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

Let be a probability space; let be a filtration of F; let be an F_-adapted stochastic process on the set S. Then X is called an F_-local martingale if there exists a sequence of F_*-stopping times such that

Examples

Example 1

[[File:Local martingale.svg|thumb|Illustration for local martingale. Up Panel: Multiple simulated paths of the process X_t which is stopped upon hitting -1. This shows gambler's ruin behavior, and is not a martingale. Down Panel: Paths of X_t with an additional stopping criterion: the process is also stopped when it reaches a magnitude of k = 2.0. This no longer suffers from gambler's ruin behavior, and is a martingale.]] 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). A time change leads to a process The process X_t 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 \tau_k = k. This sequence diverges almost surely, since \tau_k = 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.

Example 2

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 \delta (strictly speaking, not a function), being used in place of f, leads to a process defined informally as and formally as where The process Y_t is continuous almost surely (since W_1 \ne 0 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 Z_t be the complex-valued Wiener process, and The process X_t is continuous almost surely (since Z_t 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 \ln|u-1| over the circle |u|=r tends to infinity as. (In fact, it is equal to \ln r for r ≥ 1 but to 0 for r ≤ 1).

Martingales via local martingales

Let M_t 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 Thus, Condition (*) is sufficient for a local martingale M_t 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 W_t is the Wiener process, and is twice continuously differentiable. The process M_t is a local martingale if and only if f satisfies the PDE However, this PDE itself does not ensure that M_t is a martingale. In order to apply (**) the following condition on f is sufficient: for every and t there exists such that for all s \in [0,t] and

Technical details