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 *W*_{t} be the Wiener process and *T* = min{ *t* : *W*_{t} = −1 } the time of first hit of −1. The stopped process *W*_{min{ 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 τ_{k} is a martingale.^{[details 1]}

Let *W*_{t} be the Wiener process and *ƒ* a measurable function such that Then the following process is a martingale:

where

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. However, it is a local martingale. A localizing sequence may be chosen as

Let be the complex-valued Wiener process, and

The process is continuous almost surely (since does not hit 1, almost surely), and is a local martingale, since the function is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as Nevertheless, the expectation of this process is non-constant; moreover,

- as

which can be deduced from the fact that the mean value of over the circle tends to infinity as . (In fact, it is equal to for *r* ≥ 1 but to 0 for *r* ≤ 1).

Let be a local martingale. In order to prove that it is a martingale it is sufficient to prove that in *L*^{1} (as ) for every *t*, that is, here is the stopped process. The given relation implies that almost surely. The dominated convergence theorem ensures the convergence in *L*^{1} provided that

- for every
*t*.

Thus, Condition (*) is sufficient for a local martingale being a martingale. A stronger condition

- for every
*t*

is also sufficient.

*Caution.* The weaker condition

- for every
*t*

is not sufficient. Moreover, the condition

is still not sufficient; for a counterexample see Example 3 above.

A special case:

where is the Wiener process, and is twice continuously differentiable. The process is a local martingale if and only if *f* satisfies the PDE

However, this PDE itself does not ensure that is a martingale. In order to apply (**) the following condition on *f* is sufficient: for every and *t* there exists such that

for all and

**^**For the times before 1 it is a martingale since a stopped Brownian motion is. After the instant 1 it is constant. It remains to check it at the instant 1. By the bounded convergence theorem the expectation at 1 is the limit of the expectation at (*n*-1)/*n*(as*n*tends to infinity), and the latter does not depend on*n*. The same argument applies to the conditional expectation.^{[vague]}