Itō formula as presented in "Stochastic Equations in Infinite Dimensions" by Giuseppe Da Prato

Question

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula:

Given Hilbert spaces $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$, a $U$-valued Brownian motion $(W_t)_{t\ge 0}$ and $$X_t=X_0+\int_0^t\varphi_s\;{\rm d}s+\int_0^t\Phi_s\;{\rm d}W_s\tag 1$$ for some $H$-valued random variable $X_0$, $H$-valued stochastic process $(\varphi_t)_{t\ge 0}$ and $\mathfrak L(U,H)$-valued$^1$ stochastic process $(\Phi_t)_{t\ge 0}$, we've got

\begin{equation} \begin{split} f(t,X_t)-f(0,X_0)&=\color{red}{\int_0^t\langle\Phi_s{\rm d}W_s,F_x(s,X_s)\rangle}\\ &\quad\color{blue}{+\text{something unimportant for this question}} \end{split}\tag 2 \end{equation}

for all $F:[0,\infty)\times H\to\mathbb R$ with partial Fréchet derivatives $F_t$, $F_x$ and $F_{xx}$.

Question: What's the definition of the $\color{red}{\text{red}}$ term? (They don't give one in the book).

The proof of the statement can be reduced to the case $\varphi_t=\varphi_0$ and $\Phi_t=\Phi_0$. If $0=t_0<\cdots<t_n=t$ is a partition of $[0,t]$, Taylor's theorem yields$^2$

\begin{equation} \begin{split} f(t,X_t)-f(0,X_0)&=\color{red}{\sum_{i=1}^n\langle\Delta X_i,L_i\rangle}\\ &\quad\color{blue}{+\text{something unimportant for this question}} \end{split}\tag 3 \end{equation}

where $\Delta t_i=t_i-t_{i-1}$, $\Delta X_i=X_{t_i}-X_{t_{i-1}}$ and $$L_i:=F_x(t_{i-1},X_{t_{i-1}})\;.$$ Using $(1)$ and our assumption, the $\color{red}{\text{red}}$ term in $(3)$ is $$\color{red}{\sum_{i=1}^n\langle\Phi_0\Delta W_i,L_i\rangle}\color{blue}{+\sum_{i=1}^n\Delta t_i\langle\varphi_0,L_i\rangle}\tag 4\;.$$ Using the definition of the adjoint operator, the $\color{red}{\text{red}}$ term in $(4)$ is $$\sum_{i=1}^n\langle\Delta W_i,\Phi_0^\ast L_i\rangle_U\;.\tag 5$$ In the middle of page 108 they state (our $\color{red}{\text{red}}$ term from $(3)$ is called $I_2$ there) that $(5)$ converges almost surely to $$\int_0^t\langle\Phi_s{\rm d}W_s,F_x(s,X_s)\rangle\tag 6$$ for $n\to\infty$.

Why?

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$^1$ Let $\mathfrak L(A,B)$ be the space of bounded, linear operators $A\to B$.

$^2$ Notice that we can make sense of $(3)$, since $L_i\in\mathfrak L(H,\mathbb R)\cong H$ by Riesz' representation theorem.

Answer 1

There is exactly a definition of the term $\int_0^t\langle\Phi_s{\rm d}W_s,F_x(s,X_s)\rangle$.

For $\Phi_s$ taking values in $\mathfrak L_2(Q^\frac{1}{2}U,H)$ and satisfying the condition that the integral of $\Phi_s$-'s square-norm in $\mathfrak L_2(Q^\frac{1}{2}U,H)$ is a.s. finite (just called the "Energe Condition" privately), and for $\Psi_s$ a $H$-valued process, one can prove that the process $\Phi_s^*\Psi_s$ defined by $$(\Phi_s^*\Psi_s)(u)=\langle\Phi_su,\Psi_s\rangle\quad\text{for }u\in Q^\frac{1}{2}U$$ has values in $\mathfrak L_2(Q^\frac{1}{2}U,\mathbb R)$ and satisfies the Energe Condition. Hence we can define $$\int_0^t\langle\Phi_s{\rm d}W_s,\Psi_s\rangle:=\int_0^t(\Psi_s^*\Phi_s){\rm d}W_s.$$

I don't know whether there is any comment like above in Prato's book, yet it is noted in another reference book --- "Stochastic Differential Equations in Infinite Dimensions" by L. Gawarecki & V. Mandrekar, in page 61.

Answer 2

Let me split this answer into two parts:

Part 1

Let $U$ and $H$ be arbitrary Hilbert spaces, $L\in\mathcal L(U,H)$ and $x\in H$. As Q. Huang noted in his answer, the authors of Stochastic Differential Equations in Infinite Dimensions$^3$ "define" $$(L^\ast x)u:=\langle Lu,x\rangle\;\;\;\text{for }u\in U\;.\tag 7$$ I hate that, it's awful. Why? Well, cause by definition of the adjoint operator, $$\langle Lu,x\rangle_H=\langle u,L^\ast x\rangle_U\;\;\;\text{for all }u\in U\tag 8$$ and by Riesz’ representation theorem$^4$, $\exists!T\in U'$ with $$Tu=\langle u,L^\ast x\rangle_U\;\;\;\text{for all }u\in U\;.\tag 9$$ Thus, $L^\ast x\in U$ can be identified with $T\in\mathfrak L(U,\mathbb R)$. So, the mapping defined by $(7)$ equals $T$. I hate $(7)$, cause it redefines the symbol sequence $L^\ast x$, wresting the meaning of the individual symbols and hides what actually is going on.

So, with $U$ and $H$ be separable, $Q\in\mathfrak L(U)$ being nonnegative and symmetric with finite trace, $U_0:=Q^{1/2}U$ and $(\Phi_t)_{t\ge 0}$ being $\operatorname{HS}(U_0,H)$-valued, they define $$\int_0^t\langle\Phi_s{\rm d}W_s,\varphi_s\rangle_H:=\int_0^t\langle\;\cdot\;,\Phi_s^\ast\varphi_s\rangle_{U_0}{\rm d}W_s\;\;\;\text{for }t\ge 0\;.\tag{10}$$ However, I'm sure we can make sense of the equality in $(1)$ without defining it (any related comment is welcome).

Part 2

I think that the considerations above and the definition of $(10)$ in the book is needlessly complicated. Since $L_1,\ldots,L_n\in\mathfrak L(H,\mathbb R)$ and (see my other question I will mention in a moment) $$L_i\int_{t_{i-1}}^{t_i}\Phi_s\;{\rm d}W_s=\int_{t_{i-1}}^{t_i}L_i\Phi_s\;{\rm d}W_s\;\;\;\text{for all }i\in\left\{1,\ldots,n\right\}\;,\tag{11}$$ we can conclude that $$S_n:=\sum_{i=1}^nL_i(\Phi_0\Delta W_i)=\sum_{i=1}^nL_i\left(\int_{t_{i-1}}^{t_i}\Phi_0\;{\rm d}W_s\right)=\sum_{i=1}^n\int_{t_{i-1}}^{t_i}L_i\Phi_0\;{\rm d}W_s\;.$$

If we can show (and I hope that we can) that $$\operatorname P\left[\lim_{n\to\infty}S_n=\int_0^tF_x(s,X_s)\Phi_0\;{\rm d}W_s\right]=1\;,\tag{12}$$ we would be done with the proof of this special case and would not need an extra definition for the term $(S_n)_{n\in\mathbb N}$ converges to.

I've asked for $(12)$ in a new question.

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$^3$ While this book has the same title as the book in the question, they are different.

$^4$ $U'=\mathfrak L(U,\mathbb R)$ is the topological dual space of $U$.