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 complete and right-continuous filtration of $\mathcal A$
- $X:\Omega\times[0,\infty)\to H$ be an almost surely continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ with $X_0=0$ almost surely and $$X^n:=\langle X,e_n\rangle_H\;\;\;\text{for }n\in\mathbb N$$
How can we show that$^1$ $\left([X^n]_t\right)_{n\in\mathbb N}$ is summable for all $t\ge0$ almost surely?
If $M$ is an $L^2$-bounded strict $\mathcal F$-martingale, then the claim can be proved easily:
- Let $$S:=\sum_{n=1}^\infty[X^n]$$ and $t\ge0$
- Since $X$ is an $\mathcal F$-martingale, $$\operatorname E\left[\left(\left|X^n\right|^2-[X^n]\right)_t\right]=0\tag1$$ and hence $$\operatorname E[S_t]\xleftarrow{N\to\infty}\operatorname E\left[\sum_{n=1}^N[X^n]_t\right]=\operatorname E\left[\sum_{n=1}^N\left|\langle X_t,e_n\rangle_H\right|^2\right]\xrightarrow{N\to\infty}\operatorname E\left[\left\|X_t\right\|_H^2\right]\tag2$$
- By assumption, $X_t\in L^2(\operatorname P,H)$ and hence $$S_t<\infty\;\;\;\text{ almost surely}\tag3$$
- Since $[X^n]$ is nondecreasing for all $n\in\mathbb N$ and $t$ was arbitrary, we obtain $$S_t<\infty\;\;\;\text{for all }t\ge0\text{ almost surely}\tag4\;,$$ which is equivalent to the desired claim
If $M$ is not $L^2$-bounded and only a local $\mathcal F$-martingale, then $(1)$ doesn't hold. Since this equation is the central argument of the proof above, I don't know how we're able to obtain the claim in the general case.
However, the statement should be correct, since the desired summability result is crucial in the construction of the quadratic variation of $X$.
$^1$ If $M:\Omega\times[0,\infty)\to\mathbb R$ is an almost surely continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ with $M_0=0$ almost surely, then $[M]:\Omega\times[0,\infty)\to\mathbb R$ denotes the quadratic variation of $M$, i.e. $[M]$ is the $\mathcal F$-adapted continuous stochastic process on $(\Omega,\mathcal A,\operatorname P)$ of locally bounded variation (and unique up to indistinguishability) such that $M^2-[M]$ is a local $\mathcal F$-martingale.
