Computing an SDE using Ito's lemma

Question

Let $dS_t= \sigma_t dt +w_t dW_t +r_t dR_t$ where W and R are both independent brownian motions and $dS_t$ is the SDE for S. I am trying to compute $dS_t ^2$ using Ito's Lemma but I am having a hard time doing it, if anyone could help tha²t would be awesome

Answer 1

This is one way to do it by applying Itô's lemma to the process $S_t^2$:

Let $ t \in \mathbb{R}_+$, then, as $ x \rightarrow x²$ is $C²$:

$$S_t^2 = S_0^2 + \int_0^t2S_s\;dS_s + \frac{1}{2}\int_0^t2\;d<S>_s$$

$$= S_0^2 + 2\int_0^tS_s \sigma_s\;ds + 2\int_0^tS_s \omega_s\;dW_s + 2\int_0^tS_s r_s\;dR_s + \int_0^t(\omega^2_s + r^2_s)\;ds $$

$$= S_0^2 + 2\int_0^tS_s \omega_s\;dW_s + 2\int_0^tS_s r_s\;dR_s + \int_0^t\omega^2_s + r^2_s + 2S_s \sigma_s\;ds $$

And then, $$dS^2_t =2S_t(\omega_tW_t + r_tR_t) + (\omega^2_t + r^2_t + 2S_t \sigma_t)dt$$

Where $d<S>_t = (\omega^2_t + r^2_t)dt$ as the two Brownian motions are independant.

I think I have made no mistake(s), if so, please let me know.