# Definition of the Stratonovich integral in Hilbert spaces

Let

• $T>0$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
• $B$ be a (standard, real-valued) Brownian motion with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$

If $\Phi:\Omega\times[0,T]\to\mathbb R$ is

1. $\mathcal F$-progressively measurable with $$\operatorname P\left[\int_0^T\Phi_t^2\:{\rm d}t<\infty\right]=1\tag 1$$
2. an $\mathcal F$-semimartingale (e.g. an Itō process)

then the Stratonovich integral of $\Phi$ with respect to $B$ up to time $t\in[0,T]$ can be defined to be $$\int_0^t\Phi_s\circ{\rm d}B_s:=\int_0^t\Phi_s\:{\rm d}B_s+[\Phi,B]_t\;,\tag 2$$ where the stochastic integral on the right-hand side has to be understood in the Itō sense and $[\Phi,B]$ denotes the covariation (also known as cross quadratic variation) of $\Phi$ and $B$.

Now, let

• $U$ be a separable $\mathbb R$-Hilbert space
• $Q$ be a bounded, linear, nonnegative and self-adjoint operator on $U$ and $$Q^{-1}:=\left(\left.Q\right|_{\left(\ker Q\right)^\perp}\right)^{-1}$$ denote the pseudo inverse of $Q$
• $U_0:=\sqrt Q$ be equipped with $$\langle u_0,v_0\rangle_{U_0}:=\langle Q^{-1/2}u_0,Q^{-1/2}v_0\rangle_U\;\;\;\text{for }u_0,v_0\in U_0$$
• $W$ be a $Q$-Wiener process with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$
• $\operatorname{HS}(U_0,H)$ denote the space of Hilbert-Schmidt operators from $U_0$ to $H$
• $\Psi:\Omega\times[0,T]\to\operatorname{HS}(U_0,H)$ be $\mathcal F$-predictable with $$\operatorname P\left[\int_0^T\left\|\Psi_t\right\|_{\operatorname{HS}(U_0,\:H)}^2{\rm d}t\right]<\infty\tag 3$$

Question: How can we define the Stratonovich integral of $\Phi$ with respect to $W$?

I don't see an obvious generalization of $(1)$. The problem is that in the setting described above, $[\Psi,W]$ takes values in the space of bounded linear operators from $U$ to $\operatorname{HS}(U_0,H))$ (see, e.g., the book of Da Prato for a definition of $[\Phi,W]$) while the Itō integral process of $\Psi$ with respect to $W$ is $H$-valued.

I don't want to confuse anybody, but let me share a last thought: I've found plenty of papers, e.g. On the relation between the Itō and Stratonovich integrals in Hilbert spaces, which define the Itō-Stratonovich correction term in a way which led me to the assumption that $$\int_0^t\Psi_s\circ{\rm d}W_s:=\int_0^t\Psi_s\:{\rm d}B_s+\sum_{n\in\mathbb N}\left(\left[\Psi,W\right]_te_n\right)\left(\sqrt{\lambda_n}e_n\right)\tag 4$$ for $t\in[0,T]$, where $(e_n)_{n\in\mathbb N}$ is an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N\tag 5$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq[0,\infty)$ (which exists by the Hilbert-Schmidt theorem) and hence $\left(\sqrt{\lambda_n}e_n\right)_{n\in\mathbb N}$ is an orthonormal basis of $U_0$. The series on the right-hand side of $(5)$ looks like some kind of trace and hence might have a more suggestive representation. But maybe I'm wrong.