Let
- $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
- $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
- $M:\Omega\times[0,\infty)\times\mathbb R^d\to\mathbb R$ such that $M(x)=M(\;\cdot\;,\;\cdot\;,x)$ is a continous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ for all $x\in\mathbb R^d$
- $A:\Omega\times[0,\infty)\times\mathbb R^d\times\mathbb R^d\to\mathbb R$ such that
- $A(x,y)=A(\;\cdot\;,\;\cdot\;,x,y)$ is $\mathcal F$-adapted for all $x,y\in\mathbb R^d$
- $\partial^\alpha_x\partial^\beta_yA(\omega,t,x,y)$ exists for all $|\alpha|,|\beta|\le1$ for all $\omega\in\Omega$, $t\ge0$ and $x,y\in\mathbb R^d$
- $\partial^\alpha_x\partial^\beta_yA(\omega,t,\;\cdot\;,\;\cdot\;)$ is separately continuous for all $|\alpha|,|\beta|\le1$ for all $\omega\in\Omega$ and $t\ge0$
- $A_t(x,y)=[M(x),M(y)]_t$ almost surely for all $t\ge0$ and $x,y\in\mathbb R^d$
Now, let $i\in\left\{1,\ldots,d\right\}$ and $$N(x,\theta):=\frac{M(x+\theta e_i)-M(x)}\theta\;\;\;\text{for }x\in\mathbb R^d\text{ and }\theta\in\mathbb R\setminus\left\{0\right\}.$$ By assumption $$[N(x,\theta),N(y,\vartheta)]_t=\int_0^1\int_0^1\frac{\partial^2 A_t}{\partial x_i\partial y_i}(x+\theta\sigma e_i,y+\vartheta\tau e_i)\:{\rm d}\tau\:{\rm d}\sigma=:B_t((x,\theta),(y,\vartheta))$$ almost surely for all $t\ge0$ and $(x,\theta),(y,\vartheta)\in\mathbb R^d\times\mathbb R\setminus\left\{0\right\}$.
How can we conclude that $N_t(x,\theta)$ converges uniformly in $(x,t)$ on compact sets as $\theta\to0$ almost surely?
The question is based on theorem 3.1.2 in the famous book "Stochastic flows and stochastic differential equations" by Kunita. I don't how he is conclude the desired result in his proof.
