# 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.

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 weaker condition than $\mathbb{E}[[M]_t] < \infty$ is that $\mathbb{E}[[M]^{1/2}_t] < \infty$. This is actually equivalent to $\mathbb{E}[\sup_{s\le t}\vert M_s\vert] < \infty$, implying that $M$ is a martingale, and follows from the Burkholder-Davis-Gundy inequality. – George Lowther May 15 '11 at 21:36

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.

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

• I'm not sure exactly which criteria you are referring to from my blog, but the one you have here labeled 'Tenth' is mentioned in one of my posts, almostsure.wordpress.com/2010/08/31/… – George Lowther May 16 '11 at 12:31
• @ George Lowther: My mistake this is not the criteria I wanted to mention. I edit the post to give the right one, by the way this (your) criteria is quoted as a result from two papers. First - Delbean and Shirakawa "No arbitrage condition for positive diffusion price processes", Second - Kotani "On a condition that one-dimensional diffusion processes are martingales" – TheBridge May 16 '11 at 13:37