Criteria for being a true martingale Could you kindly list here all the criteria you know which guarantee that a continuous local martingale is in fact a true martingale? Which of these are valid for a general local martingale (non necessarily continuous)? Possible references to the listed results would be appreciated.
 A: Here is the other answer I was mentionning in my preceding post :
Tenth :
Given $X_t$ a continuous Itô diffusion starting at $x$ and taking values in $[l,\infty)$ with $l\in \mathbb{R}$. $X_t$ is moreover supposed to satisfy the following SDE :  
$X_t=x+\int_{0}^{t}a(X_s)dB_s$  
with :
- $B$ a standard Brownian Motion
- $a(x)^2>0$ for all $x\in (l,\infty)$
- $a^{-2}$ is locally integrable
This entails that $X$ has a weak and unique solution and that it is a local martingale (with absorbing boundary $l$). 
Then the Second order equation $$\frac{1}{2}a^2(x)u''(x)=\alpha.u(x)$$ has two positve (linearly independent) unique solutions $\phi_{\alpha} $ and $\psi_{\alpha}$ up to a multiplicative constant ($\alpha$ is a fixed strictly positive constant).   
Those solutions are respectively decreasing and increasing with boundary conditions in accordance with the boundary behaviour of $X$.  
Under those assumptions $X$ is a martingale if and onyl if :
$$lim_{z\to +\infty}\psi_{\alpha}'(z)=+\infty$$
This is from Hulley and Palten "A Visual Criterion for Identifying Itô diffusions as Martingales or Strict Local Maritngale".   
NB : The Criteria from George Lowther's blog is also given in the paper (with two references) and the connections are explicitly described.
Eleventh :
Another criteria with different tools and set of hypothesys (I would say that it is a little more specialised that the preceeding criteria) can be found in the paper by Mijatovic and Urusov "On the Martingale Property of Certain Local Martingale". I do not reproduce it here because it would be too long. 
Regards
A: Here you are :
From Protter's book "Stochastic Integration and Differential Equations" Second Edition (page 73 and 74)
First :
Let $M$ be a local martingale. Then $M$ is a martingale with 
$E(M_t^2) < \infty, \forall t > 0$, if and only if $E([M,M]_t) < \infty, \forall t > 0$. If $E([M,M]_t) < \infty$, then $E(M_t^2) = E([M,M]_t)$. 
Second :
If $M$ is a local martingale and $E([M, M]_\infty) < \infty$, then $M$ is a 
square integrable martingale (i.e. $sup_{t>0} E(M_t^2) = E(M_\infty^2) < \infty$). Moreover $E(M_t^2) = E([M, M]_t), \forall t \in [0,\infty]$. 
Third :
From George Lowther's Fantastic Blog, for positive Local Martingales that are (shall I say) weak-unique solution of some SDEs.
Take a look at it by yourself :
http://almostsure.wordpress.com/category/stochastic-processes/
Fourth :
For a positive continuous local martingales $Y$ that can written as Doléans-Dade exponential of a (continuous)-local martingale $M$, if $E(e^{\frac{1}{2}[M,M]_\infty})<\infty$ is true (that's Novikov's condition over $M$), then  $Y$ is a uniformely integrable martingale.(I think there are some variants around the same theme)
I think I can remember I read a paper with another criteria but i don't have it with me right now. I 'll to try find it and give this last criteria when I find it.
Regards
A: I found by myself other criteria that I think it is worth adding to this list.
5) $M$ is a local martingale of class DL iff $M$ is a martingale
6) If $M$ is a bounded local martingale, then it is a martingale.
7) If $M$ is a local martingale and $E(\sup_{s \in [0,t]} |M_s|) < \infty \, \forall t \geq 0$, then $M$ is a martingale.
8) Let $M$ be a local martingale and $(T_n)$ a reducing sequence for it. If $E(\sup_{n} |M_{t \wedge T_n}|) < \infty \, \forall t \geq 0$, then $M$ is a martingale.
9) Suppose we have a process $(M_t)_{t\geq 0}$ of the form $M_t=f(t,W_t)$. Then $M$ is a local martingale iff $(\frac{\partial}{\partial t}+\frac{1}{2}\frac{\partial^2}{\partial x^2})f(t,x)=0$.
If moreover $\forall \, \varepsilon >0$ $\exists C(\varepsilon,t)$ such that $|f(s,x)|\leq C e^{\epsilon x^2} \, \forall s \geq 0$, then $M$ is a martingale.
