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In mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the case that the process is a non-negative martingale, but the result is also valid for non-negative submartingales.

The inequality is due to the American mathematician Joseph Leo Doob.
Let X be a submartingale taking non-negative real values, either in discrete or continuous time. That is, for all times s and t with s < t,

(1)
\begin{align} \mathbf{E} \big[ X_{t} \big| \mathcal{F}_{s} \big] \geq X_{s}. \end{align}

(For a continuous-time submartingale, assume further that the process is sample continuous.) Then, for any constant C > 0 and p ≥ 1,

(2)
\begin{align} \mathbf{P} \left[ \sup_{0 \leq t \leq T} X_{t} \geq C \right] \leq \frac{\mathbf{E} \big[ X_{T}^{p} \big]}{C^{p}}. \end{align}