In Bernt Oksendal's Stochastic Differential Equations, Chapter 4, one has the following stochastic differential equation (whose solution is geometric Brownian motion):
$$dN_t=rN_tdt+\alpha N_tdB_t\;\;\;\text{ ie } \;\;\; N_t-N_0=r\int_0^t N_sds+\alpha\int_0^tN_sdB_s,$$
where $\alpha,r\in\mathbb{R}$ and $B_t$ is standard Brownian motion (ie $B_0=0$). After assuming that $N_t$ solves the above equation, the author abruptly deduces that
$$\frac{dN_t}{N_t}=rdt+\alpha dB_t \;\;\;\text{ie}\;\;\; \int_0^t\frac{1}{N_s}dN_s= rt+\alpha B_t. \;\;\;(*)$$
I don't understand how he obtained this directly.
What I understand for sure is that if we seek to compute
$$\int_0^t\frac{1}{N_s}dN_s $$
we apply Itô's formula for $Y_t=\ln(N_t)$ (assuming $N_t$ satisfies all the needed conditions). After some computation this yields
$$\frac{1}{N_t}dN_t=d\ln N_t+\frac{1}{2}\alpha^2dt \;\;\text{ i.e } \;\;\int_0^t\frac{1}{N_s}dN_s=\ln(N_t)-\ln(N_0)+\frac{1}{2}\alpha^2t.\;\;\;(**)$$
But at first glance, it does not seem that $(**)$ implies $(*)$. How did he obtain $(*)$? Did he use a method other than the Ito formula or am I missing something?
Thank you for the helpful answers! I've upvoted both and will accept whichever has more votes (in case of tie I'll just leave them be).
It turns out that what I was missing is the definition of the Itô integral with respect to an Itô process, which I could not find in the book. So actually by definition one has that for any Itô process of the form
$$dX_t=\alpha dt+\sigma dB_t,$$
and $Y_t$ an appropriate integrand that
$$\boxed{\int_0^tY_sdX_s:=\int_0^t\alpha Y_s ds + \int_0^t\sigma Y_sdB_s}$$
This justifies the formal notation
$$\frac{1}{N_t}dN_t=\frac{1}{N_t}(rN_tdt+\sigma N_tdB_t)= \alpha dt+\sigma dB_t,$$
and automatically gives $(*)$ when $Y_t=1/N_t$, of course, while assuming $Y_t$ meets all the necessary requirements.
Best Answer
Typically associativity of the integral is proved early on. If $X$ is a semimartingale and the integral $K \cdot X = \int K dX$ makes sense then the integral $(HK) \cdot X = \int HK dX$ makes sense if and only if the integral $H \cdot (K\cdot X) = \int H d(\int K dX)$ makes sense, in which case they are equal. In "differential form", this is $H\, d(K dX) = (HK) dX$. I am assuming you are already comfortable with linearity of stochastic integrals.
Thus taking $dN_t = N_t r dt + N_t\alpha dB_t$ and integrating $1/N_t$ with respect to the semimartingales defined by either side gives $$ (1/N_t) dN_t = (1/N_t) d(N_t r dt + N_t\alpha dB_t) = (1/N_t) d(N_t r dt)+ (1/N_t)d( N_t\alpha dB_t) = r dt + \alpha dB_t. $$
There is no need to use Ito here, it is simply associativity of the integral. The key idea is that we are not "dividing by $N_t$", instead we are integrating $1/N_t$ with respect to two (equal) semimartingales. In differential form, it just looks like dividing.