# 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 (Ω, *F*, **P**) be a probability space; let *F*_{∗} = { *F*_{t} | *t* ≥ 0 } be a filtration of *F*; let X : [0, +∞) × Ω → *S* be an *F*_{∗}-adapted stochastic process on set *S*. Then *X* is called an *F*_{∗}**-local martingale** if there exists a sequence of *F*_{∗}-stopping times *τ*_{k} : Ω → [0, +∞) such that

- the
*τ*_{k}are almost surely increasing:**P**[*τ*_{k}<*τ*_{k+1}] = 1; - the
*τ*_{k}diverge almost surely:**P**[*τ*_{k}→ +∞ as*k*→ +∞] = 1; - the stopped process

- is an
*F*_{∗}-martingale for every*k*.

## Examples

### Example 1

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 τ_{k} = *k*. This sequence diverges almost surely, since τ_{k} = *k* 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]}

### Example 2

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

here

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

### Example 3

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* ≥ 1 but to 0 for *r* ≤ 1).

## 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 *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

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

## Technical details

- ↑
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.

## References

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