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.



Itô integral with respect to Brownian sheet
See [Differentiation formulas for stochastic integrals in the plane](http://www.sciencedirect.com/science/article/pii/0304414978900315 "Differentiation formulas for stochastic integrals in the plane") by Wong and Zakai (and references therein) for such an Itô-style calculus in two dimensions. 


