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, a local martingale is not in general 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 Itô 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
Let W<sub>t</sub> be the Wiener process and T = min{ t : W<sub>t</sub> = −1 } the time of first hit of −1. The stopped process W<sub>min{ t, T }</sub> 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 ÃÂ<sub>k</sub> is a martingale.
Let W<sub>t</sub> 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,
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 âÂÂ¥ 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<sup>1</sup> (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<sup>1</sup> provided that
Thus, Condition (*) is sufficient for a local martingale being a martingale. A stronger condition
is also sufficient.
Caution. The weaker condition
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