I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process.
The well known Black scholes PDE is given by $$ \frac{1}{2}\sigma(x)^2x^2\frac{\partial^2}{\partial x^2}C(x,\tau)+\mu x\frac{\partial}{\partial x}C(x,\tau)-rC(x,\tau)-\frac{\partial}{\partial \tau}C(x,\tau)=0. $$ where the initial condition $C(x,0)=max(x-K,0)$ and $\sigma(x)=\delta x^\beta$.
Taking the Laplace transform with respect to $\tau$, we obtain the following ODE: $$ \frac{1}{2}\delta x^{2\beta+2}\frac{\partial^2}{\partial x^2}\hat{C}(x,\lambda)+\mu x\frac{\partial}{\partial x}\hat{C}(x,\lambda)-(\lambda+r)\hat{C}(x,\lambda)=-max(x-K,0). $$ where $\hat{C}(x,\lambda)=\int_0^\infty e^{-\lambda \tau}C(x,\tau)d\tau$
and the initial condition is transformed to $$ \hat{C}(x,\lambda)=\int_0^\infty e^{-\lambda \tau}C(x,0) d\tau=max(x-K,0)/\lambda $$(is this right??? it seems wrong..)
Then, $\hat{C}(x,\lambda)$ can be analytically formulated by the case $x>K$ and $x\leq K$.
How to get explicit formula for $\hat{C}(x,\tau)$? I can't proceed from this stage.
I know one paper, "(2001 Dmitry) Pricing and Hedging Path-Dependent Options under the CEV", related to this question. However, there's big jumps for me to understand readily. Could you explain it step by step?