3
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

See Differentiation formulas for stochastic integrals in the plane by Wong and Zakai (and references therein) for such an Itô-style calculus in two dimensions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.