My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly see where the problem lies.
Let a diffusion $X_t$ be defined as: $$ dX_t = 3dt + 2\sqrt{X_t}dB_t $$
The final objective is to find $\mathbb{P}\left(\tau_0 < \tau_5 \right)$, which by definition is computed as: $$ \mathbb{P}\left(\tau_0 < \tau_5 \right) = \frac{s(5) - s(x)}{s(5) - s(0)} $$ where $s(x)$ is the scale function, defined as: $$ s(x) = \int_{x_0}^{x}exp\left(-{\int_{y_0}^{y}\frac{2\mu(z)}{\sigma^2(z)}dz}\right)dy $$
Because the scale function is unique up to an affine transformation, one can take any $x_o$ and $y_o$ in the real line. Evidently, $\mu(x) = 3$ and $\sigma(x) = 2\sqrt{x}$. For convenience I take $y_0 = 1$ and obtain the final expression for $s(x)$:
$$ s(x) = \int_{x_0}^{x}exp\left(-{\int_{1}^{y}\frac{3}{2z}dz}\right)dy = \int_{x_0}^{x}y^{-\frac{3}{2}}dy $$
And here comes the problem:
- if I choose $x_0 = 0$, then I don't have a problem calculating $s(0)$ (which is clearly $0$), but now when I calculate $$s(5) = \int_{0}^{5}y^{-\frac{3}{2}}dy = -\frac{2}{\sqrt{5}} + \frac{2}{0}$$ which is an indeterminate form, which means something earlier has gone wrong.
- if I choose $x_0 \ne 0$, then I face the same problem when calculating $s(0)$.
Now the question is: how can I correctly separate the calculation of $s(x)$ so that I can obtain two different valid expressions for the scale function, one for the case where $x = 0$ and $x\ne 0$?
Let's see if somebody can help.