS dX S^2

    =(delta-r)df .1 (when you divide through with the S from part 1)
    dX S

    =(delta-r)df. X part 3


    .5(theta^2)S^2d^2F=.5(theta^2)S^3d^2 .dX^2
    dS^2 dX^2 dS^2

    =(.5*theta^2).S^3.d^2f . -2
    dX^2 X^3

    =theta^2.d^2f .X where the .X comes from dividing through by S from part 1

    = part 4

    Part 4 clearly is wrong but I'm not sure where I've made the mistake?
  2. jcsd
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