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Concept
Local martingale
Summary
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.
Local martingales are essential in stochastic analysis (see Itō calculus, semimartingale, and Girsanov theorem).
Let be a probability space; let be a filtration of ; let be an -adapted stochastic process on the set . Then is called an -local martingale if there exists a sequence of -stopping times such that
the are almost surely increasing: ;
the diverge almost surely: ;
the stopped process is an -martingale for every .
Let Wt be the Wiener process and T = min{ t : Wt = −1 } the time of first hit of −1. The stopped process Wmin{ t, T } is a martingale; its expectation is 0 at all times, nevertheless its limit (as t → ∞) is equal to −1 almost surely (a kind of gambler's ruin). A time change leads to a process
The process is continuous almost surely; nevertheless, its expectation is discontinuous,
This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as if there is such t, otherwise . This sequence diverges almost surely, since for all k large enough (namely, for all k that exceed the maximal value of the process X). The process stopped at τk is a martingale.
Let Wt be the Wiener process and ƒ a measurable function such that Then the following process is a martingale:
here
The Dirac delta function (strictly speaking, not a function), being used in place of leads to a process defined informally as and formally as
where
The process is continuous almost surely (since almost surely), nevertheless, its expectation is discontinuous,
This process is not a martingale.
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