Let $x_1,\ldots, x_n$ be independent continuous martingales with filtrations $\mathcal{F}^1_{t},\ldots, \mathcal{F}^n_{t}$. Let $Y_i=(x_i)^2$ be the associated sub-martingale with each $x_i$ (we can derive this from the fact that $x^2$ is a convex function, and apply Jensen’s inequality).
Given the independence of the $x_i$’s, $S=\sum_{i=1}^n Y_i$ is also a sub-martingale. What is the filtration it is a sub-martingale with respect to?
Further, suppose $x_i$’s are a martingale with respect to some common filtration $\mathcal{G}$, which is the filtration associated with some predictable process. Each $x_i$ can be thought of as the process viewed on some subset of its support. Can we drop the independence assumption then, and say $S$ is a sub-martingale wrt filtration $\mathcal{G}$?
I think that my question is related to Martingale preservation under independent enlargement of filtration, but could use some help with an intermediate explanation.
Best Answer
Since the $x_1,\dotsc,x_n$ are independent they are also martingales wrt. the filtration $\mathcal{F}=(\mathcal{F}_t)$ where $\mathcal{F}_t=\sigma(\mathcal{F}^1_t,\dotsc,\mathcal{F}^n_t)$. Hence, $Y_1,\dotsc,Y_n$ are sub-martingales wrt. $\mathcal{F}$ and so is the sum $S$.
In the above, if $x_1,\dotsc,x_n$ are no longer assumed to be independent independent then they are no longer guaranteed to be $\mathcal{F}$-martingales. However, since you assume that they are instead $\mathcal{G}$-martingales it follows that $Y_1,\dotsc,Y_n$ are $\mathcal{G}$-sub-martingales and ultimately that $S$ is a $\mathcal{G}$-sub-martingale. Note that independence is not needed for a sum of sub-martingales (wrt. the same filtration) to be a sub-martingale.