Ito's lemma in stochastic volatility models [closed]

Question

I couldn't help but notice that in all stochastic volatility models articles I consulted, whenever Ito lema is applied with a process of the sort $$\frac{d S_t}{S_t} = \sigma_t d W_t $$ With $(\sigma_t)$ being a stochastic process.

It's considered that $$d<S_t> = S_t^2 \sigma_t^2 dt$$ Is this justified? Given that $\sigma_t$ is stochastic?

You can find such statment for instance in the original Heston's article (page 14 of the pdf document). https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.139.3204&rep=rep1&type=pdf

Thank you

Answer 1

If we work on a probability space $(\Omega,\mathfrak{F},\mathbb{R})$ endowed with a filtration $\mathbb{F}=(\mathfrak{F}_t)_{t\geq0}$, Itô's Lemma is applicable to Itô processes, requiring the stochastic process $(\sigma_t)_{t\geq0}$ to be:

1. Adapted to the filtration $\mathbb{F}$ i.e. $\sigma_t$ is mesurable w.r.t to $\mathfrak{F}_t$ for any $t\geq0$; and
2. Integrable i.e. $\int_{[0,t]}\sigma^2_s\text{d}s<\infty$ for any $t\geq0$.