The following is an excerpt from page 62, chapter 5 in Oksendal's textbook on Stochastic Differential Equations: $$dN_t = r N_t dt + \alpha N_t dB_t$$ or, $$ \dfrac{dN_t}{N_t} = r dt + \alpha dB_t$$ Hence,$$ \int_0^t \dfrac{dN_s}{N_s} = rt + \alpha B_t$$
What does the term $dN_t/N_t$ mean in the second line? The textbook says $$ dX_t = u dt + v dB_t$$ is shorthand for $$X_t(\omega) = X_0 (\omega) + \int_0^t u(s, \omega) ds + \int_0^t v (s, \omega) dB_s$$ but it does not attribute any particular value to $dN_t(\omega)$. Just from this I cannot see any way to give a proper meaning to equations like $ \dfrac{dX_t}{X_t} = u dt + v dB$, or see how dividing by $N_t$ in the first equation is justified. The textbook does not elaborate on this further.
Any help would be appreciated.
SDEs of that type have solutions that stay away from zero, hence, the division by $N_t$ is justified. By Ito's formula the explicit solution of the SDE is $$ N_t=N_0\exp(\alpha B_t-\alpha^2t/2-rt) $$ which is never zero (let's exclute the trivial case $N_0=0\Rightarrow N_t\equiv 0).$