Let
- $H$ be a separable $\mathbb R$-Hilbert space
- $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$
- $X$ be a $H$-valued continunous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$
Let $$X^n:=\langle X,e_n\rangle_H$$ for $n\in\mathbb N$ and$^1{^3}$ $$Z^{MN}:=\sum_{m=1}^M\sum_{n=1}^N\underbrace{[X_m,X_n]e_m\otimes e_n}_{=:\:Y^{mn}}$$ for $M,N\in\mathbb N$. How can we show that $(Z^{MN})_{M,\:N\in\mathbb N}$ converges in $\mathfrak L(H)$$^2$ to an $\mathfrak L_1(H)$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)$?
Clearly, $$\left\|e_m\otimes e_n\right\|_{\mathfrak L(H)}=1\;\;\;\text{for all }m,n\in\mathbb N\tag5$$ and hence $$\sum_{m=1}^M\sum_{n=1}^N\left\|Y^{mn}\right\|_{\mathfrak L(H)}=\sum_{m=1}^M\sum_{n=1}^N[X^m,X^n]\;.\tag6$$ If $X$ would be square-integrable, then $$\operatorname E\left[\sum_{n=1}^N[X^n]_t\right]=\operatorname E\left[\sum_{n=1}^N\left|\langle X_t,e_n\rangle_H\right|^2\right]\xrightarrow{n\to\infty}\operatorname E\left[\left|X_t\right|\right]^2<\infty\tag7\;.$$
$^1$ If $X$ is a real-valued continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$, there is a real-valued $\mathcal F$-adapted stochastic process $[X]$ on $(\Omega,\mathcal A,\operatorname P)$, unique up to indistinguishability, with
- $[X]_0=0$
- $[X]$ is continuous
- $[X]$ is of locally bounded variation
- $X^2-[X]$ is a local $\mathcal F$-martingale
- $[X]$ is nondecreasing
If $(\sigma_n)_{n\in\mathbb N}$ is a localizing sequence for $X$, then $(\sigma_n\wedge n)_{n\in\mathbb N}$ is a localizing sequence for both $X$ and $X^2-[X]$. If $X$ is an $\mathcal F$-martingale, then $X^2-[X]$ is an $\mathcal F$-martingale. If $\tau$ is an $\mathcal F$-stopping time on $(\Omega,\mathcal A)$, then $$[X^\tau]=[X]^\tau\tag1\;.$$ If $X_0=0$, then $$\operatorname E\left[[X]_t\right]\le\operatorname E\left[\sup_{s\in[0,\:t]}\left|X_s\right|^2\right]\le 4\operatorname E\left[[X]_t\right]\;\;\;\text{for all }t\ge0\tag2$$ and $$\operatorname E\left[\left(X^2-[X]\right)_t\right]=0\;\;\;\text{for all }t\ge0\tag3\;.$$ If $Y$ is another real-valued continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$, then $$\left[X,Y\right]^2\le[X][Y]\tag4\;.$$
$^2$ $\mathfrak L(H)$ denotes the space of bounded linear operators and $\mathfrak L_1(H)$ denotes the space of nuclear operators on $H$.
$^3$ If $H_i$ is a $\mathbb R$-Hilbert space and $a_i\in H_i$, then $a_1\otimes a_2:=a_1\langle\;\cdot\;,a_2\rangle_{H_2}$.
