Local martingale

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).

-adapted stochastic process on the set

-local martingale if there exists a sequence of

such that 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).

is continuous almost surely; nevertheless, its expectation is discontinuous, This process is not a martingale.

A localizing sequence may be chosen as

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.

[details 1] Let Wt 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

almost surely), nevertheless, its expectation is discontinuous, This process is not a martingale.

A localizing sequence may be chosen as

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

The dominated convergence theorem ensures the convergence in L1 provided that Thus, Condition (*) is sufficient for a local martingale

A stronger condition is also sufficient.

The weaker condition is not sufficient.

Moreover, the condition is still not sufficient; for a counterexample see Example 3 above.

is a local martingale if and only if f satisfies the PDE However, this PDE itself does not ensure that

In order to apply (**) the following condition on f is sufficient: for every