Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

Abstract

Let $f=(f_n)_{n\geq 0}$ be a nonnegative submartingale starting from $x$ and let $g=(g_n)_{n\geq 0}$ be a sequence starting from $y$ and satisfying $$|dg_n|\leq |df_n|,\quad |\mathbb{E}(dg_n|\mathcal{F}_{n-1})|\leq \mathbb{E}(df_n|\mathcal{F}_{n-1})$$ for $n\geq 1$. We determine the best universal constant $U(x,y)$ such that $$\mathbb{P}(\sup_ng_n\geq 0)\leq ||f||_1+U(x,y).$$ As an application, we deduce a sharp weak type $(1,1)$ inequality for the one-sided maximal function of $g$ and determine, for any $t\in [0,1]$ and $\beta\in\mathbb{R}$, the number $$ L(x,y,t,\beta)=\inf\{||f||_1: \mathbb{P}(\sup_ng_n\geq \beta)\geq t\}.$$ The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.