Question on the proof of the existence of the covariation presented in the book of Kallenberg

Question

I'm trying to understand the proof of the existence of the covariation presented in the book of Kallenberg and got some questions about the involved objects:

Let

• $T>0$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A)$
• $(M_t)_{t\in[0,\:T]}$ be an almost surely continuous $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ with $M_0=0$ almost surely
• $n\in\mathbb N$, $$\tau_0^n:=0$$ and $$\tau_k^n:=\inf\left\{t>\tau_{k-1}^n:\left|M_t-M_{\tau_{k-1}^n}\right|=\frac1{2^n}\right\}\wedge T\;\;\;\text{for }k\in\mathbb N$$

Question 1:$\;\;\;$How can we show that $\tau_k^n$ is an $\mathcal F$-stopping time for all $k\in\mathbb N$?

Now, let $$V_t^n:=\sum_{k\in\mathbb N}1_{\left(\tau_{k-1}^n,\:\tau_k^n\right]}(t)M_{\tau_{k-1}^n}\;\;\;\text{for }t\in[0,T]\tag 1$$ and $$Q_t^n:=\sum_{k\in\mathbb N}\left|M_{\tau_{k-1}^n\:\wedge\:t}-M_{\tau_k^n\:\wedge\:t}\right|^2\;\;\;\text{for }t\in[0,T]\;.\tag 2$$

Question 2:$\;\;\;$Why do the series in $(1)$ and $(2)$ converge?

Answer 1

Q1: (Assume the filtration satisfies the usual conditions.) Consider, more generally, a real-valued adapted process $X$ with $[0,T]\mapsto X_t$ continuous a.s., and a stopping time $\sigma\le T$ such that $X_\sigma=0$. Then for a fixed real $b>0$, $\tau:=\inf\{t>\sigma:X_t=b\}\wedge T$ is a stopping time. With this fact in hand, a simple induction show that $\tau_1^n, \tau_2^n,\ldots$ are stopping times.

To prove the assertion, we can (because of the "usual conditions") assume that $t\mapsto X_t(\omega)$ is continuous for every $\omega\in\Omega$. (Else, let $B:=\{\omega:t\mapsto X_t(\omega)$ fails to be continuous at some $t\in[0,T]\}$. Then $B\in\mathcal F_0$ and $\operatorname P(B)=0$. The process $X'_t(\omega):=1_{B^c}(\omega)X_t(\omega)$ is indistinguishable from $X$ and has everywhere continuous sample paths. The function ${\tau_k^n}'$ defined as was $\tau_k^n$ but in terms of $X'$ will be shown to be a stopping time in what follows, and because $\operatorname P(\tau_k^n={\tau_k^n}')=1$, you see that $\tau_k^n$ is also a stopping time.) Define $Y_t:=\sup_{0\le s\le t}1_{\{\sigma\le s\}}X_s$. By path continuity, this is the same as $\sup_{0\le s\le t,s\in\Bbb Q}1_{\{\sigma\le s\}}X_s$, which shows that $Y$ is an adapted process. Finally, $\{\tau\le t\}=\{Y_t\ge b\}\in\mathcal F_t$, for each $t\in[0,T)$, so $\tau$ is indeed a stopping time.

Q2: Fix $n\in\Bbb N$. Each of the sums (1) and (2) has only finitely many non-zero terms, a.s., because the assumed path-continuity of $M$ implies that $\operatorname P(\cup_k\{\tau_k^n=T\})=1$. (For a sample point $\omega\in\Omega$ with $\tau^n_1(\omega)<\tau^n_2(\omega)<\cdots<\tau_k^n(\omega)<\tau^n_{k+1}(\omega)<\cdots<T$ for all $k$, the limit $\lim_{t\uparrow T}M_t(\omega)$ fails to exist.)

Answer 2

1. My definition of $\tau$ is correct as written: in your alteration, what is the value of $\tau$ should it happen that $X_t<b$ for all $t\in[\sigma,T]$? And $Y$ is defined as it should be.

2. What is $N$? I have reduced to the case in which $t\mapsto X_t(\omega)$ is continuous for all $\omega$.

3. If $\tau_k^n(\omega)=T$ then $\tau_j^n(\omega)=\tau_k^n(\omega)$ for all $j>k$, and all of the associated terms in your sums vanish for that \omega$.

4. As I noted, if $\omega$ is such that $\tau^n_1(\omega)<\tau^n_2(\omega)<\cdots<\tau_k^n(\omega)<\tau^n_{k+1}(\omega)<\cdots<T$, then the limit $\lim_{t\uparrow T}M_t(\omega)$ fails to exist.)