Chapter 3
Doléans measures
1.
Introduction
The definition of the isometric stochastic integral (in Chapter 2) with respect
to L 2 -martingales depends on the existence of the Doléans measure, the
sigma-finite measure µ M on the predictable sigma-field P characterized by the
properties
(i) P F M02 = µ M {s = 0, ω ∈ F} for each F ∈ F0
2
(ii) µ M [[0, τ ∧ t]] = PM(τ ∧ t)2 = PM02 + P Mτ ∧t − M0 for each t ∈ R+
and stopping time τ .
Notice that the measure µ M depends on M only through M02 and the
2
submartingale Mt − M0 . In fact, analogous measures can be defined for any
submartingale S. To simplify the basic construction, I will initially consider
only cadlag submartingales indexed by [0, 1] with S0 ≡ 0. For the general case,
the measure will be built from countably many finite pieces.
<1>
Definition. Say that the cadlag submartingale S has a Doléans measure µ,
defined on the predictable sigma-field of [0, 1] × , if µ[[0, τ ]] = PSτ for
every τ in the collection T of all stopping times taking values in [0, 1].
The assumption S0 ≡ 0 forces µ[[0]] = PS0 = 0, which means that µ
concentrates on ((0, 1]], with finite total mass, µ((0, 1]] = PS1 < ∞. The
countable additivity of µ imposes (Problem [2]) a subtle extra requirement
on S, known somewhat cryptically as property [D]:
[D]
{Sτ : τ ∈ T} is uniformly integrable.
The property will let us calculate expectations at stopping times by passing to
the limit from calculations with simpler stopping times. The square integrable
submartingales involved in the construction of the isometric stochastic integral all
have property [D]: if Mt := P(M1 | Ft ), for 0 ≤ t ≤ 1, then Mτ = P(M1 | Fτ )
and
2
2
for each τ in T.
0 ≤ M τ − M 0 ≤ P M 1 − M 0 | Fτ
(Remember that {P(X | G) : G ⊆ F} is uniformly integrable for each integrable
random variable X .)
2.
Existence of the Doléans measure under property [D]
Let {St : 0 ≤ t ≤ 1} be a cadlag submartingale with property [D] and S0 ≡ 0.
In this Section, I will construct the Doléans measure µ as an increasing linear
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Section 3.2
Existence of the Doléans measure under property [D]
functional on the set H+ of all adapted processes H for which there exist finite
constants C H and C H with
0 ≤ H (t, ω) ≤ C H
|H (t, ω) − H (s, ω)| ≤
C H |s
for all t ∈ [0, 1] and ω ∈ − t|
for all s, t ∈ [0, 1] and ω ∈ .
It is easy to show that H+ generates the predictable sigma-field P on [0, 1] × .
The Daniell existence theorem (UGMTP, §A.6) asserts that finite measures on P
correspond precisely to those increasing linear functionals µ on H+ with the
“sigma-smoothness” property: µHk → 0 whenever 1 ≥ Hk ↓ 0 pointwise.
<2>
Theorem. For a cadlag submartingale {St : 0 ≤ t ≤ 1} with property [D],
the Doléans measure (as in Definition <1>) exists.
Proof. Construct µ as a limit of simpler increasing linear functionals on H+ ,
with property [D] ensuring its sigma-smoothness. For each n in N define
i,n := S((i + 1)/2n ) − S(i/2n ) and write Pi,n (· · ·) for expectations conditional
on F(i/2n ). Note that Pi,n i,n ≥ 0 almost surely, by the submartingale property.
For each H in H+ , define linear functionals
µn H :=
P H (i/2n )i,n =
P H (i/2n )Pi,n i,n .
0≤i<2n
0≤i<2n
The second form ensures that µn is an increasing functional.
To prove that µH := limn→∞ µn H exists for each H ∈ H+ , I will show
that the sequence {µn H : n ∈ N} is Cauchy. For n ≤ m, write t j for the largest
multiple of 2−n smaller than j/2m . Then
|µm H − µn H | = |
P H ( j/2m )i,n − P H (t j ) j,m |
0≤ j<2m
≤
P |H ( j/2m ) − H (t j )| P j,m j,m
j
≤ C H 2−n P
j,m = C H 2−n PS1 ,
j
which tends to zero as n tends to infinity.
The functional µ inherits linearity and the increasing property from
the {µn }. For each fixed > 0, property [D] will give a convenient upper
bound for µH in terms of the stopping time
<3>
τ (H, ) := inf{t : H (t, ω) ≥ } ∧ 1.
Temporarily write τn for the discretized stopping time obtained by rounding
τ (H, ) up to the next multiple of 2−n . Then
P {i/2n < τn }Pi,n i,n + C H {i/2n ≥ τn }Pi,n i,n
µn H ≤
0≤i<2n
≤
Pi,n + C H P
{i/2n ≥ τn }i,n
0≤i<2n
0≤i<2n
≤ PS1 + C H P S1 − Sτn .
<4>
Let n tend to infinity. Uniform integrability of the sequence {Sτn } together with
right-continuity of the sample paths of S lets us deduce in the limit that
µH ≤ PS1 + C H P S1 − Sτ (H,) .
Now suppose {Hk : k ∈ N} is a sequence from H+ for which 1 ≥ Hk ↓ 0
pointwise. For a fixed > 0, temporarily write σk for τ (Hk , ). By compactness
of [0, 1], the pointwise convergence of the continuous functions, Hk (·, ω) ↓ 0,
is actually uniform. For each ω, the sequence {σk (ω)} not only increases to 1,
it actually achieves the value 1 at some finite k (depending on ω). Uniform
integrability of {Sσk : k ∈ N} and the analog of <4> for each Hk then give
as k → ∞.
µHk ≤ PS1 + P S1 − Sσk → PS1
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Chapter 3
Doléans measures
The sigma-smoothness of µ follows. The functional corresponds to the integral
with respect to a finite measure on P, with total mass µ[[0, 1]] = limn µn 1 = PS1 .
It remains to prove that µ[[0, τ ]] = PSτ for τ ∈ T. For given > 0,
approximate [[0, τ ]] by the continuous process
H (t, ω) = min 1, (τ (ω) + − t)+ /
for 0 ≤ t ≤ 1.
It is adapted because {H (t, ω) ≤ c} = {τ ≤ t − (1 − c)} ∈ Ft , for each fixed t
and constant 0 ≤ c < 1. It belongs to H+ , with C H = 1 and C H = 1/, and
[[0, τ ]] ≤ H ≤ [[0, τ + ]].
Hε
1–Hε
ε
τ
0
τ+ε
1
−n
When 2 < , direct calculation shows that µn H ≤ PSτ +2 , which in the
limit implies µH ≤ PSτ +2 and, by Dominated Convergence,
µ[[0, τ ]] = lim→0 µH ≤ PSτ .
Inequality <4> applied with τ := τ (1 − H , ) gives
µ((τ + , 1]] ≤ µ 1 − H ≤ PS1 + P S1 − Sτ ,
which, in the limit as tends to zero, implies µ((τ, 1]] ≤ P S1 − Sτ . From
the fact that
PS1 = µ[[0, 1]] = µ[[0, τ ]] + µ((τ, 1]] ≤ PSτ + P S1 − Sτ ,
<5>
3.
conclude that µ[[0, τ ]] = PSτ .
Example. Let {Bt : 0 ≤ t ≤ 1} be a Brownian motion with continuous
sample paths. The submartingale St := Bt2 has a very simple Doléans measure,
characterized for F ∈ Fα by
µ S (α, β] ⊗ F = PF Bβ2 − Bα2 = PF(β − α).
That is, µ S = m ⊗ P, with m equal to Lebesgue measure on B[0, 1]. Of course
µ S has a further extension to the product sigma-field B[0, 1] ⊗ F.
A Poisson process {Nt : 0 ≤ t ≤ 1} with intensity 1 shares with Brownian
motion the independent increment property, but the increment Nt − Ns has a
Poisson(t − s) distribution. The sample paths are constant, except for jumps of
size 1 corresponding to points of the process. The process {Nt : 0 ≤ t ≤ 1}
is a submartingale with respect to its natural filtration, with Doléans measure
m ⊗ P, the same as the square of Brownian motion.
Clearly the Doléans measure does not uniquely determine the submartingale: both squared Brownian motion and the Poisson process have Doléans
measure m ⊗ P. But the only square integrable martingale M with continuous
sample paths and Doléans measure µ M = m ⊗ P is Brownian motion: if F ∈ Fs
and s < t then PF(Mt2 − Ms2 ) = µ M (s, t] ⊗ F = (t − s)PF, from which it
follows that Mt2 − t is a martingale with respect to {Ft }. It follows from Lévy’s
characterization that M is a Brownian motion.
Doléans measure for submartingales
Once more consider processes indexed by R+ . Redefine T to be the set of all
stopping times taking values in [0, ∞]. Say that a submartingale {St : t ∈ R+ }
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Section 3.3
Doléans measure for submartingales
has property [D] if
{Sτ {τ < ∞} : τ ∈ T} is uniformly integrable.
For such a submartingale, the variables St converge almost surely and in L 1 to
a limit S∞ , and {St : 0 ≤ t ≤ ∞} is a submartingale (Problem [3]). Clearly
{S∞ {τ = ∞} : τ ∈ T} is uniformly integrable. Thus we may reinterpret the
property [D] for S as an assertion that
{Sτ : τ ∈ T} is uniformly integrable.
Via any one-to-one increasing map ψ from [0, ∞] onto [0, 1], we may
identify S with a submartingale indexed by [0, 1] having the property [D] as
it was understood in Section 2. The corresponding Doléans measure µ0 puts
zero mass on [[1]], because µ0 ((ψ(n), 1]] = P(S∞ − Sn ) → 0 as n → ∞.
The inverse map ψ −1 transfers µ0 back to a finite measure on the predictable
sigma-field of R+ × for which µ[[0, τ ]] = PSτ for each τ ∈ T.
Remark. The construction from Section 2 could have been carried out
directly for R+ × , at the cost of some extra precautions to keep mass
from straying off to infinity. Alternatively, we could apply an analog of
the construction from Section 2 to build Doléans measures for each [[0, n]],
with n ∈ N, which could then be pieced together in much the same way as
described below.
The construction of Doléans measures also extends to processes that are
only locally submartingales with property [D], that is, processes X for which
there exist an sequence of stopping times τk ↑ ∞ for which each X ∧τk is a
submartingale with property [D]. The sequence {τk } is said to reduce X to a
submartingale with property [D]. The extension is important partly because of
the following result.
<6>
<7>
Extend to local submartingales?
Lemma.
Every cadlag submartingale has property [D] locally.
Proof. For a cadlag submartingale {St : t ∈ R+ }, define stopping times
τk = k ∧ inf{t : |St | ≥ k}. It suffices to show that Sτk is integrable, for each
fixed k, for then uniform integrability will follow from the fact that each |Sτ ∧τk |
is bounded by the same integrable random variable k + |Sτk |.
Let {Mt : 0 ≤ t ≤ k} be the cadlag version of the uniformly
integrable
martingale P(Sk | Ft ). The random variable Mτk = P Mk | F(τk ) is integrable.
The process X t := Mt − St , for 0 ≤ t ≤ k, is a positive supermartingale. By
the Stopping Time Lemma, 0 ≤ PX τk ≤ PX 0 < ∞. The random variable |Sτk |
is bounded by the sum X τk + |Mτk |, which is integrable.
Theorem. Let {St : t ∈ R+ } be a cadlag submartingale, reduced to a process
with property [D] by stopping times {τk }. Suppose S0 ≡ 0. Then there exists a
sigma-finite measure µ on the predictable sigma-field of R+ ⊗ characterized
by the property µ[[0, τ ]] = limk→∞ PSτ ∧τk for each stopping time τ .
Proof. Write µk for the Doléans measure generated on P by S∧τk . First note
that µk [[0, ∞]] = PS∞∧τk = µn [[0, τk ]], which implies that the measure µk
concentrates on the stochastic interval [[0, τk ]]. Then note that, for each stopping
time τ ,
µk [[0, τ ]] = PSτ ∧τk = PS(τ ∧ τk ∧ τk+1 ) = µk+1 [[0, τ ]] ∩ [[0, τk ]] ,
which implies that the restriction of µk+1 to [[0, τk ]] equals µk .
For each nonnegative, predictable H the sequence {µk H } is increasing.
Its limit, µH , defines a linear functional with the monotone convergence
property; the limit is the Doléans measure of S. By definition, µ[[0, τ ]] =
limk µk [[0, τ ]] = lim PSτ ∧τk for every stopping time τ .
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Chapter 3
Doléans measures
To see that the limit does not depend on the particular choice for
the reducing sequence, let {σk } be another reducing sequence of stopping
times, with associated Doléans measures {νk } and limit ν. For each j and k,
and each stopping time τ ,
ν j [[0, τ ∧ τk ]] = PSτ ∧τk ∧σj = µk [[0, τ ∧ σ j ]].
4.
Let j tend to infinity. The interval [[0, τ ∧ σ j ]] increases pointwise to [[0, τ ]],
even if τ is allowed to take infinite values (a benefit of excluding {∞} ⊗ from
the stochastic interval). From the definition of ν and the countable additivity
of µk , deduce that ν[[0, τ ∧ τk ]] = µk [[0, τ ]]. Then let k tend to infinity to
deduce that ν and µ give the same value to each stochastic interval [[0, τ ]],
for τ ∈ T. Also ν[[0]] = 0 = µ[[0]]. The equality for all sets in P follows by a
generating class argument.
Problems
[1]
Let {St : 0 ≤ t ≤ 1} be a cadlag martingale with S0 ≡ 0. For a fixed λ ∈ R+ ,
define stopping times σ := inf{t : St ≤ −λ} ∧ 1 and τ := inf{t : St ≥ λ} ∧ 1.
(i) Show that
λP{supt St > λ} ≤ λP{Sτ ≥ λ} ≤ PSτ {Sτ ≥ λ} ≤ PS1 {Sτ ≥ λ} ≤ P|S1 |.
(ii) Show that
λP{inft St < −λ} ≤ P(−Sσ ){Sσ ≤ λ} ≤ −PSσ + PS1 {Sσ ≥ −λ} ≤ P|S1 |.
(iii) Deduce that λP{supt |St | > λ} ≤ 2P|S1 |.
[2]
Suppose a cadlag martingale {St : 0 ≤ t ≤ 1}, with S0 ≡ 1, has a Doléans
measure µ in the sense of Definition <1>, that is, µ[[0, τ ]] = PSτ for
every τ ∈ T. Show that S has property [D] by following these steps.
(i) Invoke the continuous version of the Stopping Time Lemma (from
Chapter 1) to show that 0 ≤ PSτ and PSτ+ ≤ PS1+ for each τ ∈ T. Deduce
that supτ ∈T P|Sτ | ≤ κ := 2PS1+ < ∞.
(ii) For each C ∈ R+ , show that PSτ {Sτ > C} ≤ PS1 {Sτ > C}. (Hint: STL
again.) Deduce, via the integrability of S1 and the bound P{|Sτ | > C} ≤
κ/C that supτ ∈T PSτ {Sτ > C} → 0 as C → ∞.
(iii) Show that every cadlag function on [0, 1] is bounded in absolute value.
Deduce that the stopping time σC := inf{t : St < −C} ∧ 1 has σC (ω) = 1
for all C large enough. Deduce that µ((σC , 1]] → 0 as C → ∞.
(iv) For a given τ ∈ T and C ∈ R+ , define Fτ := {Sτ < −C}. Show that
τ := τ Fτc + Fτ is a stopping time for which
P S1 − Sτ Fτ = P Sτ − Sτ = µ((τ, τ ]] ≤ µ((σC , 1]],
then deduce that supτ ∈T P − Sτ {Sτ < −C} → 0 as C → ∞. Hint:
Show that if ω ∈ Fτ then σC (ω) ≤ τ (ω) and if ω ∈ Fτc then τ (ω) = τ (ω).
[3]
Let {St : t ∈ R+ } be a submartingale of class
[D]. Show that there exists an
integrable random variable S∞ for which P S∞ | Ft ≥ St → S∞ almost surely
and in L 1 by following these steps.
(i) Show that the uniformly integrable submartingale
{Sn : n ∈ N} converges
almost surely and in L 1 to an S∞ for which P S∞ | Fn ≥ Sn .
(ii) For t ≤ n, show that St ≤ P P(S∞ | Fn ) | Ft = P S∞ | Ft .
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Section 3.4
Problems
(iii) For t ≥ n, show that
P(St − Sn )− ≤ P(St − Sn )+ ≤ P(S∞ − Sn )+ → 0
as n → ∞.
(iv) For each k ∈ N, choose n(k) for which P|S∞ − Sn(k) | ≤ 4−k . Invoke
Problem [1] to show that k P{supt≥n(k) |St − Sn(k) | > 2−k } < ∞. Deduce
that St → S∞ almost surely
5.
Notes
My exposition in this Chapter is based on ideas drawn from a study of
Métivier (1982, §13), Dellacherie & Meyer (1982, Chapter VII), and Chung
& Williams (1990, Chapter 2). The construction in Section 2 appears new,
although it is clearly closely related to existing methods.
References
Chung, K. L. & Williams, R. J. (1990), Introduction to Stochastic Integration,
Birkhäuser, Boston.
Dellacherie, C. & Meyer, P. A. (1982), Probabilities and Potential B: Theory
of Martingales, North-Holland, Amsterdam.
Métivier, M. (1982), Semimartingales: A Course on Stochastic Processes,
De Gruyter, Berlin.
Pollard, D. (2001), A User’s Guide to Measure Theoretic Probability, Cambridge
University Press.
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