Stochastic integral with respect to a random field

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!

• Let's see what happens if we set some things constant that you haven't ruled out could be constant. If $\sigma\equiv 1$ and $W(s,u)=W_s$ (so $W$ does not depend on $u$) then $c\equiv 1$ and it looks like your Ito isometry does not hold: $$t^2 (T_2-T_1)=E\left(\left(t(W_{T_2}-W_{T_1})\right)^2\right)=E\left(\left(\int_0^t\int_{T_1}^{T_2} dW_s du\right)^2\right) \ne E\left(\int\int\int du\,dv\,ds\right)= t(T_2-T_1)^2$$ Jul 7, 2020 at 6:39
• @BjørnKjos-Hanssen Thank you Jul 7, 2020 at 19:06

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.
• Thank you this helps a lot. This is probably a very basic questions but I don't fully understand how you interpret $c$. On the left side of the second equation you have $c$ evaluated at two measures and on the right side $c$ is evaluated at to real numbers. Why is this possible? Jul 7, 2020 at 19:22
• @MartinHairer Usually there is some linear operator $Q$ with finite trace in $C(\mathbb{R})$ so that $W_t-W_s\sim N(0,(t-s)Q)$. What is $Q$ in terms of $c$? Jul 11, 2020 at 18:35
• Maybe @MartinHairer can correct me but I think if we were to think of a Hilbert space valued Weiner process, $Qf(\cdot)=\int c(s,\cdot) f(s)ds$, $f\in H$ and $Q\in L(H).$ Jul 11, 2020 at 23:48