Doob's martingale inequality

In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a submartingale 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 martingale, but the result is also valid for submartingales. The inequality is due to the American mathematician Joseph L. Doob. The setting of Doob's inequality is a submartingale relative to a filtration of the underlying probability space. The probability measure on the sample space of the martingale will be denoted by P. The corresponding expected value of a random variable X, as defined by Lebesgue integration, will be denoted by E[X]. Informally, Doob's inequality states that the expected value of the process at some final time controls the probability that a sample path will reach above any particular value beforehand. As the proof uses very direct reasoning, it does not require any restrictive assumptions on the underlying filtration or on the process itself, unlike for many other theorems about stochastic processes. In the continuous-time setting, right-continuity (or left-continuity) of the sample paths is required, but only for the sake of knowing that the supremal value of a sample path equals the supremum over an arbitrary countable dense subset of times. Let X1, ..., Xn be a discrete-time submartingale relative to a filtration of the underlying probability space, which is to say: The submartingale inequality says that for any positive number C. The proof relies on the set-theoretic fact that the event defined by max(Xi) > C may be decomposed as the disjoint union of the events Ei defined by Xi > C and Xj ≤ C for all j < i. Then having made use of the submartingale property for the last inequality and the fact that for the last equality. Summing this result as i ranges from 1 to n results in the conclusion which is sharper than the stated result.