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:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)