Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a separable $\mathbb R$-Hilbert space $U$. I've frequently read that the Itō-Stratonovich correction term would be of the form $$\frac 12{\rm D}\sigma(t,x)\sigma(t,x)\operatorname{tr}Q\;,\tag 2$$ see, for example, A Wong-Zakai theorem for stochastic PDEs (page 2). That doesn't make sense to me, cause the correction term should be the integrand of the drift and hence take values in $H$ while ${\rm D}\sigma(t,x)\in\mathfrak L(H,\mathfrak L(U_0,H))$ ($\mathfrak L(A,B)$ denotes the space of bounded linear operators from $A$ to $B$ and $U_0:=Q^{1/2}U$), $\sigma(t,x)\in\text{HS}L(U_0,H)$ (Hilbert-Schmidt operators from $U_0$ to $H$) and my understanding of the symbol sequence $\operatorname{tr}Q$ is $$\operatorname{tr}Q:=\sum_{n\in\mathbb N}\langle Qe_n,e_n\rangle_U$$ for some orthonormal basis $(e_n)_{n\in\mathbb N}$ of $U$. Thus, the element $(2)$ is in $\mathfrak L(U_0,\mathfrak L(U_0,H))$.

So, what am I missing? Maybe they have a different understanding of $\operatorname{tr}Q$ than I have.


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