An equality in SDE.

Question

I read an example in Shreve:

---

---

How to get the equality in the last line?

Answer 1

Applying Itô's formula to $f(x):=x^2$ yields:

$$\begin{align*} \Delta_t^2 &= 2 \int_0^t \Delta_s \, d\Delta_s + \underbrace{\langle \Delta \rangle_t}_{0} \\ &=2 \int_0^t \Delta_s (b(s,X_s^1)-b(s,X_s^2)) \, ds \end{align*}$$

using that $$\Delta_t = X_t^1-X_t^2 = \int_0^t (b(s,X_s^1)-b(s,X_s^2)) \, ds.$$