TITLE: Criteria for being a true martingale
QUESTION [29 upvotes]: 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.

REPLY [5 votes]: 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