Let $\mu_t$ and $\sigma_t$ be strictly positive bounded predictable processes and $W_t$ a Wiener process. Consider for $\Delta>0$
$$ X_{\Delta} = \int_0^{\Delta}\mu_s\,ds+\int_0^{\Delta}\sigma_s\,dWs $$
and
$$ Q_{\Delta} =\left\{ \begin{array}{ll} \frac{X_{\Delta}-\mu_0\,\Delta}{\sigma_0\,\sqrt{\Delta}}. & \text{ if }\Delta>0\\ 1 & \text{ if }\Delta=0 \end{array} \right. $$
If I am not wrong I can write
$$ \int_0^{\Delta}\mu_s\,ds = \mu_0\,\Delta+o_P(\Delta),\quad \int_0^{\Delta}\sigma_s\,dWs = \sigma_0\,\sqrt{\Delta}+o_P(\sqrt{\Delta}) $$ whence for $\Delta>0$ $$ Q_{\Delta} = 1+o_p(1). $$ Now consider the random variable ("instantaneous drift")
$$ D_{\Delta} = \frac{Q_{\Delta}-Q_{0}}{\Delta} $$
My doubt is: does $D_{\Delta}$ has a finite limit in probability when $\Delta\rightarrow 0$ ? I get lost since I do not know if the residual $Q_{\Delta}-1$ goes to zero in probability faster or slower than $\Delta$.