I am trying to express the following one-dimensional quadratic Brownian motion process in terms of a one-dimensional squared Bessel process: $$Z_{t}=\alpha^{2}W_{t}^{2}-2\beta W_{t}$$ with $\alpha>0$ and $\beta>0$.
I would like to know in particular what its specific scale and starting point would be. I would expect a squared Bessel with scale $\alpha^2$ that starts from $\beta^2/\alpha^2$ but I am not sure how to get there.
Same question then but with time-dependent parameters: $$Z_{t}=\int_0^t\alpha^{2}_{s}dW_{s}^{2}-2\int_0^t\beta_{s}dW_{s}$$ with $\alpha_{t}>0$ and $\beta_{t}>0$ and adapted (with good conditions).
Addendum: based on jakobdt's remarks, here is what I am thinking...
We have: $$Z_{t} =\alpha^{2}W_{t}^{2}-2\beta W_{t} =\alpha^{2}\left(W_{t}-\frac{\beta}{\alpha^{2}}\right)^{2}-\frac{\beta^{2}}{\alpha^{2}}$$
with: $$Z_{t}+\frac{\beta^{2}}{\alpha^{2}}\geq 0$$
and thus: $$\left|W_{t}-\frac{\beta}{\alpha^{2}}\right|=\frac{1}{\alpha}\sqrt{Z_{t}+\frac{\beta^{2}}{\alpha^{2}}}$$
Deriving $Z_t$ in time: $$dZ_{t} =\alpha^{2}\left[dt+2\left(W_{t}-\frac{\beta}{\alpha^{2}}\right)d\left(W_{t}-\frac{\beta}{\alpha^{2}}\right)\right] =\alpha^{2}\left[dt+2\left|W_{t}-\frac{\beta} {\alpha^{2}}\right|\mathrm{sgn}\left(W_{t}-\frac{\beta}{\alpha^{2}}\right)d\left(W_{t}-\frac{\beta}{\alpha^{2}}\right)\right]$$
where: $$\mathrm{sgn}(w)=\begin{cases} 1 & w\geq0\\ -1 & w<0 \end{cases}$$
Introducing: $$B_{t}=\int_{0}^{t}\mathrm{sgn}\left(W_{s}-\frac{\beta}{\alpha^{2}}\right)d\left(W_{s}-\frac{\beta}{\alpha^{2}}\right)$$
we have: $$\langle B\rangle_{t}=\int_{0}^{t}\mathrm{sgn}^{2}\left(W_{s}-\frac{\beta}{\alpha^{2}}\right)ds=t$$
whence $B_{t}$ is a Brownian motion. Therefore, we can write $$dZ_{t} =\alpha^{2}\left[dt+2\left|W_{t}-\frac{\beta}{\alpha^{2}}\right|dB_{t}\right] =\alpha^{2}\left[dt+\frac{2}{\alpha}\sqrt{Z_{t}+\frac{\beta^{2}}{\alpha^{2}}}dB_{t}\right]$$
Now, if we define $Z'_{t}=Z_{t}+\frac{\beta^{2}}{\alpha^{2}}$, we can as well re-write: $$dZ'_{t} =\alpha^{2}\left[dt+\frac{2}{\alpha}\sqrt{Z'_{t}}dB_{t}\right]$$ with $Z'_{0}=\frac{\beta^{2}}{\alpha^{2}}$. Then $Z'_{t}$ is a squared Bessel process starting from $\frac{\beta^{2}}{\alpha^{2}}$ with some $\alpha$ scaling.