I have a function $\displaystyle f(S,t)$, which satisfies some PDE, with the condition that $\displaystyle f(S,T) = max\{S-1,0\}$. I introduce two new variables $\displaystyle x$ and $\displaystyle \tau$ as follows:

$\displaystyle S = e^x$ and $\displaystyle t=T-\frac{2\tau}_{\sigma^2}$, and define the function $\displaystyle W$ by $\displaystyle f(S,t) = W(x,\tau)$. Is the new initial condition going to be:

$\displaystyle f(S,T)=W(x,\tau(t)=T)=W(x,0)=\{e^{x}-1,0\}?$

Thanks.