Wiki
Wiki
Courses and public working groups
Courses and public working groups
Private Working Groups
Private Working Groups
- New!!! Reading Group
- Theatre
- Admin
- Research
- Teaching
Sidebar
world:martingale
Table of Contents
Supermartingale convergence results
Theorem. Let be a filtration and let be an -adapted sequence of -random variables such that
- for all , we have .
that is, is a -supermartingale, with negative part bounded in . Then, almost surely, exists and is in .
Proof
Let and define and for , In words, the first time flags is when . Then it flags until goes above . Then it flags until goes below . So consecutive sequences of are linked with upcrossings of for . Now, define
Define the number of upcrossings of for . Then, From the fact that is a -supermartingale and is -previsible, we deduce that is also a -supermartingale so that . Then, Letting goes to infinity, the monotone convergence theorem yields: and thus, for all . Now, which shows that exits almost surely.
Moreover, so that which implies by Fatou's lemma that . The proof is completed.
Corollary: Submartingale convergence results
As a consequence, the conclusion also holds if is a -submartingale, with positive part bounded in : indeed, we only need to apply the previous result to .
Corollary Assume that is a filtration and let be an -adapted sequence of random variables such that
- for all , we have .
that is, is a -submartingale, with positive part bounded in . Then, almost surely, exists and is in .
world/martingale.txt · Last modified: 2022/03/16 07:40 (external edit) · Currently locked by: 114.119.134.93
