# Ito's lemma in stochastic volatility models [closed]

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 $$d = S_t^2 \sigma_t^2 dt$$ 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

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 $$t\geq0$$.
• Indeed, but in this case, can we state that $d<S>_t = S_t^2\sigma_t^2dt$, shouldn't this be valid only conditionally on the paths taken by $\sigma_t$?