The product property for random variables (RVs) [1] (Page 20) provides a way to transform symmetric $\alpha$-stable ($S\alpha S$) random variables to ones with different indices. In particular, let $X$ and $Y$ be independent, strictly stable variables with characteristc exponents $\alpha^{\prime}$ and $\alpha$ respectively. Assume $Y$ to be a positive variable ($\alpha<1$). The product $XY^{1/\alpha^{\prime}}$ has a stable distribution with exponenent $\alpha^{\prime}\alpha$. Choosing one of these RVs to be Gaussian, i.e., $\alpha^{\prime}=2$, one could sample from any $S\alpha S$ distribution by first sampling from a canonical $S\alpha S$ and then using it to modulate the variance of a conditional Gaussian.
Now assume that $X(t)$ and $Y(t)$ are Wiener ($B(t)$) and symmetric Lévy processes ($L(t)$). May I follow a similar procedure and write down: \begin{align} \mathop{}\!\mathrm{d} L(t) &= \sqrt{\frac{1}{2}\phi_t}\mathop{}\!\mathrm{d}B(t), \end{align} where $\phi_t$ is some kind of a process / differential equation, controlling the variance? (We know that the variance of the Levy process is $\frac{1}{2}\phi_t$.)
If this can be written, what is a suitable $\phi_t$?
[1] Samoradnitsky, Gennady. Stable non-Gaussian random processes, Gennady Samorodnitsky, Murad S. Taqqu. 1996.