martingale
Martingales definition
Definition. Let be a filtered probability space and be a stochastic process such that is integrable (http://planetmath.org/Integral2) for all . Then, is called a submartingale if
and a supermartigale if
A submartingale that is also a supermartingale is called a martingale, i.e., a martingale satisfies
Similarly, if the form a decreasing collection of -subalgebras of , then is called a reverse submartingale if
and a reverse supermartingale if
Remarks
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The martingale property captures the idea of a fair bet, where the expected future value is equal to the current value.
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The submartingale property is equivalent to
and similarly for the other definitions. This is immediate from the definition of conditional expectation.
Title | martingale |
Canonical name | Martingale |
Date of creation | 2013-03-22 13:33:09 |
Last modified on | 2013-03-22 13:33:09 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 25 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 60G46 |
Classification | msc 60G44 |
Classification | msc 60G42 |
Related topic | LocalMartingale |
Related topic | DoobsOptionalSamplingTheorem |
Related topic | ConditionalExpectationUnderChangeOfMeasure |
Related topic | MartingaleConvergenceTheorem |
Defines | martingale |
Defines | supermartingale |
Defines | submartingale |
Defines | reverse submartingale |
Defines | reverse supermartingale |