I have two questions.
1) I was reading a presentation by Prof. Protter about some technicalities in stochastic calculus and, at one point, in the slides, I see stated that
"P. A. Meyer (1973) showed that there are no local martingales in discrete time; they are a continuous time phenomenon."
I wonder if anyone could either refer me to the specific paper, or textbook, or if anyone knows the specific example Prof. Protter is mentioning.
2) There are several examples of local martingales which are not martingales (in continuous-time). Does anyone know an interesting example not involving Brownian motion or specific stochastic processes requiring to use specific properties of those processes? I am looking for a ``vanilla'' example that requires some thinking.
I am aware of one case where, if the probability space is $([0,1], {\cal B}[[0,1]), \lambda), $ and one chooses the filtration $\mathbb{F} = \{{\cal F}_t:t \in \mathbb{R}^+\} $ with ${\cal F}_t = {\cal B}([0, 1]) $ for all $t, $ and $Z $ is a non-integrable r.v. then $\{X_t:= Z: t \in \mathbb{R}^+\} $ is not a martingale as integrability is lost from the start. However, it is a local martingale, because we would have that $X-X_0 \equiv 0. $ But this seems rather uninteresting as there is not even the need to introduce a localizing sequence. I would really appreciate one, where at least a localizing sequence has to be suggested and some simple calculations are involved.
I seem to remember one example I saw some time ago. I think it went along the lines of introducing a filtration $\mathbb{F} = \{{\cal F}_t: t \in \mathbb{R}^+\} $ and a r.v. $Z $ supposed to be ${\cal F}_1$-measurable but having infinite first moment (which takes care of the fact that we will not end up with a martingale). Then one introduces another r.v., say $W, $ which is the classic $\pm 1 $ with probability $1/2, $ and independent of ${\cal F}_{2-} $ and finally we let $X_t = WZ\cdot 1_{\{t\ge 2\}}. $ Then if I am not mistaken one ends up with a local martingale by introducing the localizing sequence given
by
$$T_n = 1\cdot 1_{\{Z> n\}} + n\cdot 1_{\{Z\le n\}}. $$
Could anyone show me the key details on how to prove that $X_t $ so defined is a local martingale? I think it is even bounded since by our definition of $T_n $
we would end up with an upper limit $n $ for $X_{T_n\wedge t} $ for each choice of $n. $
Thank you and sorry for the long question. I am just trying to clean some examples I collected in the old (sigh!) days of graduate school.
Maurice