Difference between backward martingales and martingales.

Question

I can't see a difference between a martingale and a backward martingale except for the index. What is the difference besides this? If we consider the reverse filtration then it is a martingale. In other words by going upwards.

Answer 1

The index set $\{0,1,2,\ldots\}$ of a (discrete time) martingale has a first element but is "infinite to the right". The index set $\{\ldots,-2,-1,0\}$ of a (discrete time) backward martingale has a last element but is "infinite to the left". In the latter case, call the martingale $(Y_n)$; you then have $Y_n =\Bbb E[Y_0\mid\mathcal F_n]$ for each $n<0$. In particular, the sequence $(Y_n)$ is uniformly integrable, and this is the key difference between the two notions.