III Stochastic Calculus and Applications

2Semi-martingales

2.2 Local martingale

From now on, we assume that (Ω

, F,

(

)

, P

) satisfies the usual conditions,

namely that

(i) F

contains all P-null sets

(ii) (F

)

is right-continuous, i.e. F

= (F

s>t

for all t ≥ 0.

We recall some of the properties of continuous martingales.

Theorem

(Optional stopping theorem)

Let

be a c`adl`ag adapted integrable

process. Then the following are equivalent:

(i) X is a martingale, i.e. X

∈ L

for every t, and

E(X

| F

) = X

for all t > s.

(ii)

The stopped process

= (

) = (

T ∧t

) is a martingale for all stopping

times T .

(iii)

For all stopping times

T, S

with

bounded,

∈ L

and

(

| F

) =

T ∧S

almost surely.

(iv) For all bounded stopping times T , X

∈ L

and E(X

) = E(X

For X uniformly integrable, (iii) and (iv) hold for all stopping times.

In practice, most of our results will be first proven for bounded martingales,

or perhaps square integrable ones. The point is that the square-integrable

martingales form a Hilbert space, and Hilbert space techniques can help us say

something useful about these martingales. To get something about a general

martingale

, we can apply a cutoff

inf{t >

0 :

≥ n}

, and then

will be a martingale for all

. We can then take the limit

n → ∞

to recover

something about the martingale itself.

But if we are doing this, we might as well weaken the martingale condition

a bit — we only need the

to be martingales. Of course, we aren’t doing

this just for fun. In general, martingales will not always be closed under the

operations we are interested in, but local (or maybe semi-) martingales will be.

In general, we define

Definition

(Local martingale)

A c`adl`ag adapted process

is a local martingale

if there exists a sequence of stopping times

such that

→ ∞

almost surely,

and X

is a martingale for every n. We say the sequence T

reduces X.

Example.

(i)

Every martingale is a local martingale, since by the optional stopping

theorem, we can take T

= n.

(ii) Let (B

) to be a standard 3d Brownian motion on R

. Then

)

t≥1





t≥1

is a local martingale but not a martingale.

To see this, first note that

sup

t≥1

< ∞, EX

→ 0.

Since

→

0 and

≥

0, we know

cannot be a martingale. However,

we can check that it is a local martingale. Recall that for any f ∈ C

= f(B

) −f(B

) −

∆f(B

) ds

is a martingale. Moreover, ∆

|x|

= 0 for all

x 6

= 0. Thus, if

|x|

didn’t have

a singularity at 0, this would have told us

is a martingale. Thus, we

are safe if we try to bound |B

| away from zero.

Let

= inf



t ≥ 1 : |B

| <



and pick

∈ C

such that

(

) =

|x|

for

|x| ≥

. Then

− X

t∧T

. So X

is a martingale.

It remains to show that

→ ∞

, and this follows from the fact that

→ 0.

Proposition.

Let

be a local martingale and

≥

0 for all

. Then

is a

supermartingale.

Proof. Let (T

) be a reducing sequence for X. Then

E(X

| F

) = E



lim inf

n→∞

t∧T

| F



≤ lim

n→∞

E(X

t∧T

| F

)

= lim inf

→∞

s∧T

= X

Recall the following result from Advanced Probability:

Proposition. Let X ∈ L

(Ω, F, P). Then the set

χ = {E(X | G) : G ⊆ F a sub-σ-algebra}

is uniformly integrable, i.e.

sup

Y ∈χ

E(|Y |1

|Y |>λ

) → 0 as λ → ∞.

Recall also the following important result about uniformly integrable random

variables:

Theorem

(Vitali theorem)

. X

→ X

iff (

) is uniformly integrable and

→ X in probability.

With these, we can state the following characterization of martingales in

terms of local martingales:

Proposition. The following are equivalent:

(i) X is a martingale.

(ii) X is a local martingale, and for all t ≥ 0, the set

= {X

: T is a stopping time with T ≤ t}

is uniformly integrable.

Proof.

–

(a)

⇒

(b): Let

be a martingale. Then by the optional stopping theorem,

(

| F

) for any bounded stopping time

T ≤ t

. So

is uniformly

integrable.

–

(b)

⇒

(a): Let

be a local martingale with reducing sequence (

), and

assume that the sets

are uniformly integrable for all

t ≥

0. By the

optional stopping theorem, it suffices to show that

(

) =

(

) for any

bounded stopping time T .

So let T be a bounded stopping time, say T ≤ t. Then

E(X

) = E(X

T ∧T

)

for all

. Now

T ∧ T

is a stopping time

≤ t

, so

T ∧T

}

is uniformly

integrable by assumption. Moreover,

∧T → T

almost surely as

n → ∞

hence

T ∧T

→ X

in probability. Hence by Vitali, this converges in

E(X

) = E(X

Corollary.

Z ∈ L

is such that

| ≤ Z

for all

, then

is a martingale. In

particular, every bounded local martingale is a martingale.

The definition of a local martingale does not give us control over what the

reducing sequence

}

is. In particular, it is not necessarily true that

will be bounded, which is a helpful property to have. Fortunately, we have the

following proposition:

Proposition. Let X be a continuous local martingale with X

= 0. Define

= inf{t ≥ 0 : |X

| = n}.

Then

is a stopping time,

→ ∞

and

is a bounded martingale. In

particular, (S

) reduces X.

Proof. It is clear that S

is a stopping time, since (if it is not clear)

≤ t} =

k∈N



sup

s≤t

| > n −



k∈N

[

s<t,s∈Q



| > n −



∈ F

It is also clear that S

→ ∞, since

sup

s≤t

| ≤ n ↔ S

≥ t,

and by continuity and compactness, sup

s≤t

| is finite for every (ω, t).

Finally, we show that

is a martingale. By the optional stopping theorem,

∧S

is a martingale, so

is a local martingale. But it is also bounded by

n. So it is a martingale.

An important and useful theorem is the following:

Theorem.

Let

be a continuous local martingale with

= 0. If

is also a

finite variation process, then X

= 0 for all t.

This would rule out interpreting

as a Lebesgue–Stieltjes integral

for

a non-zero continuous local martingale. In particular, we cannot take

to be Brownian motion. Instead, we have to develop a new theory of integration

for continuous local martingales, namely the Itˆo integral.

On the other hand, this theorem is very useful. We will later want to define

the stochastic integral with respect to the sum of a continuous local martingale

and a finite variation process, which is the appropriate generality for our theorems

to make good sense. This theorem tells us there is a unique way to decompose a

process as a sum of a finite variation process and a continuous local martingale

(if it can be done). So we can simply define this stochastic integral by using the

Lebesgue–Stieltjes integral on the finite variation part and the Itˆo integral on

the continuous local martingale part.

Proof.

Let

be a finite-variation continuous local martingale with

= 0. Since

is finite variation, we can define the total variation process (

) corresponding

to X, and let

= inf{t ≥ 0 : V

≥ n} = inf



t ≥ 0 :

|dX

| ≥ n



Then

is a stopping time, and

→ ∞

since

is assumed to be finite

variation. Moreover, by optional stopping,

is a local martingale, and is also

bounded, since

≤

t∧S

|dX

| ≤ n.

So X

is in fact a martingale.

We claim its

-norm vanishes. Let 0 =

< t

< ··· < t

be a

subdivision of [0

, t

]. Using the fact that

is a martingale and has orthogonal

increments, we can write

E((X

)

) =

i=1

E((X

− X

i−1

)

Observe that

is finite variation, but the right-hand side is summing the

square of the variation, which ought to vanish when we take the limit

max |t

−

i−1

| → 0. Indeed, we can compute

E((X

)

) =

i=1

E((X

− X

i−1

)

≤ E

max

1≤i≤k

− X

i−1

i=1

− X

i−1

≤ E



max

1≤i≤k

− X

i−1

t∧S



≤ E



max

1≤i≤k

− X

i−1



Of course, the first term is also bounded by the total variation. Moreover, we

can make further subdivisions so that the mesh size tends to zero, and then the

first term vanishes in the limit by continuity. So by dominated convergence, we

must have

((

)

) = 0. So

= 0 almost surely for all

. So

= 0 for all

t almost surely.