Let \(M=(M_ t)_{t\in\mathbb{R}^ 2_ +}\) be a right continuous, two- parameter strong martingale such that \(\mathbb{E}\sup_{s\leq t}M_ s<\infty\) for all \(t=(t_ 1,t_ 2)\in\mathbb{R}^ 2_ +\). The authors prove that the quadratic variation \(([M]_ t)\) of \(M\) exists and that the Davis inequalities hold: \[ C\mathbb{E}[M]_ t^{1/2}\leq\mathbb{E}\sup_{s\leq t} M_ s\leq D\mathbb{E}[M]_ t^{1/2}. \]