Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces

Question

I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.

In the notation of the paper, let

• $H,H_1$ be separable $\mathbb R$-Hilbert spaces
• $Q\in\mathfrak L(H)$ be nonnegative and self-adjoint with finite trace
• $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$
• $B:H_1\to\mathfrak L(H,H_1)$

They show (and that is easy to see) that for any $y\in H_1$ and $L\in\mathfrak L(H,\mathfrak L(H,H_1))$, there is a unique $\tilde L(y)\in\mathfrak L(U)$ with $$\langle y,(Lu)v\rangle_H=\langle v,\tilde L(y)u\rangle_U\;\;\;\text{for all }u,v\in H\tag 1$$ and a unique $\tilde{\operatorname{tr}}\:QL\in H_1$ with $$\operatorname{tr}Q\tilde L(y)=\langle y,\tilde{\operatorname{tr}}\:QL\rangle_H\;\;\;\text{for all }y\tag 2$$ given by $$\tilde{\operatorname{tr}}\:QL=\sum_{n\in\mathbb N}(Le_n)(\lambda_ne_n)\;.\tag 3$$

In particular, $$\tilde{\operatorname{tr}}\:Q{\rm D}B(x)B(x)=\sum_{n\in\mathbb N}({\rm D}B(x)B(x)e_n)(\lambda_ne_n)\;\;\;\text{for all }x\in H_1\;.\tag 4$$

In the paper, they consider a SDE $${\rm d}z_t=[Az_t+C(z_t)]{\rm d}t+B(z_t){\rm d}W_t$$ where only the operator $C$ on $H_1$ is of (marginal) interest in the following. They state the following assumptions:

(A3): $\exists K>0$ with $$\left\|C(x)\right\|_{H_1}^2+\operatorname{tr}B(x)Q{B(x)}^\ast\le K(1+\left\|x\right\|_{H_1}^2)\;\;\;\text{for all }x\in H_1\tag 5$$ and $$\left\|C(x)-C(y)\right\|_{H_1}^2+\operatorname{tr}(B(x)-B(y))Q(B(x)-B(y))^\ast\le K\left\|x-y\right\|_{H_1}^2\tag 6\;.$$

(A4): $B$ is continuously Fréchet differentiable with bounded, Lipschitz-continuous derivative.

Question: Now they state that (A4) and the convergence of the series in $(4)$ would imply (A3). Why? I absolutely don't understand this.

Answer 1

To paraphrase what you said, the paper you cite introduces an Ito SDE in (2.3) and a second Ito SDE in (2.4), which has an Ito-Stratonovich correction term. The paper then states that assumptions (A1)-(A4) imply existence and uniqueness of a mild solution for both (2.3) and (2.4). They claim that the latter statement follows from the fact that the Ito-Stratonovich correction term appearing in the drift of (2.4) satisfies their (restrictive) assumptions on the infinite-dimensional drift given in (A3).

This latter conclusion sounds fishy at first glance, because the Ito-Stratonovich correction term does not appear to be globally Lipschitz (i.e., satisfies (A3)). To see what the issue is, suppose $B$ was in $C^2$, then the Frechet derivative of the correction term involves $B$ itself, which is not necessarily bounded by their assumptions. In much simpler terms, if a $C^2$ function $f$ has bounded first derivative then $f'(x) f(x)$ (the correction term) does not always have bounded first derivative; take for instance $f(x) = x + \cos(x)$. However, I think their point is that even though the term $D^2B(x) B(x)$ is not necessarily bounded, its convolution with $Q$ as defined by that series is bounded.