Sharp maximal inequality for stochastic integrals

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

$$\sup_{t\geq 0}\vert\vert Y_t\vert\vert _1 \leq \beta_0 \vert\vert\sup_{t\geq 0}X_t\vert\vert _1,$$

where $\beta_0=2+(3e)^{-1}=2,1226\ldots$. A discrete-time version of this inequality is also established.