I have an equation of the form:

$$dX_{t}=\mu(X_{t})X_{t}dt+\sigma(X_{t})X_tdZ_{t}$$

I know that if I wrote it as $dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dZ_{t}$, I would need strong assumptions on the functions $\mu(\cdot)$ and $\sigma(\cdot)$ to guarantee that a strong solutions exists (usually a Lipschitz thing).

But since I am putting more structure on my equation, by assuming that the drift and volatility can be written as $\mu(X_{t})X_{t}$ and $\sigma(X_{t})X_t$, I was wondering if there is any other theorem (with weaker conditions on $\mu(\cdot)$ and $\sigma(\cdot)$) that guarantees a strong solution.

Thanks!