I have two questions related to the random time change introduced in Oksendal's textbook on SDEs (page 155-156). Specifically, for Lemma 8.5.6., I have no clue as to why we should define $t_j$ in terms of $\alpha_t$ in such a strange way. Perhaps the biggest confusion comes from the chain of identities in the proof of Lemma 8.5.6., it seems that the author used $$\sum_j \int_{\alpha_j}^{\alpha_{j+1}} f(s,\omega)\,\mathrm{d}B_s = \int_0^{\alpha_t} f(s,\omega)\,\mathrm{d}B_s,$$ but this relation is not that obvious in my mind (perhaps it is linked to the weird way that $t_j$ is defined...) My second question concerns a step in the proof of Theorem 8.5.7, in which the author used the relation $\Delta \tilde{B}_j = \sqrt{c(\alpha_j,\omega)}\,\Delta{B}_{\alpha_j}$. However, this relation is not clear to me, as from (8.5.13), we should have $$\Delta \tilde{B}_j = \tilde{B}_{j+1} - \tilde{B}_j = \int_{\alpha_j}^{\alpha_{j+1}} \sqrt{c(s,\omega)}\,\mathrm{d}B_s.$$ So for the claimed identity to hold, we need at least that $\alpha_{j+1} - \alpha_j$ to be sufficiently small (say of order $\mathrm{d}t$). My deepest thanks for any help on these questions.
Notice that in (8.5.12), the RHS is an integral evolving up to time $\alpha_t$, with the clock of the Brownian motion ticking in time $t$; the Brownian motion on the LHS is evolving in time $\alpha_t$. The definition for $t_j$ is simply a formula that makes the clocks on the LHS and RHS match (on dyadic times, in particular).
The equation $$\sum_j \int_{\alpha_j}^{\alpha_{j+1}}f(s, \omega) dB_s = \int_0^{\alpha_t} f(s,\omega )dB_s$$ is simply linearity of integration plus the fact that $\alpha$ is increasing and continuous.
Finally, the author does not assert that $\Delta \tilde{B}_j = \sqrt{c(\alpha_j, \omega)} \Delta B_{\alpha_j}$. Notice, in particular, that this substitution is made in the context of $k \to \infty$, in which case it is true, precisely because of the reason you state: for large $k$, $\alpha_{j+1} - \alpha_j$ is of the order $dt$.