Continuous-Time Martingalesocw.nctu.edu.tw/upload/classbfs1209013347185177.pdf · CHAPTER 7...

CHAPTER 7

Continuous-Time Martingales

7.1. Stochastic processes

Let (Ω,F , P) be a probability space and let I ⊂ [0,∞) be an interval.

Definition 7.1. A real-valued stochastic process X = (Xt)t∈I is a family of random

variables (Xt : t ∈ I) on (Ω,F).

��, �� probability space.

Remark 7.2. (1) We may regard the stochastic process X as a function of two

variables

X : I × Ω −→ R

(t, ω) �−→ Xt(ω).

(�� stochastic process ��.)

(2) For fixed ω ∈ Ω, then

t �−→ Xt(ω)

is a function: I −→ R, which is called a path of X.

(3) For fixed t ∈ I, then

ω �−→ Xt(ω)

is a function: Ω −→ R, which is a random variable.

��: �� path? path ��? �� discrete time ��

��.

157

158 7. CONTINUOUS-TIME MARTINGALES

Definition 7.3. (1) A filtration F = (Ft)t∈I is a family of σ-algebras satisfying

Fs ⊆ Ft

for all s, t ∈ I with s ≤ t.

(2) A filtration is called right-continuous1 if

Ft = Ft+ :=⋂s>t

Fs.

(3) The σ-algebra F is complete if A ∈ F with P(A) = 0 and B ⊂ A implies B ∈ F .

(�� σ-algebra.)

(4) The filtration F is complete if F is complete and every null set in F is contained

in Ft for all t ∈ I (or contained in F0).2

(5) F is said to satisfy the usual condition if F is right-continuous and complete.

� continuous time �� measurability ��.

Definition 7.4. (1) The stochastic process X is adapted to F if, for each t ≥ 0,

Xt is Ft-measurable.

1��. ��, ��

��. �� right-continuous ��

��, � ��, ��. ��

�� right-continuous ��. In fact, filtration � right-continuous ��, �� left-continuous �

continuous ��, ��, � ��.2F is complete �� F �� null set �� Ft �. Moreover, we have

F is complete =⇒ Ft is complete for all t ∈ I.

��.

7.1. STOCHASTIC PROCESSES 159

(2) The stochastic process X is called measurable if the mapping

([0,∞) × Ω,B([0,∞)) ⊗F) −→ (R,B(R))

(t, ω) �−→ Xt(ω)

is measurable, i.e., for each A ∈ B, the set

{(t, ω) : Xt(ω) ∈ A} ∈ B([0,∞)) ⊗F .

(3) The stochastic process X is called progressively measurable with respect to F if

([0, t] × Ω,B([0, t]) ⊗Ft) −→ (R,B(R))

(s, ω) �−→ Xs(ω)

is measurable for all t ≥ 0, i.e., for each t ≥ 0, A ∈ B(R), the set

{(s, ω) : 0 ≤ s ≤ t, ω ∈ Ω, Xs(ω) ∈ A} ∈ B([0, t]) ⊗F

Remark 7.5. (1) Any progressively measurable process is measurable and adapted.

(2) If X is measurable and adapted, then it has a progressively measurable modifi-

cation3.

(3) If the stochastic process X is F-adapted and every sample path is right-continuous

or else every sample path is left-continuous, then X is progressively measurable

with respect to F.

Definition 7.6. A random variable T : Ω −→ I ∪ {∞} is called a stopping time with

respect to F if

{T ≤ t} ∈ Ft, for all t ∈ I.

3A modification Y of the stochastic process X is a stochastic process on the same probability space,

with the same parameter set I such that

P(Xt = Yt) = 1

for all t ∈ I.


� �� stopping time discrete stopping time ��. Discrete

stopping time ��

{T = t} ∈ Ft for all t ∈ I.

�� continuous time ��. �� remark.

Remark 7.7. If T is a stopping time, then {T < t} ∈ Ft for all t ∈ I. But the

converse does not hold in general.

Proof. (1) For all t ∈ I,

{T < t} =⋃n∈N

{T ≤ t − 1

n

}︸︷︷︸

∈ Ft− 1n

∈ Ft.

(2) Consider a filtration

Ft =

⎧⎪⎨⎪⎩

{∅, Ω}, t ≤ 1,

power set of Ω, t > 1,

and a random time T given by

T =

⎧⎪⎨⎪⎩

1, if ω ∈ A,

2, otherwise,

where A is a nontrivial subset of Ω. Then

• for t ≤ 1, {T < t} = ∅ ∈ Ft,

• for 1 < t ≤ 2, {T < t} = A ∈ Ft,

• for t > 2, {T < t} = Ω ∈ Ft.

This implies that {T < t} ∈ Ft for all t ≥ 0. However, {T ≤ 1} = A �∈ F1. This means

that T is not a stopping time. �

��?


Lemma 7.8. Suppose that F is right-continuous. Then T is a stopping time if and

only if {T < t} ∈ Ft, for all t ∈ I.

Proof. “=⇒”: By Remark 7.7.

“⇐=”: Since

{T ≤ t} =⋂

n≥m

{T < t +

1

n

}︸︷︷︸∈ Ft+ 1

n⊆ Ft+ 1

m

∈ Ft+ 1m

for all m ∈ N, we have

{T ≤ t} ∈⋂

m∈N

Ft+ 1m

= Ft+ = Ft

due to the right-continuity of F. �

Example 7.9. Consider a set A ⊆ R (Rn ��). Define the first hitting time of A by

TA := inf{t ≥ 0 : Xt ∈ A}.

1 dimensional � 2 dimensional first hitting time �� Figure 7.9 � Figure 7.9.

A

X0

TA TA

Figure 7.1. 1 dimensional first hitting time


A X0

XTA

Figure 7.2. 2 dimensional first hitting time

(1) If F is right-continuous, X is adapted and right-continuous, A is open, then TA

is a stopping time.

(2) If X is adapted and continuous, A is closed, then TA is a stopping time.

Proof. For every t ∈ I, since X is right-continuous, A is open

{TA < t} =⋃s<t

{Xs ∈ A} =⋃

r<t,r∈Q

{Xr ∈ A}︸︷︷︸∈ Fr ⊆ Ft

∈ Ft.

By Lemma 7.8, we see that TA is a stopping time. �

Definition 7.10. For stopping times S and T with S ≤ T , the stochastic interval

((S, T ]] is defined by

((S, T ]] := {(t, ω) ∈ I × Ω : S(ω) < t < T (ω)}.

[[S, T ]], ((S, T )), and [[S, T )) are defined similarly.

Definition 7.11. Let T be a stopping time,

FT := {A ∈ F : A ∩ {T ≤ t} ∈ Ft for all t ∈ I}


I

S1(w ) S2(w )

RI

Figure 7.3. Ω = R, �� [[S, T ]] ��.

is called the σ-algebra of events determined prior to the stopping time T .

�� discrete time ��, � �� discrete time ��

�.

Lemma 7.12. Suppose (Xt) is an adapted and right-continuous stochastic process and

T < ∞ is a stopping time. Then (XT )(ω) := XT (ω)(ω) is FT -measurable.

Exercise

(1) For the given sample space and probability measure, find the smallest complete

σ-algebra.

(a) Ω = {1, 2, 3, 4, 5, 6}, P({1, 2}) = P({3, 4}) = P({5, 6}) =1

3.

(b) Ω = {1, 2, 3, 4, 5, 6}, P({1, 2}) = 0, P({3, 4}) = P({5, 6}) =1

2.

(c) Ω = R, P([n − 1, n)) =1

2n, for all n ∈ N.


7.2. Uniform integrability

Definition 7.13. A family of random variables (Yα)α∈Λ is called uniformly integrable

(u.i.) if

limc→∞

supα∈Λ

∫{|Yα|>c}

|Yα| dP = 0.

��, ��.

Theorem 7.14. (Yα)α∈Λ is uniformly integrable if and only if it satisfies the following

two conditions:

(i) supα∈Λ

E|Yα| < ∞;

(ii) For all ε > 0, there exists δ = δ(ε) > 0 such that for E ∈ F ,

∫E

|Yα| dP < ε, for all α ∈ Λ,

whenever P(E) < δ.

Example 7.15. (1) If |Yα| ≤ Z for all α ∈ Λ and for a random variable Z ∈L1(P), then (Yα)α∈Λ is uniformly integrable.

Proof. Since |Yα| ≤ Z, we have

{|Yα| ≥ C} ⊆ {Z ≥ C}.

Hence,

0 ≤ limc→∞

supα∈Λ

∫{|Yα|>c}

|Yα| dP ≤ limc→∞

supα∈Λ

∫{Z>c}

|Yα| dP

≤ limc→∞

supα∈Λ

∫{Z>c}

Z dP = limc→∞

∫{Z>c}

Z dP

= limc→∞

∫Z I{Z>c} dP =

∫ (limc→∞

Z I{Z>c})

dP = 0

7.2. UNIFORM INTEGRABILITY 165

due to the dominated convergence theorem. Hence, we get that

limc→∞

supα∈Λ

∫{|Yα|>c}

|Yα| dP = 0.

�

(2) If

supα∈Λ

E[|Yα|p] < ∞

for some p > 1, then (Yα)α∈Λ is uniformly integrable.

(Note: � Theorem 7.14 �� p = 1 ��.)

Proof. Since

supα∈Λ

∫{|Yα|>c}

|Yα| dP = supα∈Λ

∫{|Yα|>c}

|Yα|p|Yα|p−1

dP

≤ supα∈Λ

1

cp−1

∫|Yα|p dP =

1

cp−1supα∈Λ

E[|Yα|p],

which approaches to 0 as c goes to ∞, we get that (Yα)α∈Λ is uniformly integrable.

�

(3) If Z ∈ L1(P), then the collection of random variables

{E[Z|G] : G ⊆ F is a σ-algebra }

is uniformly integrable.

(4) Let Yn = nI(0,1/n) and let P be the Lebesgue measure. For any c, there exists

n ∈ N such that for n > c,

∫{|Yn|>c}

|Yα| dP =

∫ 1/n

0

n dx = 1.

This implies that

supn∈N

∫{|Yn|>c}

|Yn| dP = 1, for all c.


This means

limc→∞

supn∈Λ

∫{|Yn|>c}

|Yn| dP = 1.

Thus, (Yn) is not uniformly integrable.

(5) Consider a sequence of i.i.d. random variables (ξi) with

P(ξi = 1) = P(ξi = −1) =1

2.

Let X0 = 0 and for n ≥ 1,

Xn =]ξ1 + ξ2 + · · · + ξn.

Then (Xn)n≥1 is not uniformly integrable, since

limnto∞E|Xn|√

n=

√2

π,

which implies that supn∈Λ

E|Yn| < ∞.

7.3. Martingale theory in continuous-time

Let I = [0,∞).

Definition 7.16. (1) A stochastic process X = (Xt)t≥0 is called a martingale

(with respect to P and F) if

(a) Xt ∈ L1(P) for all t ≥ 0;

(b) X is adapted;

(c) For 0 ≤ s ≤ t < ∞,

E[Xt|Fs] = Xs, P − a.s.

(2) X is called a submartingale if (i) + (ii) +

7.3. MARTINGALE THEORY IN CONTINUOUS-TIME 167

(iiia) For 0 ≤ s ≤ t < ∞,

E[Xt|Fs] ≥ Xs, P − a.s.

(3) X is called a supermartingale if (i) + (ii) +

(iiib) For 0 ≤ s ≤ t < ∞,

E[Xt|Fs] ≤ Xs, P − a.s.

Example 7.17. Let Z ∈ L1(P), then the process (Xt) defined by

Xt = E[Z|Ft],

is a martingale. In fact, (Xt) is a uniformly integrable martingale.

Theorem 7.18 (Optional Stopping Theorem). (1) Let X = (Xt) be a right-continuous,

uniformly integrable martingale and let S and T be stopping times with S ≤ T ,

then XS, XT ∈ L1(P) and

E[XT |FS] = XS, P − a.s.

(2) Let X = (Xt) be a right-continuous, uniformly integrable supermartingale and let

S and T be stopping times with S ≤ T , then XS, XT ∈ L1(P) and

E[XT |FS] ≤ XS, P − a.s.

Definition 7.19. For a stochastic process X = (Xt)t≥0 and a stopping time T , the

stopped process XT := (XTt )t≥0 is defined by

XTt := Xt∧T =

⎧⎪⎨⎪⎩

Xt, if t ≤ T,

XT , if t > T.


Corollary 7.20. (1) If X = (Xt) is a right-continuous, uniformly integrable su-

permartingale and T is a stopping time, then XT is a right-continuous, uniformly

integrable supermartingale.

(2) If X = (Xt) is a right-continuous, uniformly integrable martingale and T is a

stopping time, then XT is a right-continuous, uniformly integrable martingale.

Remark 7.21. The condition “uniformly integrable” is necessary, e.g., let (Xt)t≥0 be

a random walk4 and let

T := inf{t ≥ 0 : Xt = 1},

then XT is not a martingale, since XT∞ = 1 P-a.s. ��, �� X � Xt =

t∑i=1

ξi,

where (ξi) is a sequence of Bernoulli distributed random variables with

P(ξi = 1) = P(ξi = −1) =1

2.

By Example 7.15(5), we see that X is not uniformly integrable. �� X �

uniformly integrable ��.

Proposition 7.22. Let X = (Xt)0≤t≤∞ be adapted, right-continuous and satisfy

E[XT ] = E[X0]

for every stopping time T with XT ∈ L1(P), then X is a uniformly integrable martingale.

Remark 7.23. An alternative version of optional sampling theorem.

(1) Suppose X = (Xt)0≤t≤∞ is a right-continuous martingale with last element X∞,

S and T are stopping times with S ≤ T , then

E[XT |FS] = XS, P − a.s.

4�� random walk �� .

7.4. LOCAL MARTINGALES 169

(2) Suppose X = (Xt)0≤t≤∞ is a right-continuous supermartingale with last element

X∞, S and T are stopping times with S ≤ T , then

E[XT |FS] ≤ XS, P − a.s.

7.4. Local martingales

Definition 7.24. An adapted, right-continuous stochastic process X = (Xt)t≥0 is

called a local martingale, if there exists a sequence of stopping times (Tn) with Tn ↑ ∞P-a.s. such that the stopped process

XTnI{Tn>0} = (Xt∧TnI{Tn>0})t≥0

is a (uniformly integrable) martingale with respect to (Ft).

Notation 7.25. Mloc = the collection of all local martingales.

Mloc0 = {X ∈ Mloc : X0 = 0 P − a.s.}.

Remark 7.26. Every martingale is local martingale.

Proof. Let Tn = n, then (Xt∧n,Ft) is a martingale. this implies that (Xt) is a local

martingale. �

Remark 7.27. A local martingale may not be a martingale, c.f., Karatzas and Shreve

[19] P.168.

Remark 7.28. A local martingale with

sup0≤r≤t

|Xr| ∈ L1(P)

for all t ≥ 0, is a martingale.


Proof. Let X0 = 0 and let (Tn) be the sequence of stopping times with Tn ↑ ∞ such

that XTn is a martingale for all n. For 0 ≤ s ≤ t, we have

E[Xt∧Tn|Fs] = Xs∧Tn .

As n → ∞, Tn −→ ∞, we have

Xt∧Tn −→ Xt P − a.s.,

Xs∧Tn −→ Xs P − a.s.

Due to

supn

|Xt∧Tn| ≤ sup0≤r≤t

|Xr| ∈ L1(P),

Xs = limn→∞

Xs∧Tn = limn→∞

E [Xt∧Tn|Fs]

= E

[lim

n→∞Xt∧Tn|Fs

]= E[Xt|Fs]

by Lebesgue convergence theorem. Hence (Xt) is a martingale. �

Proposition 7.29. very nonnegative local martingale is a supermartingale.

Proof. Let (Tn) be a sequence of stopping times with Tn ↑ ∞ and XTn is a martingale

for all n. Then for 0 ≤ s < t,

Xs = limn→∞

Xs∧Tn = limn→∞

E[Xt∧Tn|Fs]

≥ E

[lim

n→∞Xt∧Tn|Fs

]= E[Xt|Fs]

due to Fatou’s lemma. �

7.5. DOOB-MEYER DECOMPOSITION 171

7.5. Doob-Meyer decomposition

Recall: (Doob decomposition)

Suppose that (Xn,Fn) is a supermartingale, then

Xn = Yn − Zn,

where (Yn) is a martingale and (Zn) is an increasing previsible process.

This decomposition is unique.

How about the case in the continuous time?

�� previsible ��. � discrete time �,

previsible ��: Xn � Fn−1-measurable. �� continuous time ��

��, �� .

Definition 7.30. Let Ω = Ω × (0,∞).

(1) P is called a previsible σ-algebra if it is generated by all left-continuous, adapted

process on Ω.

(2) A stochastic process X is called previsible if X is measurable with respect to a

previsible σ-algebra P over Ω.

Remark 7.31. previsble �� (�

��).

�� left-continuous process ��? �� stochastic process

� left-continuous ��, ��.

Note: previsible process �� left-continuous.

Remark 7.32. (1) Every previsible process is adapted.


(2) Every continuous, increasing process is previsible.

(3) If F satisfies the usual condition, every previsible process is adapted to (Ft−).

� previsible process �� Ph. Protter [24].

�� Doob-Meyer decomposition ��, �� notation.

Notation 7.33. M2 = the collection of all cadlag5 martingales (Mt) with

supt≥0

E[M2t ] < ∞; (7.1)

M20 = {M ∈ M2 : M0 = 0};

M2,c0 = {M ∈ M2

0 : M is continuous in t}.

Theorem 7.34 (Doob-Meyer decomposition). Let X = (Xt)t≥0 be a right-continuous

supermartingale and the collection of random variables

{XT : T is a stopping time with P(T < ∞) = 1}

is uniformly integrable. Then X admits a unique decomposition

Xt = X0 + Mt − At,

where M is a right-continuous, uniformly integrable martingale with M0 = 0 and A is an

increasing, right-continuous, previsible process with A0 = 0.

Corollary 7.35. Let M ∈ M2 be right-continuous. Then there exists a unique right-

continuous previsible process 〈M〉 = (〈M〉t)t≥0 with 〈M〉0 = 0 such that the process M2 −〈M〉 is a martingale.

5continu a droite, limite a gauche, � right-continuous, left-limit exists, �� RCLL.

7.5. DOOB-MEYER DECOMPOSITION 173

�� M ∈ M2 �� , �� martingale � convergence

theorem. ��, �� M∞ �

Mt = E[M∞|Ft].

Proof. By Jensen’s inequality,

E[M2t |F s] ≥ (E[Mt|Fs])

2 = M2s .

we see that M2t is a submartingale. Remain to check that (M2

t ) is uniformly integrable.

� M ∈ M2 �� M∞ exists and is integrable ��, �� (M2t ) is uniformly

integrable. Applying Doob-Meyer decomposition, we can get the desired result. �

Definition 7.36. 〈M〉 is called the quadratic variation of M .

�� t ∈ [0,∞) �� (7.1) �� quadratic variation?

For general case, �� . ��, for fixed N , then

sup0≤t≤N

E[M2t ] = E[M2

N ] < ∞,

and

Mt = E[MN |Ft].

�� , � � martingale �� (7.1), ��

quadratic variation. �� 〈M〉 ��. � discrete time case

�, ��, �!�� Brownian motion �"

��.

Lemma 7.37. Let M ∈ M2,c. For partition Π of [0, t], set

‖Π‖ := max1≤k≤m

|tk − tk−1|,


we have

lim‖Π‖→0

m∑k=1

|Mtk − Mtk−1|2 = 〈M〉t in probability,

i.e., for any ε > 0, η > 0, there exists δ > 0 such that

max1≤k≤m

|tk − tk−1| < δ =⇒ P(∣∣|Mtk − Mtk−1

|2 − 〈M〉t∣∣ > ε

)< η.

Definition 7.38. Let M,N ∈ M2. Then the process

〈M,N〉 :=1

4(〈M + n〉 − 〈M − N〉)

is called a cross variation (or quadratic covariation) of M and N .

Remark 7.39. (1) 〈M,M〉t = 〈M〉t.(2) MN − 〈M, N〉 is a martingale. Moreover, if M , N are right-continuous, 〈M, N〉

is the unique right-continuous, previsible process B of bounded variation (��

��: �� B �� nonincreasing or nondecreasing) with B0 = 0

such that MN − B is a martingale.

(3) If M, N ∈ M2,loc are right-continuous, then there exist a unique increasing,

right-continuous, previsible process 〈M〉 and a unique right-continuous, previsible

process 〈M,N〉 of bounded variation with 〈M〉0 = 〈M,N〉0 = 0 such that M2 −〈M〉 and MN − 〈M,N〉 are local martingales.

(4) lim‖Π‖→0

m∑k=1

(Mtk − Mtk−1)(Ntk − Ntk−1

) = 〈M,N〉t in probability.

Exercise

(1) Let M,N ∈ M2,c0 be independent stochastic processes on a filtered probability

space (Ω,F , F = (Ft)t≥0, P) with quadratic variation 〈M〉t = 2t and 〈N〉t = 4t,

respectively.

(a) Find the cross variation 〈M, N〉t.

7.6. SEMIMARTINGALES 175

(b) Find the quadratic variation of M + N .

(c) Find the quadratic variation of M − N .

(2) (16 points) Let M, N ∈ M2,c0 be stochastic processes on a filtered probability

space (Ω,F , F = (Ft)t≥0, P) with quadratic variation 〈M〉t = 4t and 〈N〉t = 6t,

respectively. Moreover, their cross variation 〈M, N〉t is given by 2t.

(a) Find the quadratic variation of M + N .

(b) Find the quadratic variation of M − N .

7.6. Semimartingales

�� martingale � local martingale ��.

Definition 7.40. A stochastic process X = (Xt)t≥0 is called a semimartingale (#�)

if X is an adapted process with the decomposition

Xt = X0 + Mt + At, (7.2)

where (Mt) is a local martingale with M0 = 0 and (At) is an adapted, cadlag process of

bounded variation, i.e., there exist increasing, adapted process A+, A− such that

At = A+t − A−

t .

Remark 7.41. In general, the decomposition (7.2) is not unique. But if X is contin-

uous, this this decomposition is unique.

Lemma 7.42. A continuous local martingale of bounded variation is constant P-a.s.

Remark 7.43. A continuous non-constant local martingale is not of bounded varia-

tion.


�� "�� martingale �, ��

��.

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