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.
Let (Ω, F, P) 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
the τk are almost surely increasing: P[τk < τk+1] = 1;the τk diverge almost surely: P[τk → +∞ as k → +∞] = 1;the stopped processis an
F∗-martingale for every
k.
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
X t = { W min ( t 1 − t , T ) for 0 ≤ t < 1 , − 1 for 1 ≤ t < ∞ . The process X t is continuous almost surely; nevertheless, its expectation is discontinuous,
E X t = { 0 for 0 ≤ t < 1 , − 1 for 1 ≤ t < ∞ . This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as τ k = min { t : X t = k } 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.
Let Wt be the Wiener process and ƒ a measurable function such that E | f ( W 1 ) | < ∞ . Then the following process is a martingale:
X t = E ( f ( W 1 ) | F t ) = { f 1 − t ( W t ) for 0 ≤ t < 1 , f ( W 1 ) for 1 ≤ t < ∞ ; here
f s ( x ) = E f ( x + W s ) = ∫ f ( x + y ) 1 2 π s e − y 2 / ( 2 s ) . The Dirac delta function δ (strictly speaking, not a function), being used in place of f , leads to a process defined informally as Y t = E ( δ ( W 1 ) | F t ) and formally as
Y t = { δ 1 − t ( W t ) for 0 ≤ t < 1 , 0 for 1 ≤ t < ∞ , where
δ s ( x ) = 1 2 π s e − x 2 / ( 2 s ) . The process Y t is continuous almost surely (since W 1 ≠ 0 almost surely), nevertheless, its expectation is discontinuous,
E Y t = { 1 / 2 π for 0 ≤ t < 1 , 0 for 1 ≤ t < ∞ . This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as τ k = min { t : Y t = k } .
Let Z t be the complex-valued Wiener process, and
X t = ln | Z t − 1 | . 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 u ↦ ln | u − 1 | is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as τ k = min { t : X t = − k } . Nevertheless, the expectation of this process is non-constant; moreover,
E X t → ∞ as
t → ∞ , which can be deduced from the fact that the mean value of ln | u − 1 | over the circle | u | = r tends to infinity as r → ∞ . (In fact, it is equal to ln r for r ≥ 1 but to 0 for r ≤ 1).
Let M t be a local martingale. In order to prove that it is a martingale it is sufficient to prove that M t τ k → M t in L1 (as k → ∞ ) for every t, that is, E | M t τ k − M t | → 0 ; here M t τ k = M t ∧ τ k is the stopped process. The given relation τ k → ∞ implies that M t τ k → M t almost surely. The dominated convergence theorem ensures the convergence in L1 provided that
( ∗ ) E sup k | M t τ k | < ∞ for every
t.
Thus, Condition (*) is sufficient for a local martingale M t being a martingale. A stronger condition
( ∗ ∗ ) E sup s ∈ [ 0 , t ] | M s | < ∞ for every
tis also sufficient.
Caution. The weaker condition
sup s ∈ [ 0 , t ] E | M s | < ∞ for every
tis not sufficient. Moreover, the condition
sup t ∈ [ 0 , ∞ ) E e | M t | < ∞ is still not sufficient; for a counterexample see Example 3 above.
A special case:
M t = f ( t , W t ) , where W t is the Wiener process, and f : [ 0 , ∞ ) × R → R is twice continuously differentiable. The process M t is a local martingale if and only if f satisfies the PDE
( ∂ ∂ t + 1 2 ∂ 2 ∂ x 2 ) f ( t , x ) = 0. 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 ε > 0 and t there exists C = C ( ε , t ) such that
| f ( s , x ) | ≤ C e ε x 2 for all s ∈ [ 0 , t ] and x ∈ R .
