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).

Definition

be a probability space be a filtration-adapted stochastic process-local martingale-stopping times such that

• are almost surely increasing;

• ;

• the stopped process.

Examples

Example 1

Let Wt be the Wiener process and T=min{t:Wt=1} the time of first hit of1. 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). A time change leads to a process

is continuous almost surely; nevertheless, its expectation is discontinuous,

if there is such t 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.

Example 2

Let Wt be the Wiener process and Then the following process is a martingale:

where

The Dirac delta function and formally as

where

almost surely), nevertheless, its expectation is discontinuous,

Example 3

be the complex-valued Wiener process, and

is harmonic Nevertheless, the expectation of this process is non-constant; moreover,

for r ≥ 1 but to 0 for r ≤ 1).

Martingales via local martingales

in L1) for every t almost surely. The dominated convergence theorem ensures the convergence in L1 provided that

1. for every t.

being a martingale. A stronger condition

1. for every t

is also sufficient.

Caution. The weaker condition

1. for every t

is not sufficient. Moreover, the condition

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

A special case:

is the Wiener process is twice continuously differentiable is a local martingale if and only if f satisfies the PDE

is a martingale. In order to apply (**) the following condition on f and t such that

Technical details

References