Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$
- $M$ be a local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$
- $\tau$ be an $\mathcal F$-stopping time
Is $M^\tau$ a local $\mathcal F$-martingale?
With the given assumptions, this claim can be found in the proof of Lemma 15.1 in Foundations of Modern Probability by Olav Kallenberg.
I know how we're able to prove the claim using the Optional Sampling Theorem, if $\tau$ takes only countable many values or $M$ is almost surely right-continuous, but I have no idea how I can prove the claim without one of these additional assumptions.
More concretely, he states that if $(\sigma_n)_{n\in\mathbb N}$ is an $\mathcal F$-localizing sequence for $M$, then $(M^\tau)^{\sigma_n}=(M^{\sigma_n})^\tau$ is an $\mathcal F$-martingale. So, maybe the question reduces to the question if a stopped martingale is a martingale, but, again, I'm only able to proof this with the Optional Sampling Theorem and one of its additional assumptions.