Doob's martingale inequality

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. 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 L. 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, (For a continuous-time submartingale, assume further that the process is càdlàg.) Then, for any constant C > 0, In the above, as is conventional, P denotes the probability measure on the sample space Ω of the stochastic process and E denotes the expected value with respect to the probability measure P, i.e. the integral in the sense of Lebesgue integration. F s {displaystyle {mathcal {F}}_{s}} denotes the σ-algebra generated by all the random variables Xi with i ≤ s; the collection of such σ-algebras forms a filtration of the probability space. There are further (sub)martingale inequalities also due to Doob. With the same assumptions on X as above, let and for p ≥ 1 let