Doob's Martingale Inequality - Statement of The Inequality

Statement of The Inequality

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 and p ≥ 1,

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. denotes the σ-algebra generated by all the random variables X_i with i ≤ s; the collection of such σ-algebras forms a filtration of the probability space.