Change of variable in $\varphi(s) = t$, effect in $\mathbb{d} W_t$

Question

If I have the stochastic integral $$ \int_0^T f(t)\,\mathbb{d} W_t $$ and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the integration is taken in the Itô's sense and $W_t$ is a Wiener process)?

PS: for deterministic integrals $$ \int_{\varphi(a)}^{\varphi(b)} f(t)\,\mathbb{d}t = \int_a^b f(\varphi(s))\varphi'(s)\, \mathbb{d}t $$

Answer 1

The time change formula for Itô integrals is $$ \int_0^{\alpha_t} v(s,\omega)\,\mathbb{d}W_s = \int_0^{t} v(\alpha_r,\omega)\sqrt{\alpha'_r}\,\mathbb{d}W_r $$ where $\alpha'$ is the derivative with respect to $r$ of $\alpha(r,\omega) = s$.

For references, see Oksendal's book, pg 156, Theorem 8.5.7.