Show this truncated stochastic process is a martingale

Question

Consider a filtered space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0}, P)$. Let $(M_t)$ be a uniformly integrable (UI) martinagle w.r.t. $\{\mathcal{F}_t\}_{t\geq 0}$.

Let $\tau, \nu$ be two stopping times such that $\tau\geq \nu$ for every $\omega\in \Omega$.

Let $\Gamma_C(x):=\max\{\min\{x,C\},-C\}$ be the truncated function ($x$ truncated at $C$).

Show the following process

$$N_t:=\Gamma_C(M_{t\wedge \nu})(M_{t\wedge \tau}-M_{t\wedge \nu})$$

is also a martingale.

Background. I met this prblem when reading the paper On quadratic variation of martingales (page 463, Eq. (3.12)), which a tutorial paper cited by Wikipedia to explain the quadratic variation process. The author first rewrited a process to the form of $N$, then claimed that it is a martingale.

What I think. I know that $M_{t\wedge \tau}-M_{t\wedge \nu}$ is a martinagle, since stopped martingales are martingales. But I can not figure out how the trucation $\Gamma_C(M_{t\wedge \nu})$ preserves the martingale property. Moreover, I feel the expression of $N_t$ is similar to that of a stochastic integral of elementary predictable process, which is a martingale.

UPDATE:

The situation with non-random stopping times is discussed in this post. However, the solution does not apply since we are considering random times here.

Answer 1

After some searching, I figure out the answer to my question. I post it below in case anyone needs it in the future.

In Rogers & Williams, Diffusions, Markov Processes and Martingales (Vol. 2, pg. 10, Lem. 5.3),

Fundermental Lemma. Consider a uniformly integrable (UI) martingale $M$, two stopping times $S,T$ such that $S\geq T$, and a bounded random variable $Z$ measurable to $\mathcal{F}_S$. Let $$H:=Z(M_{t\wedge T}-M_{t\wedge S})$$ be the stochastic integral of $Z$ w.r.t. $M$. Then $Z$ is also a UI martingale.

This lemma is proved using the following theorem:

Theorem (77.6). Let $(M_t)$ be a progressive process such that for any (finite or infinite) stopping time $T$, we have $E(|M_T|)<\infty$ and $E(M_T)=0$. Then $M$ is a UI martingale.

Thus, it can be extended to cases where $Z$ is a random process relying on $t$, as long as the boundedness and measurability conditions hold.