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
Famous quotes containing the words statement of, statement and/or inequality:
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (17671845)
“A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
—J.L. (John Langshaw)
“However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
—Alexis de Tocqueville (18051859)