Let $Z_\pi$ be the continuous semimartingale defined by $$d(\log Z_\pi(t)) = \gamma(t) dt + \sum_{j=1}^n a_j(t) dW_j(t) $$ where $W_j, j =1, ..., n$ is a standard $n$-dimensional Brownian motion and $$\int_0^t |\gamma(s)| + |a_j(s)|^2 ds < \infty \quad \forall t \ge 0 \text{ a.s.}$$ Why is the following true?
Ignore the part about the change in weights for my question (it is not relevant - this passage is from Stochastic Portfolio Theory, p. 130 by Fernholz). I know that regarding Ito's formula, we have $$Z_\pi^2(t+dt) - Z_\pi^2(t) = \int_t^{t+dt}2Z_\pi(s)dZ_\pi(s) + \int_t^{t+dt}d \langle Z_\pi \rangle_s$$ and that the sum of the squares over a partition of $[0,t]$ and $[0, t+dt]$ will converge to the quadratic variation, but how do we justify the little $o(dt)$ in each case? I do not see how to get here from Ito.
If I'm not mistaken, this would be equivalent to showing $$\int_t^{t+dt} Z_\pi(s) dZ_\pi(s) = Z_\pi(t) (Z_\pi(t+dt) - Z_\pi(t)) + o(dt)$$ and I have no idea how the $o(dt)$ enters here either.
Thanks for any help you can give here!
I think this is a confusion in or abuse of notation. By the "Suppose ..." sentence, $dt$ is a positive "real" time step that should be written $\newcommand{\D}{\mathit{\Delta}}$ $\D t$ to make it different from the differential notation. The following equations are likewise about the differences $\D Z_\pi$ and $\D(Z_\pi^2)$. Then the Landau $o(\D t)$ is the truncation error of that formula (or the Euler-Maruyama method).