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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.