## ::Local martingale

### ::concepts

Process::infty ''k''::mathbb ''t''::local Almost::cases Surely::every ''F''::function

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, Girsanov theorem.

**Local martingale sections**

Intro Definition Examples Martingales via local martingales Technical details References

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