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.


1 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.

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    $\begingroup$ No. They do not mean (5'') and (6''). They mean that the corrected nonlinear part of the drift still satisfies (A3). To be sure, if we label the correction term as $C_1(x)$, they mean: $\| C_1(x) \|_{H_1}^2 \le K (1+ \|x\|_{H_1}^2)$ (linear growth condition) and $\| C_1(x) - C_1(y) \|_{H_1}^2 \le K \| x - y \|_{H_1}^2$ (globally Lipschitz). With the caveat given in my answer, the proof of this result seems math.stackexchange level. $\endgroup$ Aug 25, 2016 at 13:09
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    $\begingroup$ Indeed, since the convolution defining the correction term is bounded and $C^1$, it is straightforward to prove that it is globally Lipschitz. Moreover, math.stackexchange.com/questions/1227314/… $\endgroup$ Aug 25, 2016 at 13:18
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    $\begingroup$ I'm not sure what you mean with "corrected nonlinear part of the drift", but I guess $$C_1(x)=C(x)+\frac12\tilde{\operatorname{tr}}\:Q{\rm D}B(x)B(x)\;,$$ right? $\endgroup$
    – 0xbadf00d
    Aug 26, 2016 at 19:26
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    $\begingroup$ They really mean that $w$ is a $Q$-Wiener process: please note that the left-hand-side of (2.1) involves the eigenvalues of $Q$. I think (2.4) is right, since the role of this term is to eliminate the Ito correction to the standard chain rule -- that is the essence of the Stratonovich description. $\endgroup$ Aug 27, 2016 at 13:39
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    $\begingroup$ I think that the discussion here is complete. However, I still got a question related to your last comment and asked a new question. $\endgroup$
    – 0xbadf00d
    Aug 28, 2016 at 12:58

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