super martingale which dominates

Question

Could anyone tell me what it means by ''it dominates $\{E (Z|G_k), G_k\}, E(|Z|)<\infty$'' ? in the following statement? Thanks.

Let $(Y_k, G_k)$ be a super-martingale and it dominates $\{E (Z|G_k), G_k\}, E(|Z|)<\infty$,

Answer 1

In general, we say that a random variable $Y$ dominates a random variable $X$ if \begin{align} |X(\omega)|\leq Y(\omega) \quad a.s. \end{align} This language is used for example in the Dominated Convergence Theorem.

In your case, you have a supermartingale $\{Y_k\}$ and a stochastic process $\{E[Z|G_k]\}$. Thus, that $\{Y_k\}$ dominates $\{E[Z|G_k]\}$ simply means that for all $k$ we have \begin{align} \big|E[Z|G_k]\big|\leq Y_k \quad a.s. \end{align}