Sharp maximal inequality for stochastic integrals

Abstract:

Let $X=(X_t)_{t\geq 0}$ be a nonnegative supermartingale and $H=(H_t)_{t\geq 0}$ be a predictable process with values in $[-1,1]$. Let $Y$ denote the stochastic integral of $H$ with respect to $X$. The paper contains the proof of the sharp inequality \[ \sup _{t\geq 0}||Y_t||_1 \leq \beta _0 ||\sup _{t\geq 0}X_t||_1,\] where $\beta _0=2+(3e)^{-1}=2,1226\ldots$. A discrete-time version of this inequality is also established.