Change Variable: Solving PDE with Variable Transformation

[Bazman]

Hi,

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
dX


Lastly

.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
dX^2

= part 4

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

 

[lippyka]

The correct answer is:-df = -deltaf + (delta- r)Sdf+0.5(theta^2)(S^2)d^2f dt dS dS^2I'm sorry for the long question, but I'm really stuck. Any help would be greatly appreciated.Your mistake is in part 4. You have divided through by S twice. You should divide through by S once. This should give you the correct answer. The correct answer should be:-df = -deltaf + (delta- r)Sdf+0.5(theta^2)(S^2)d^2f dt dS dX^2