Inequalities for dominated martingales
Inequalities for dominated martingales
Let $(M_n)$, $(N_n)$ be two Hilbert-space-valued martingales adapted to some filtration $(ℱ_n)$, with corresponding difference sequences $(d_n)$, $(e_n)$, respectively. We assume that $(N_n)$ weakly dominates $(M_n)$, that is, for any convex non-decreasing function $ϕ : ℝ_+→ℝ-_+$ and any $n=1,2,…$ we have, almost surely, $\mathrm{E}(ϕ(|d_n|)|ℱ_{n−1})\leqslant\mathrm{E}(ϕ(|e_n|)|ℱ_{n−1})$. We apply the Burkholder method to …