# Construction of the quadratic variation process in infinite dimensions

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\;.$$^1 If X is a real-valued continuous local \mathcal F-martingale on (\Omega,\mathcal A,\operatorname P), there is a real-valued \mathcal F-adapted stochastic process [X] on (\Omega,\mathcal A,\operatorname P), unique up to indistinguishability, with 1. [X]_0=0 2. [X] is continuous 3. [X] is of locally bounded variation 4. X^2-[X] is a local \mathcal F-martingale 5. [X] is nondecreasing If (\sigma_n)_{n\in\mathbb N} is a localizing sequence for X, then (\sigma_n\wedge n)_{n\in\mathbb N} is a localizing sequence for both X and X^2-[X]. If X is an \mathcal F-martingale, then X^2-[X] is an \mathcal F-martingale. If \tau is an \mathcal F-stopping time on (\Omega,\mathcal A), then$$[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}\$.