Local martingale

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.

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

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 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.

Example 2

Let W_t 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 } .

Example 3

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).

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 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 t

is also sufficient.

Caution. The weaker condition

sup s ∈ [ 0 , t ] E | M s | < ∞    for every t

is 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 .