I came across a generalized Black-Scholes equation formulation in this paper.

Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion and for a fixed $t$, $W$ is a continuous function. $W$ satisfies the following:

- $dW(t,u)dW(t,v)=c(u,v)dt$
- $dW(t,u)dW(t,u)=dt$

The author's define the dynamics of some asset as follows:

$$\frac{dS(t)}{S(t)}=\mu(t)dt+\int_{T_1}^{T_2}\sigma(t,u)dW(t,u)du$$

The second term on the right side intrigues me. The paper doesn't really get into explaining the doing calculus with such processes so I just want to ask the following.

If I define $dg(t)=\int_{T_1}^{T_2}\sigma(t,u)dW(t,u)du$ then does it follow that

$$dg(t)dg(s)=\int_{T_1}^{T_2}\sigma(t,u)dW(t,u)du \int_{T_1}^{T_2}\sigma(t,v)dW(t,v)dv\\=\int_{T_1}^{T_2}\int_{T_1}^{T_2}\sigma(t,u)\sigma(t,v)c(u,v)dtdudv$$

Do we have the Ito isometry?

$$E\left[\left(\int_0^t\int_{T_1}^{T_2}\sigma(s,u)dW(s,u)du\right)^2\right]=E\left[\int_0^t\int_{T_1}^{T_2}\int_{T_1}^{T_2}\sigma(s,u)\sigma(s,v)c(u,v)dudvds\right]$$

I was not able to find any literature on such integrals. If anyone can suggest any references that would also help a lot. Thanks!