Journal of the
Korean Mathematical Society JKMS

Let $X$ and $Y$ be It\^o processes with $dX_s=\varphi_sdB_s +\psi_sds$ and $dY_s=\zeta_sdB_s +\xi_sds$. Burkholder obtained a sharp bound on the distribution of the maximal function of $Y$ under the assumption that $|Y_0|\leq |X_0|$, $|\z|\leq |\varphi|$, $|\xi|\leq|\psi|$, and that $X$ is a nonnegative local submartingale. In this paper we consider a wider class of It\^o processes, replace the assumption $|\xi|\leq|\psi|$ by a more general one $|\xi|\leq\a|\psi|$, where $\a \geq 0$ is a constant, and get a weak-type inequality between $X$ and the maximal function of $Y$. This inequality, being sharp for all $\a \geq 0$, extends the work by Burkholder.

Keywords: It\^o process, $\alpha$-subordinate, It\^o's formula, Doob's optional sampling theorem, stopping time, martingale, submartingale, Brownian motion, exit time, strong Markov property, best constant

MSC numbers: Primary 60H05; Secondary 60E15