Definition
Let (Ω, F, P) be a probability space; let F∗ = { Ft | t ≥ 0 } be a filtration of F; let X : [0, +∞) × Ω → S be an F∗-adapted stochastic process. Then X is called an F∗-local martingale if there exists a sequence of F∗-stopping times τk : Ω → [0, +∞) such that
- the τk are almost surely increasing: P = 1;
- the τk diverge almost surely: P = 1;
- the stopped process
- is an F∗-martingale for every k.
Read more about this topic: Local Martingale
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)