Consider a Brownian sheet $W(t_1,\ldots,t_d)$ for some $d\in\mathbb N$. Given some process $X:\mathbb (0,\infty)^d\to\mathbb R$, and time values $T_1,\ldots,T_d$, is there a theory similar to that of Itô calculus that makes sense of/studies stochastic integrals of the form $$\int_{[0,T_1]\times\cdots\times[0,T_d]}X(t_1,\ldots,t_d)~d W(t_1,\ldots,t_d)?$$ After looking in most of the textbooks I know/googling I've only ever seen this theory developed for one time variable.
For example, in dimension two, I expect that we could approximate $X(t_1,t_2)$ by some simple process $X_n$ that is piecewise constant on intervals of the form $$[n^{-1}i,n^{-1}(i+1))\times[n^{-1}j,n^{-1}(j+1)),\qquad i,j\in\mathbb N,$$ and then obtain the above integral as some limit in probability/$L^2$ of a sum inspired by the multivariate Stieltjes integral $$\sum_{i,j} X_n(n^{-1}i,n^{-1}j)\Big( W\big(n^{-1}(i+1),n^{-1}(j+1)\big) -W\big(n^{-1}i,n^{-1}(j+1)\big) -W\big(n^{-1}(i+1),n^{-1}j\big) +W\big(n^{-1}i,n^{-1}j\big) \Big),$$ but before I try to reinvent the wheel I thought I'd ask if anyone knows of a good comprehensive reference for such results.
