Localization

Stream: Brownian motion

[Rémy Degenne]

I have questions about the definition of a stochastic process satisfying a property locally, and I'm hoping somebody more familiar with this can help. In the material that is currently being added to the blueprint, we will use notions of local (sub)martingales and locally of class D (and perhaps others).

A definition could be this (as found in the wikipedia page for local martingales for example):
For $P$ a property on stochastic processes $\mathbb{R}_+ \to \Omega \to E$ we say that a process $X$ satisfies $P$ locally if there exists a sequence $(\tau_n)_{n \in \mathbb{N}}$ of stopping times which is increasing to infinity a.s., such that the stopped processes $X^{\tau_n}$ satisfy $P$ for all $n$.

The issue with this seems to be that we want any constant process $X_t = X_0$ to be a local martingale, and that would not be the case with that definition if $X_0$ is not integrable.
Kallenberg in Foundations of Modern Probability and Le Gall in Brownian Motion, Martingales and Stochastic Calculus use the definition above for some properties and add the condition $X_0 = 0$ in lemmas where needed. For the case of local martingales they extend the definition to say that $M$ is a local martingale if $M_t - M_0$ satisfies the definition above.
On the other hand, Lowther use a different definition in the Almost Sure blog, which is also the one used by Meyer in Martingales and Stochastic Integrals I: the processes that should satisfy $P$ are not $X^{\tau_n}$ but $X^{\tau_n}\mathbb{I}_{\tau_n > 0}$. This modification ensures that the constant process example is a local martingale. It seems to allow one unique definition of "locally", that does not need for example to center the process in the case of a martingale.

I'm tempted to follow Meyer and use the definition with $X^{\tau_n}\mathbb{I}_{\tau_n > 0}$ because it looks like it needs less side conditions and the blog page demonstrates that it has nice properties. But I'm not very familiar with these notions. So if you know about this and have opinions, I'd love to read them!

[Sébastien Gouëzel]

I don't know anything about these (much less than you do) but I agree that the definition with indicators is nicer.

[Kexing Ying]

A downside with the $M_t - M_0$ definition is if $X_t = X_0 > 0$, then it is not locally non-zero despite being globally non-zero. This is not an issue with the indicator definition so I think it is better.

[Alessio Rondelli]

I also think the definition using indicators is a bit more flexible. In some sense both achieve the result of throwing away a bad edge case that we would like to include. The $M_t-M_0$ definition is useful in this case but creates strange phenomena when used in other situations.
Just as an example consider the process $M_t$ that has the property $P$ of "uniformly in $t>0$ it is bounded" but is unbounded as a process in $0$. Unfortunately while it is $P$ it is not locally $P$ using the $M_t-M_0$ definition. The flexibility of the second definition sidesteps the problem.
It should be noted that even the definition using the indicator does not work in every case (just consider a non integrable nonzero constant process and the property of being "integrable and nonzero", for the same reason a constant process should be a local martingale this should be locally integrable and nonzero) but it covers without changing anything all the "sane" cases.

[Alessio Rondelli]

I would also add that localizing is more of a precedure than an hard rule (just like we wanted constant processes to be local martingales and we bent the definition a little) and so in my opinion the right notion of localizing a property depends upon the function that you choose ($x-M_0$,$x\mathbb{1}_{\cdot>0}$ or something else). Having said that I don't think implementing this much flexibility is wise since then it should be proved on a case by case basis all the nice results of the Almost Sure Blog.

[Rémy Degenne]

Thanks for all your comments. I'm mostly looking for a way to reduce code duplication, so if the indicator definition is good for most of the cases we will consider, then that's good enough.