I came across a generalized Black-Scholes equation formulation in [this](https://www.researchgate.net/publication/260912555_A_random_field_LIBOR_market_model) 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!



Stochastic integral with respect to a random field
The notations in the question are ambiguous (Bjørn Kjos-Hanssen showed that the other interpretation cannot be correct). I assume that the expression of interest is given by
$$
g(t) = \int\_0^t \Sigma(s)\,dW(s)\;,
$$
where $W$ is a $C(\mathbb{R})$-valued Wiener process with covariance $\hat c$ (at time $1$) and $\Sigma(s) \in C(\mathbb{R})^\*$ is the finite measure given by $\Sigma(s) f = \int\_{T\_1}^{T\_2}\sigma(s,u)f(u)\,du$. Here, the covariance $\hat c$ is the bilinear map on $C(\mathbb{R})^\*$ such that, for measures $\mu$ and $\nu$,
$$
\hat c(\mu,\nu) = \int c(u,v)\, \mu(du)\,\nu(dv)\;.
$$
Itô isometry then indeed reads
$$
\mathbb{E} g(t)^2 = \int\_0^t \mathbb{E} \hat c(\Sigma(s),\Sigma(s))\,ds \;,
$$
assuming of course that $\Sigma$ is adapted and square integrable.
Regarding references, any book on SPDEs would do, for example "Stochastic Equations in Infinite Dimensions" by Da Prato & Zabczyk or Section 3 of [my lecture notes](http://www.hairer.org/notes/SPDEs.pdf).


