A sharp weak-type bound for Itō processes and subharmonic functions

Abstract

Let $\alpha \ge 0$, and let $X$, $Y$ be Itō processes

$$d{X}_t={\varphi }_{t}d{B}_t+{\psi }_{t}dt,d{Y}_t={\zeta }_{t}d{B}_t+{\xi }_{t}dt$$

such that $\left|X_0\right|\ge \left|Y_0\right|$, $\left|\varphi \right|\ge \left|\zeta \right|$, and $\alpha \psi \ge \left|\xi \right|$. The purpose of the paper is to determine the optimal universal constant $C_{\alpha }$ in the weak-type estimate

$${sup\hspace{0.17em}}_{\lambda }\lambda \mathbb{P}\left({sup\hspace{0.17em}}_t\right|Y_t|\ge \lambda )\le C_{\alpha }{sup\hspace{0.17em}}_t\mathbb{E}\left|X_t\right|.$$

Then the inequality is extended, with unchanged constant, to the more general setting when $X$ is a submartingale and $Y$ is $\alpha$-strongly differentially subordinate to $X$. As an application, a related estimate for subharmonic functions is established. The inequalities generalize and unify the earlier results of Burkholder, Choi, and Hammack for Itō processes, stochastic integrals, and smooth functions on Euclidean domains.