Let $X_{n}, Y_{n}$ be independent martingals. Is it also the case that $X_{n} + Y_{n}$ is a martingale? And if we drop the independence condition does ist also define a martingal?
Asked
Active
Viewed 177 times
0
-
okay...thanks. But what if $X_{n}$ and $Y_{n}$ are not independet? – May 30 '18 at 11:28
-
Then, in general, the sum fails to be a martingale. Just take a look at the other question; it's all explained there. (....I doubt that you read it within 3 minutes.....) – saz May 30 '18 at 11:30
-
It will, of course, depend on your filtration – blanchey May 30 '18 at 13:17
-
can you give an exampple of a filtration such that $X_{n}, Y_{n}$ are dependent and the sum is a martingal? – May 30 '18 at 13:19
-
$X_n = Y_n = X$ for some integrable random variable $X$; $\mathcal{F}_n := \sigma(X)$. – saz May 30 '18 at 18:37