1
$\begingroup$

I want to be sure I understand the definition correctly, say as in here:

http://en.wikipedia.org/wiki/Local_martingale

I have grown accustomed to thinking of there being 3 classes of martingales. Those satisfying the $L^1$ condition are most common, but the nonnegative ones, and the ones for which the integral exists (most general, this one means that the positive and negative integrals are not both infinite). All 3 classes have been useful. However, when people say "martingale" in the definition of "local martingale", i.e. where the stopped process has to be a martingale, they mean $L^1$ always, or at least always unless there is further clarification? However, it is also true that the RVs within the stochastic process itself may not be $L^1$?

Are other modifications of the notion as were used for regular martingales also considered often? (For instance, adding prefixes like sub- and super-, backward martingales, discrete time...)

$\endgroup$
4
  • 2
    $\begingroup$ The terminology is somewhat deceptive since local martingales are not (always) martingales. This remark might be answering most of your question. $\endgroup$
    – Did
    Sep 3, 2013 at 6:29
  • $\begingroup$ @Did I have proceeded with my assumptions and the document I'm reading seems to be consistent with my interpretation of this definition. However, one oddity that arises from my definition that makes me a little suspicious is it does not seem like a sum of local martingales must be a local martingale. $\endgroup$
    – Jeff
    Sep 3, 2013 at 7:52
  • $\begingroup$ A sum of local martingales is definitely a local martingale. $\endgroup$
    – Did
    Sep 3, 2013 at 8:02
  • $\begingroup$ Oh right, you have to combine the two sequences of stopping times well (Taking min) $\endgroup$
    – Jeff
    Sep 3, 2013 at 17:00

1 Answer 1

0
$\begingroup$

The terminology is somewhat deceptive since local martingales are not (always) martingales. This remark might be answering most of your question.

On the other hand, a sum of local martingales is definitely a local martingale.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .