Local martingale

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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[edit]

Let (Ω, FP) be a probability space; let F = { Ft | 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 τk : Ω → [0, +∞) such that

is an F-martingale for every k.

Examples[edit]

Example 1[edit]

Let Wt be the Wiener process and T = min{ t : Wt = −1 } the time of first hit of −1. The stopped process Wmin{ tT } 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 τk = k. This sequence diverges almost surely, since τk = k 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]

Example 2[edit]

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 (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[edit]

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,

  as

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[edit]

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

   for every t.

Thus, Condition (*) is sufficient for a local martingale being a martingale. A stronger condition

   for every t

is also sufficient.

Caution. The weaker condition

   for every t

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[edit]

  1. ^ 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[edit]