# Properties of the trace term in the Itō formula

Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where

• $U,H$ are separable $\mathbb R$-Hilbert spaces
• $Q\in\mathfrak L(U)$ is nonnegative and self-adjoint with finite trac
• $W$ is a $Q$-Wiener process
• $u\in C^{1,\:2}(\mathbb R_{\ge 0}\times H,H)$
• $\xi:\mathbb R_{\ge 0}\times H\to\operatorname{HS}(U_0,H)$ with $U_0:=\sqrt QU$ being equipped with the usual inner product

By the Itō formula, we obtain $${\rm d}u^{(n)}(t,X_t)=\left[\frac{∂u^{(n)}}{∂t}(t,X_t)+{\rm D}u^{(n)}(t,X_t)u(t,X_t)+\frac12\text{tr}\left[{\rm D}^2u^{(n)}(t,X_t)\left(ξ(t,X_t)\sqrt Q\right)\left(ξ(t,X_t)\sqrt Q\right)^*\right]\right]{\rm d}t+{\rm D}u^{(n)}(t,X_t)ξ(t,X_t){\rm d}W_t\tag 2$$ for all $n\in\mathbb N$ and hence $${\rm d}u(t,X_t)=\left[\frac{∂u}{∂t}(t,X_t)+\underbrace{{\rm D}u(t,X_t)u(t,X_t)}_{=:\:C_1(t,X_t)}+\frac12\underbrace{\sum_{n∈ℕ}{\rm D}^2u(t,X_t)\left(ξ(t,X_t)\sqrt Qe_n\right)\left(ξ(t,X_t)\sqrt Qe_n\right)}_{=:\:C_2(t,X_t)}\right]{\rm d}t+{\rm D}u(t,X_t)ξ(t,X_t){\rm d}W_t\;,\tag 3$$ where $$u^{(n)}:=\langle u,f_n\rangle_H\;\;\;\text{for }n\in\mathbb N$$ for some orthonormal basis $(f_n)_{n\in\mathbb N}$ of $H$.

Let $C:=C_1+\frac12C_2$. The question is: In analogy to my question about the Itō-Stratonovich correction term, can we show that $$\left\|C(t,x)-C(t,y)\right\|_H\le K\left\|x-y\right\|_H\;\;\;\text{for all }t\ge 0\text{ and }x,y\in H\tag 4$$ and $$\left\|C(t,x)\right\|_H\le K(1+\left\|x\right\|_H)\;\;\;\text{for all }t\ge 0\text{ and }x\in H\tag 5$$ for some $K>0$?

• @NawafBou-Rabee I'm not sure how exactly I need to reformulate it. Can you share your thoughts? – 0xbadf00d Aug 27 '16 at 10:28
• @NawafBou-Rabee $u_t(X_t)$ is the drift of $(1)$. You can imagine that it is the determinstic part of the velocity of the perturbed particle trajectory $t\mapsto X_t$ at time $t$. In the application of the Itō formula, $u$ still denotes the $u$ from $(1)$, cause I want to obtain an expression for the differential velocity ${\rm d}u_t(X_t)$. I hope that answers your question. "I suggest to simply incorporate the Ito correction term to the usual chain rule into the drift": Could you please explain what you mean in more detail? – 0xbadf00d Aug 27 '16 at 12:20
• @NawafBou-Rabee (a) Are you sure? The Itō formula yields$${\rm d}u^{(n)}(t,X_t)=\left[\frac{∂u^{(n)}}{∂t}(t,X_t)+{\rm D}u^{(n)}(t,X_t)u(t,X_t)+\frac12\text{tr}\left[{\rm D}^2u^{(n)}(t,X_t)\left(ξ(t,X_t)\sqrt Q\right)\left(ξ(t,X_t)\sqrt Q\right)^*\right]\right]{\rm d}t+{\rm D}u^{(n)}(t,X_t)ξ(t,X_t){\rm d}W_t$$and hence$${\rm d}u(t,X_t)=\underbrace{\left[\frac{∂u}{∂t}(t,X_t)+{\rm D}u(t,X_t)u(t,X_t)+\frac12\sum_{n∈ℕ}{\rm D}^2u(t,X_t)\left(ξ(t,X_t)\sqrt Qe_n\right)\left(ξ(t,X_t)\sqrt Qe_n\right)\right]}_{=:\:C(t,X_t)}{\rm d}t+{\rm D}u(t,X_t)ξ(t,X_t){\rm d}W_t\;.$$ – 0xbadf00d Aug 27 '16 at 19:25
• @NawafBou-Rabee So, the drift of the SDE for $u$ should be $C$, shouldn't it? (b) In my real application, I know another expression for ${\rm d}u(t,X_t)$ and consider the SDE obtained by equating these two expressions. My problem is that I don't see that the nonlinear part of $C$ satisfies the usual growth and Lipschitz conditions as, for example, presented in Da Prato's book (Google books link) in (7.25) and (7.26). – 0xbadf00d Aug 27 '16 at 19:26
• @NawafBou-Rabee I've completely updated the question. – 0xbadf00d Aug 28 '16 at 13:16

For any $t>0$, the function $u(t,\cdot)$ is twice-differentiable and the function $\xi(t,\cdot)$ is differentiable, in agreement with the context set by the OP. Moreover, these functions satisfy:
(A1) For all $x \in H$, there exists a real constant $C>0$ such that $$\| D^3 u(t,x) \|_H \vee\| D^2 u(t,x) \|_H \vee \| Du(t,x)\|_H \vee \| u(t,x) \|_H \le C \;.$$ (A2) For all $x \in H$, there exists a real constant $C>0$ such that $$\| D\xi(t,x) \|_{L_0^2} \le C \;.$$ where $L_0^2$ is standard notation for the set of linear operators from the Cameron-Martin space $U_0$ to $H$.
Recall that a sufficient condition for a differentiable function to be globally Lipschitz is that its derivative is uniformly bounded. By differentiating $C_1(t,\cdot)$, and invoking (A1), its clear that $C_1(t,\cdot)$ is globally Lipschitz. Similarly, by differentiating $C_2(t,\cdot)$, invoking (A1) and (A2), and using the fact that $\xi(t,x)$ is a Hilbert-Schmidt operator, its clear that $C_2(t,\cdot)$ is globally Lipschitz. The linear growth condition follows directly from (A1) and (A2).