Let
- $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
- $T>0$
- $I:=(0,T]$
- $(\mathcal F_t)_{t\in\overline I}$ be a complate and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
- $M$ be a real-valued continuous square-integrable $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$
- $\mu_M$ denote the Doléans measure corresponding to $M$
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be open
Now, let $F:\Omega\times\overline I\times\Lambda\to\mathbb R$ with $$F_t\in C^1(\Lambda)\;\;\;\text{almost surely for all }t\in\overline I\tag1$$ and $$F(x)\in\mathcal L^2(\mu_M)\;\;\;\text{for all }x\in\Lambda\tag2.$$ Moreover, let $i\in\left\{1,\ldots,d\right\}$ and assume that $$\int\sup_{x\in K}\left|\frac{\partial F}{\partial x_i}(x)\right|^2\:{\rm d}\mu_M<\infty\;\;\;\text{for all compact }K\subseteq\Lambda\tag3.$$
Let $$N(x):=F(x)\cdot M$$ denote the Itō integral process of $F(x)$ with respect to $M$ for $x\in M$. Are we able to conclude that $N$ is partially differentiable with respect to the $i$th-variable?
For the sake of simplicity, assume $\Lambda=\mathbb R^d$. Let $x\in\Lambda$ and $$G(h):=\frac{N(x+he_i)-N(x)}h\;\;\;\text{for }h\in\mathbb R\setminus\left\{0\right\}.$$ Clearly, $$G(h)=\frac{F(x+he_i)-F(x)}h\cdot M\;\;\;\text{almost surely for all }h\in\mathbb R\setminus\left\{0\right\}\tag4$$ and $$\frac{F_t(x+he_i)-F_t(x)}h\xrightarrow{h\to0}\frac{\partial F_t}{\partial x_i}(x)\;\;\;\text{almost surely for all }t\in\overline I\tag5$$ From $(3)$ and $(5)$, we should be able to conclude that the convergence in $(5)$ holds in $L^2(\mu_M)$ by applying Lebesgue's dominated convergence theorem. This should yield that the corresponding Itō integral processes converge in the space of square-integrable martingales.
Am I missing something?
By the definition of the Doléans measure, we have
$$\begin{align*} \left\| G(h)- (\partial_{x_i} F(x)) \bullet M \right\|_{L^2(\mathbb{P})}^2 &= \left\| \left( \frac{F(x+h e_i)-F(x)}{h}- \partial_{x_i} F(x) \right) \bullet M \right\|_{L^2(\mathbb{P})}^2 \\ &= \int \left| \frac{F(x+h e_i)-F(x)}{h}- \partial_{x_i} F(x) \right|^2 \, d\mu_M. \tag{6} \end{align*}$$
Since
$$\frac{F_t(x+h e_i)-F_t(x)}{h}- \partial_{x_i} F_t(x) \xrightarrow[]{h \to 0} 0 \quad \text{a.s.}$$
and, by the mean value theorem,
$$\begin{align*} \left| \frac{F_t(x+h e_i)-F_t(x)}{h}- \partial_{x_i} F_t(x) \right|^2 &\leq \left( \left|\frac{F_t(x+h e_i)-F_t(x)}{h} \right| + |\partial_{x_i} F_t(x)| \right)^2 \\ &\leq 4 \sup_{y \in B(x,1)}|\partial_{x_i} F_t(y)|^2 \in L^1(\mu_M) \end{align*}$$
for any $h \in (0,1)$, it follows from the dominated convergence theorem that the right-hand side of $(6)$ converges to $0$ as $h \to 0$, and therefore $N(x) = F(x) \bullet M$ is partially differentiable (in the $L^2(\mathbb{P})$-sense) and
$$\partial_{x_i} N(x) = (\partial_{x_i} F(x)) \bullet M$$