I am a bit skeptical about the use of Ito formula on the following stochastic process (it's a fractional brownian motion process). Suppose that $W_s$ is a typical Brownian motion with $1/2<\alpha<1$ and the parameters of $\kappa, \theta, \nu$. We have this process \begin{equation} V_t = V_0 + \frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha-1} \kappa(\theta-V_s)ds +\frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1}\nu \sqrt{V_s} dW_s \end{equation} Can the usual Ito lemma be used in this case, since there is a singular integral? I saw proof on letting $g(s) = \frac{1}{\Gamma(\alpha)} (t-s)^{\alpha-1}$ for $s\in [0,t)$ and the above equation becomes \begin{equation} V_t = V_0 + \int_0^t g(s) \kappa(\theta-V_s)ds + \int_0^t g(s)\nu \sqrt{V_s} dW_s \end{equation} Then, the author of the proof apply Ito's formula on $\sigma_t = \sqrt{V_t}$, which gives \begin{equation} \sigma_t = \sigma_0 + \int_0^t (\frac{\kappa \theta}{2}\cdot g(s) -\frac{\nu^2}{8})\frac{1}{\sigma_s} - \frac{\kappa \sigma_s}{2}\cdot g(s) ds + \int_0^t \frac{\nu}{2} \cdot g(s) dWs \end{equation} I do know how to apply the usual Ito lemma, but the question is - is this even the right way of applying Ito lemma on a singular stochastic process / volterra process? Or is there some other way to construct the Ito formula. The formula has been used as an application in finance called rough Heston model.
Any pointers/guidance are much appreciated!
Yes, I think it's illegal to directly apply the traditional Ito's lemma to the (singular) stochastic Volterra equation. As far as I learn, we can find a semimartingale proxy of $V_t$ to make everything work. For example, let $$V_s^t=\int_{0}^s (t-u)^{\alpha-1}\kappa (\theta-V_u)du+ \int_{0}^s (t-u)^{\alpha-1}\sigma\sqrt{V_u}dW_u,$$ which is a semimartingale that coincides with $V_t$ at $s=t$. Thus, we can apply the Ito's lemma to the proxy process, and we can get an expression of $h(V_t)$.
However, by following this approach, the formula I obtain is pretty involved. I personally did some researches about it recently, and using this method would bring some unwanted properties. So, I'm also seeking for some better approach.
BTW, have you solved this problem? Could I know how you thought about it? Many thanks!