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, 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.

Definition

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

Example 1

Let Wt be the Wiener process and T = min the time of first hit of −1. The stopped process Wmin is a martingale; its expectation is 0 at all times, nevertheless its limit is equal to −1 almost surely. 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 τk = k. This sequence diverges almost surely, since τk = k for all k large enough. The process stopped at τk is a martingale.

Example 2

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 , being used in place of leads to a process defined informally as and formally as
where
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

Example 3

Let be the complex-valued Wiener process, and
The process is continuous almost surely, and is a local martingale, since the function is harmonic. 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..

Martingales via local martingales

Let be a local martingale. In order to prove that it is a martingale it is sufficient to prove that in L1 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 L1 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

Technical details