You don't need to consider the problem in that complicated way.
Anyway, your thinking is correct. What need to be shown is that:
Under risk neutral measure where D=exp(rt) is used as a numeraire, the stock process should follow:
dS=r*S*dt+sigma*S*dW (we have done it)
while under the so-called forward risk neutral (FRN) measure where the stock St is used as a numeraire, the stock process should follow:
dS=(r+sigma^2)S*dt+sigma*S*dW
So the problem is how can we show this?
Your problem now is C=exp(-rT)*EQ(ST*I{ST>K}) ( Q is a risk neutral measure), and we can construct a measure, say S, in which all the assets price C use S as a numeraire is a martingale.
C0/S0=ES(CT/ST);
In our case is C0/S0=ES(ST*I{ST>K}/ST)=ES(I{ST>K}), you know that there should be only one price for the option, so: C0=exp(-rT)*EQ(ST*I{ST>K})=S0*ES(I{ST>K}), which gives us
EQ(ST*exp(-rT)*I{ST>K})=ES(S0*I{ST>K}/ST) (the expectation is conditional at t=0 so S0 can be put into it)
compare left side to the right side, we know that EQ(ST*exp(-rT)/S0 *X)=ES(X), where X is any random variable in our case its the indicator function(not 特征函数)。
From change of measure theorem, we know that the Likelihood ratio, or the Radon–Nikodym derivative
dS/dQ=ST*exp(-rT)/S0, (of course, here ST is the process under Q measure)
You now have the likelihood ratio ( or R-N derivative) which can be simplified as:
LR(likelihood ratio) =exp(-0.5sigma^2*T+sigma*WT)
=exp(-∫(0 to T)0.5 (-sigma)^2 dt-∫(0 to T) -sigma*dW )
set -sigma=theta
From Girsanov theorem, if a W(t) is a brownian motion under A measure, then W(t)+theta*t is also a browian motion under B measure, where the change in the two measures are linked with the likelihood ratio
=exp(-∫(0 to T) 0.5(theta)^2 dt-∫(0 to T) theta*dW )
In our case W(t) is a brownian motion under Q( risk neutral measure), then, W(t)-sigma*t is a brownian motion under FRN measure where St is used as a numeraire.
Then every thing is simple. dW(t)_S=dW(t)_Q-sigma*dt=>dW_Q=dW(t)_S+sigma*dt
under S measure: dS=r*S*dt+sigma*S*(dW(t)_S+sigma*dt)
so dS=(r+sigma^2)*S*dt+sigma*S*dW(t)_S
done!
the following links are for reference
1. Change of numeraire:
http://en.wikipedia.org/wiki/Numéraire
2. Radon-Nikodym theorem
http://en.wikipedia.org/wiki/Radon–Nikodym_theorem
3. Girsanov theorem
http://en.wikipedia.org/wiki/Girsanov_theorem
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