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 Xi with i ≤ s; the collection of such σ-algebras forms a filtration of the probability space.
Read more about this topic: Doob's Martingale Inequality
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