Abstract
Set-indexed martingales and submartingales are defined and studied. The admissible function of a submartingale is defined and some class (D) conditions are given which allow the extension of the function to a σ-additive measure on the predictable σ-algebra. Then, we prove a Doob-Meyer decomposition: A set-indexed submartingale can be decomposed into the sum of a weak martingale and an increasing process. A hypothesis of predictability ensures the uniqueness of this decomposition. An explicit construction of the increasing process associated with a submartingale is given. Finally, some remarks, about quasimartingales are discussed.
Original language | English |
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Pages (from-to) | 499-525 |
Number of pages | 27 |
Journal | Journal of Theoretical Probability |
Volume | 7 |
Issue number | 3 |
DOIs | |
State | Published - Jul 1994 |
Externally published | Yes |
Keywords
- Doob-Meyer decomposition
- Lattice
- admissible measure
- class (D)
- increasing process
- predictable σ-algebra
- quasimartingale
- right-continuity
- set-indexed martingale
- stopping set
- submartingale