Convergence of values in optimal stopping

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by  Sandrine Toldo

Convergence of values in optimal stopping
Sandrine Toldo
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Convergence of values in optimal stopping
∗
Sandrine TOLDO
IRMAR, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
Abstract: Underthehypothesisofconvergenceinprobabilityofasequenceofca`dla`gprocesses
(Xn) to a ca`dla`g process X, we are interested in the convergence of corresponding values in
n
optimal stopping. We give results under hypothesis of inclusion of filtrations or convergence of
filtrations.
Keywords : Values in optimal stopping, Convergence of stochastic processes, Convergence of
filtrations.
1 Introduction
4
0 Let us consider a ca`dla`g process X. Let us denote by FX its natural filtration and by F the
0 right-continuous associated filtration (∀t,F = FX). We denote by T the set of F stopping
2 t t+ L
times bounded by L.

y Let γ : [0,+∞[×R → R a bounded continuous function. We define the value in optimal
a stopping of horizon L of the process X by :
M
6 Γ(L)= supE[γ(τ,Xτ)].
τ∈TL
2

- Remark 1 As it is written in Lamberton and Pag`es (1990), the value of Γ(L) only depends

1 on the law of X.

n
o We are interested in the following problem : let us consider a sequence (Xn)n of processes
i which converges in probability to a limit process X. For all n, we denote by Fn the natural
s
r filtration of Xn and by Tn the set of Fn stopping times bounded by L. Then, we define the
L
e values in optimal stopping Γ (L) by Γ (L)= supE[γ(τ,Xn)]. The main aim of this paper is
v n n τ
τ∈Tn
L
, to give conditions under which (Γ (L)) converges to Γ(L).
5 n n
In his unpublished manuscript (Aldous, 1981), Aldous proved that if X is quasi-left con-
2
6 tinuous and if there is extended convergence (in law) of ((Xn,Fn))n to (X,F), then (Γn(L))n
1 convergestoΓ(L). Intheirpaper(LambertonandPag`es,1990),LambertonandPag`esobtained
0 the same result under the hypothesis of weak extended convergence of ((Xn,Fn)) to (X,F),
n
0
quasi-left continuity of the Xn’s and Aldous’ criterion of tightness for (Xn) .
0 n
As a first step, we are going to prove in section 3 that, under very weak hypothesis, holds
0
- the inequality Γ(L)6liminfΓn(L).
d Then,toprovethat(Γ (L)) convergestoΓ(L),itremainstoshowthatΓ(L)>limsupΓ (L).
s n n n
c This inequality is more difficult and both papers (Aldous, 1981) and (Lamberton and Pag`es,
c 1990) need weak extended convergence to prove it. Here, we prove that it happens under the
hypothesis of inclusion of filtrations Fn ⊂F or under convergence of filtrations.
ThemainideainourproofoftheinequalityΓ(L)>limsupΓ (L)isthefollowing. Webuild
n
a sequence (τn) of (Fn) stopping times bounded by L. Then, we extract a convergent subse-
quence of (τn) to a random variable τ and, at the same time, we wish to compare E[γ(τ,X )]
τ
and Γ(L). We are going to do that through two methods.
∗Email address : [email protected]
1
First, we will enlarge the space of stopping times, by considering the randomized stopping
times and the topology introduced in (Baxter and Chacon, 1977). Baxter and Chacon have
shown that the space of randomized stopping times for a right continuous filtration with the
associated topology is compact. We are going to use this method in section 5 when we have
the inclusion of the filtrations Fn ⊂F (it means that ∀t∈[0,T],Fn ⊂F ).
t t
When we do not have the previous inclusion, we enlarge the filtration F associated to the
limiting process X. This method is used, in a slightly different way, in (Aldous, 1981) and
in (Lamberton and Pag`es, 1990). In section 6, we enlarge (as little as possible) the limiting
filtration so that the limit τ∗ of a convergent subsequence of the randomized (Fn) stopping
times associatedto the (τn) is a randomized stopping time for this enlargedfiltration and we
n
use convergence of filtrations instead of extended convergence.
For technical reasons, we need Aldous’ criterion of tightness for the sequence (Xn) . In
n
P
section 4, we are going to show that, if Xn −→X, Aldous’ criterion of tightness for (Xn) and
n
quasi-left continuity of the limiting process X are equivalent.
Finally, in section 7, we give applications of the convergence of values in optimal stopping
to discretizations and also to financial models.
In what follows, we are given a probability space (Ω,A,P). We fix a positive real T and
also L between 0 and T. Unless otherwise specified, every σ-field is supposed to be included
in A, every process will be indexed by [0,T] and taking values in R and every filtration will
be indexed by [0,T]. D =D([0,T]) denotes the space of ca`dla`g functions from [0,T] to R. We
endow D with the Skorokhod topology.
For technical background about Skorokhod topology, the reader may refer to (Billingsley,
1999) or (Jacod and Shiryaev, 2002).
2 Statement of the result of convergence of the optimal
values
The main purpose of this paper is to prove the following Theorem :
Theorem 2 Letusconsideraca`dla`g continuousin probability processXandasequence(Xn)
n
of ca`dla`g processes. Let F be the right continuous filtration associated to the natural filtration
P
of X and (Fn) the natural filtrations of the processes (Xn) . We assume that Xn −→ X and
n n
that one of the following assertions holds :
- for all n, Fn ⊂F,
- Fn −→w F.
Then, Γ (L)−−−−→Γ(L).
n
n→∞
The notion of convergence of filtrations has been defined in (Hoover, 1991) :
Definition 3 We say that (Fn) converges weakly to F if for every A ∈ F , (E[1 |Fn])
T A . n
converges in probability to E[1 |F] for Skorokhod topology. We denote Fn −→w F.
A .
The proof of Theorem 2 will be given through two steps :
- Step 1 : we show that Γ(L)6liminfΓ (L),
n
- Step 2 : we show that Γ(L)>limsupΓ (L).
n
3 Proof of the inequality Γ(L) 6 liminf Γ (L)
n
Theorem 4 Letusconsideraca`dla`gprocessX suchthatP[∆X 6=0]=0,itsnaturalfiltration
L
FX, a sequence of ca`dla`g processes (Xn) and their natural filtrations (Fn) . We suppose that
n n
2
P
Xn −→X. Then Γ(L)6liminfΓ (L).
n
Proof
The proof is broken in several steps.
Lemma 5 Let τ be a FX stopping time bounded by L taking values in a discrete set {t }
i i∈I
such that P[∆X 6= 0] = 0, ∀i. For all i, we consider A ={τ =t }. We define τn by :
ti i i
τn(ω)=min{t :i∈{j :E[1 |Fn](ω)>1/2}}, ∀ω. Then, (τn) is a sequence of (Tn) such
i Aj tj L
P
that (τn,Xn )−→(τ,X ).
τn τ
Proof
(τn) is, by definition, a sequence of (Fn)-stopping times.
n
Moreover,for all ω, τn(ω)6max{t ,i∈I}6L because τ is bounded by L. So, (τn) is a
i n
sequence of (Fn) stopping times bounded by L.
P
Let us show that τn −→τ.
To prove that, we are going to use the convergence of σ-fields (also defined in (Hoover,
1991)) :
Definition 6 We say that (An) converges to A and denote An → A if for every A ∈ A,
n
E[1 |An]−→P 1 .
A A
We have Fn →FX,∀i according to the following Lemma :
ti ti
Lemma 7 Let(Xn) beasequenceofca`dla`g processes thatconverges inprobability toaca`dla`g
n
process X, (Fn) the natural filtrations of the Xn’s and FX the natural filtration of X. Then,
for all t, Fn →FX.
t t
Proof
Take t ∈ [0,T]. Let us fix t < ... < t 6 t such that for all i, P[∆X 6= 0] = 0 and let
1 k ti
f :Rk →R be a bounded continuous.
Xn −→P X and for all i=1,...,k, P[|∆X |=6 0]=0, so
ti
P
(Xn,...,Xn)−→(X ,...,X ).
t1 tk t1 tk
f is bounded continuous so :
f(Xn,...,Xn)−L−→1 f(X ,...,X ). (1)
t1 tk t1 tk
Take ε>0.
P[|E[f(X ,...,X )|Fn]−f(X ,...,X )|>ε]
t1 tk t t1 tk
6 P[|E[f(X ,...,X )|Fn]−E[f(Xn,...,Xn)|Fn]|>ε/2]
t1 tk t t1 tk t
+P[|E[f(Xn,...,Xn)|Fn]−f(X ,...,X )|>ε/2]
t1 tk t t1 tk
4
6 E[|f(Xn,...,Xn)−f(X ,...,X )|]
ε t1 tk t1 tk
using Markov’s inequality
−−−−→ 0 using (1).
n→∞
The conclusion comes with the following characterizationof the convergenceof σ-fields, whose
proofuseexactlysameargumentsasintheproofofLemma3in(Coquet,M´eminandS lomin´ski,
2001) :
3
Lemma 8 Let Y be a ca`dla`g process, A=σ({Y ,t>0}) and (An) a sequence of σ-fields. The
t
following conditions are equivalent :
i) An →A,
ii) E[f(Y ,...,Y )|An] −→P f(Y ,...,Y ) for every continuous bounded function f : Rk → R
t1 tk t1 tk
and t ,...,t continuity points of Y.
1 k
Lemma 7 is proved. (cid:3)
With this Lemma, we can prove the convergence in probability of (τn) to τ.
n
Let us consider a subsequence (τϕ(n)) of (τn) . For every i, the convergence of the σ-fields
n n
(Fn) toFX impliesE[1 |Fϕ(n)]−→P 1 .Bysuccessiveextractionsfori∈I finite,thereexists
ti n ti Ai ti Ai
ψ such that for every i, E[1 |Fϕ◦ψ(n)]−a−.s→. 1 . For n large enough, we have τϕ◦ψ(n) =τ a.s.
Ai ti Ai
Then, τϕ◦ψ(n) −a−.s→. τ. It follows that τn −→P τ.
P
It remains to show that Xn −→X .
τn τ
P P
Xn −→X sowecanfindasequence(Λn) ofrandomtimechangessuchthatsup |Xn −X |−→0
n t Λn(t) t
P
and sup |Λn(t)−t|−→0. Fix ε>0 and η >0. We have :
t
P[|Xn −X |>η]
τn τ
6 P[|Xτnn −X(Λn)−1(τn)|>η/2]+P[|X(Λn)−1(τn)−Xτ|>η/2].
There exists n suchthat for every n>n , P[sup |Xn −X |>η/2]6ε by choice of (Λn) .
0 0 t Λn(t) t n
In particular, for every n>n ,
0
P[|Xτnn −X(Λn)−1(τn)|>η/2]6ε. (2)
On the other hand, for everyi∈I (recall that I is finite), P[∆X 6=0]=0. Then, there exists
ti
α>0 such that for every i∈I, for every s,
|s−t |6α⇒P[|X −X |>η/2]6ε. (3)
i ti s
P P P
τn −→τ and sup |Λn(t)−t|−→0, so |τ −(Λn)−1(τn)|−→0. Then, there exists n such that for
t 1
every n>n ,
1
P[|τ −(Λn)−1(τn)|>α]6ε. (4)
Then, for every n>n ,
1
P[|X(Λn)−1(τn)−Xτ|>η/2] (5)
= P[|X(Λn)−1(τn)−Xτ|1|τ−(Λn)−1(τn)|>α >η/2]
+P[|X(Λn)−1(τn)−Xτ|1|τ−(Λn)−1(τn)|<α >η/2]
6 P[2sup|Xt|1|τ−(Λn)−1(τn)|>α >η/2]+ε using (3)
t
6 P[|τ −(Λn)−1(τn)|>α]+ε
6 2ε using (4).
So, using (2) and (5), for every n>max(n ,n ),
0 1
P[|Xn −X |>η]63ε.
τn τ
P
Finally, (τn,Xn )−→(τ,X ).
τn τ
Lemma 5 is proved. (cid:3)
WiththisLemma,wecanprovethatTheorem4istrueforstoppingtimesthattakesafinite
number of values.
4
Letusconsiderasubdivisionπ of[0,T]suchthatnofixedtimeofdiscontinuityofX belongs
toπ. WedenotebyTπ thesetofF stoppingtimestakingvaluesinπ andboundedbyL. Then,
L
we define :
Γπ(L)= supE[γ(τ,X )].
τ
τ∈Tπ
L
Lemma 9 Γπ(L)6liminfΓ (L).
n
Proof
Fix ε>0. There exists a FX stopping time τ bounded by L taking values in π such that
E[γ(τ,X )]>Γπ(L)−ε.
τ
According to Lemma 5,there exists a sequence(τn) ofFn stopping times bounded byL such
n
that
P
(τn,Xn )−→(τ,X ).
τn τ
E[γ(τn,Xn )] → E[γ(τ,X )] because γ is bounded continuous. Moreover, by definition, for
τn τ
every n, E[γ(τn,Xn )]6Γ (L). Il follows that
τn n
liminfE[γ(τn,Xn )]6liminfΓ (L).
τn n
But, liminfE[γ(τn,Xn )]=E[γ(τ,X )]>Γπ(L)−ε. So,
τn τ
Γπ(L)−ε6liminfΓ (L),∀ε>0.
n
Then, Γπ(L)6liminfΓ (L). (cid:3)
n
It remains to link the values of optimal stopping for stopping times taking values in finite
subdivisions and Γ(L).
Lemma 10 Let us consider an increasing sequence (πk) of subdivisions without fixed times
k
of continuity of X such that L ∈ πk for every k (it is possible because P[∆X 6= 0] = 0) and
L
|πk|−−−−−→0. Then Γπk(L)−−−−−→Γ(L).
k→+∞ k→+∞
Proof
(Γπk(L)) is anincreasingsequence boundedfromaboveby Γ(L). So (Γπk(L)) convergesto a
k k
limit l with l6Γ(L). Let us show that l=Γ(L).
Fix ε>0.
We can find τ ∈T such that
L
E[γ(τ,X )]>Γ(L)−ε.
τ
We denote πk ={tk,...,tk }. Then, let us consider
1 Kk
Kk−1
τk = tk 1 .
i+1 tk<τ6tk
i i+1
Xi=1
Foreveryk,τk ∈Tπk becauseτ isboundedbyLandL∈πk. Since |πk|→0,wehaveτk −→P τ.
L
P
Moreover,τk >τ and X is right-continuous,so X −→X . γ is bounded continuous, so
τk τ
E[γ(τk,X )]−−−−→E[γ(τ,X )].
τk τ
k→∞
But, for every k, Γπk(L)>E[γ(τk,X )]. It follows that
τk
l>E[γ(τ,X )]>Γ(L)−ε.
τ
This is true for every ε>0, so l>Γ(L).
Then, Γπk(L)−−−−−→Γ(L) and Lemma 10 si proved. (cid:3)
k→+∞
At last, Theorem 4 follows from Lemmas 9 and 10. (cid:3)
5
Remark 11 If P[∆X 6= 0] > 0, the result may not hold any longer. Let us give an example
L
when L=1/2. We consider some processes x and (xn) defined on [0,1] by x =1 (t) and
t [1/2,1]
xn = 1 (t), ∀t. Let us consider γ : [0,+∞[×R → R such that γ(t,y) = y∧2. γ is a
t [1/2+1/n,1]
continuous bounded function. We want to compare Γ(1/2) and the limit of Γ (1/2) when n
n
goes to +∞.
We have : Γ(1/2)= sup E[γ(τ,x )]= sup x =1.
τ t
τ∈T1/2 t61/2
On the other hand, for every n, Γ (1/2)= sup xn =0.
n t
t61/2
So liminfΓ (1/2)=0<1=Γ(1/2).
n
Remark 12 The Theorem remains true if we replace FX by the right continuous filtration
associated to F (∀t,F =FX) and if we take the Γ(L) associated to F.
t t+
4 Aldous’ criterion for tightness
In his papers (Aldous, 1978) and (Aldous, 1989), Aldous deals with the following criterion for
tightness :
∀ε>0,lim limsup sup P[|Xn−Xn|>ε]=0. (6)
S T
δ↓0 n→+∞ S,T∈Tn,S6T6S+δ
L
Hegivesmanyresultswhichlinksthatcriterionandweakconvergenceofsequencesofprocesses.
Inhis unpublishedmanuscript(Aldous, 1981),Aldousshowsthe followingresult(Corollary
16.23) which links convergence of stopping times to convergence of processes :
Proposition 13 Let us consider a sequence of ca`dla`g processes (Xn) that converges in law
n
to a ca`dla`g process X. We denote by Fn the natural filtrations of the processes Xn and by F
the right continuous natural filtration of the process X. Let us consider a sequence (τn) of
n
(Fn)-stopping times that converges in law to a random variable V. We suppose that we have
the join convergence in law of ((τn,Xn)) to (V,X) and that Aldous’ criterion for tightness (6)
n
is filled. Then (τn,Xn )−→L (V,X ).
τn V
Proof
We just give the sketch of Aldous’ proof.
If P[∆X 6= 0] = 0, using the Skorokhod representation Theorem, we can prove that
V
(τn,Xn )−→L (V,X ).
τn V
IfP[∆X 6=0]6=0,we canfinda decreasingsequence(δ ) ofrealsthatconvergesto0 and
V k k
such that for every k, P[∆X 6=0]=0.
V+δk
Let us take f :R2 →R bounded and continuous.
|E[f(τn,Xn )−f(V,X )]|
τn V
6 |E[f(τn,Xn )−f(τn+δ ,Xn )]|
τn k τn+δk
+|E[f(τn+δ ,Xn )−f(V +δ ,X )]|
k τn+δk k V+δk
+|E[f(V +δ ,X )−f(V,X )]|.
k V+δk V
But :
- ∀k, limsup E[f(τn+δ ,Xn )−f(V +δ ,X )]=0 because P[∆X 6=0]=0,
- lim En[f→(+V∞+δ ,X k)−τfn(+Vδ,kX )]=0 bekcausVe+Xδk is right-continuous, V+δk
k→+∞ k V+δk V
- limk→+∞limsupn→+∞E[f(τ∗,n,Xτn∗,n)−f(τ∗,n+δk,Xτn∗,n+δk)]=0 using Aldous’ criterion.
The result follows. (cid:3)
Remark 14 We will see in Proposition 19 an analogous result in the case of randomized
stopping times.
6
The following caracterization of Aldous’ Criterion is probably widely known, however I do
not know of any reference to a proof of it, so I give one of my own here.
Proposition 15 Let us consider a sequence of ca`dla`g processes (Xn) and a ca`dla`g process X
n
P
such that Xn −→X. The following conditions are equivalent :
i) X is continuous in probability everywhere, ie for every t P[∆X 6=0]=0,
t
ii) Aldous’ criterion for tightness (6) is filled.
Proof
i) ⇒ ii). Let δ > 0. Let (Tn) and (S ) be two sequences of Tn such that for every n,
n n n L
Sn 6Tn 6Sn+δ. Let ε>0 and η >0.
P P
Xn −→X sowecanfindasequenceofrandomtimechanges(Λn) suchthatsup |Xn −X |−→0.
n t Λn(t) t
Then there exists n such that
0
∀n>n ,P[sup|Xn −X |>η/3]6ε.
0 Λn(t) t
t
We have :
P[|Xn −Xn |>η]
Sn Tn
6 P[|XSnn −X(Λn)−1(Sn)|>η/3]
+P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|>η/3]
+P[|X(Λn)−1(Tn)−XTnn|>η/3] (7)
But, for every n>n ,
0
P[|XSnn −X(Λn)−1(Sn)|>η/3]6P[sup|XΛnn(t)−Xt|>η/3]6ε, (8)
t
and
P[|X(Λn)−1(Tn)−XTnn|>η/3]6P[sup|XΛnn(t)−Xt|>η/3]6ε. (9)
t
It remains to show that :
limlimsupP[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|>η/3]=0.
δ↓0 n→+∞
X is a ca`dla`g process, so there exists θ >0 such that
P[w′(X,θ)>η/12]6ε,
where ∀x ∈ D,w′(x,δ) = inf max w(x,[t ,t [), F is the set of subdivisions
{ti}∈Fδ 16i6v i−1 i δ
{t } of[0,T]suchthat∀i,t −t >δ andwisthemodulusofcontinuityw(x,[t ,t [)=
i 16i6v i i−1 i−1 i
sup{|x −x |,t <s<t<t } (see e.g. Billingsley, 1999, Section 12).
t s i−1 i
By defintion of w′, there exists a subdivision {t } such that
k
∀k,|t −t |>θ and P[maxw(X,[t ,t [)>η/12]62ε.
k+1 k k k+1
k
On the other hand,
P[|(Λn)−1(Tn)−(Λn)−1(Tn+δ)|>θ]
6 P[|(Λn)−1(Tn)−Tn|>θ/3]+P[|Tn−(Tn+δ)|>θ/3]
+P[|Tn+δ−(Λn)−1(Tn+δ)|>θ/3]
6 2P[sup|(Λn)−1(t)−t|>θ/3] for every δ <θ/3.
t
P
sup |(Λn)−1(t)−t|−→0 , so there exists n such that
t 1
∀n>n ,P[sup|(Λn)−1(t)−t|>θ/3]6ε.
1
t
7
Then, for every n>n , for every δ <θ/3,
1
P[|(Λn)−1(Tn)−(Λn)−1(Sn)|>θ]63ε. (10)
So, for every n>n , for every δ <θ/3,
1
P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|>η/3]
= P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|1|(Λn)−1(Tn)−(Λn)−1(Sn)|<θ >η/3]
+P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|1|(Λn)−1(Tn)−(Λn)−1(Sn)|>θ >η/3].
But,
P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|1|(Λn)−1(Tn)−(Λn)−1(Sn)|<θ >η/3]
6 P[(2maxw(X,[t ,t [)+max|∆X |)>η/3]
k k k+1 k tk
6 P[maxw(X,[t ,t [)>η/12]+P[max|∆X |>η/6]
k k k+1 k tk
6 2ε+ P[|∆X |>η/6]
tk
Xk
6 2ε because X has no fixed time of discontinuity
and
P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|1|(Λn)−1(Tn)−(Λn)−1(Sn)|>θ >η/3]
6 P[2sup|Xt|1|(Λn)−1(Tn)−(Λn)−1(Sn)|>θ >η/3]
t
6 P[|(Λn)−1(Tn)−(Λn)−1(Sn)|>θ]
6 3ε using (10).
So for every n>n , for every δ <θ/3,
1
P[|X(Λn)−1(Sn)−X(Λn)−1(Tn)|>η/3]65ε. (11)
Finally, using inequalities (7), (8), (9) and (11), for every n>max(n ,n ), for every δ <θ/3,
0 1
P[|Xn −Xn |>η]67ε.
Sn Tn
n , n and θ do not depend on (T ) and (S ) . Then, for every n > max(n ,n ), for every
0 1 n n n n 0 1
δ <θ/3,
sup P[|Xn−Xn|>η]67ε.
S T
S,T∈Tn,S6T6S+δ
L
Aldous’ criterion follows.
ii)⇒i). We suppose that there exists t such that P[∆X 6=0]>0.
0 t0
Let ε>0 and η >0 be such that P[|∆X |>2ε]>2η.
t0
P P P
Xn −→X sowecanfindarandomsequence(tn) suchthattn −→t and∆Xn −→∆X .There
n 0 tn t0
exists n such that for every n>n ,
0 0
P[|tn−t |>δ/2]6η/2 and P[|∆Xn −∆X |>ε]6η/2. (12)
0 tn t0
We are going to show that for every n>n , for δ large enough,
0
P[|Xn −Xn |>ε/3]>η/2.
t0+δ/2 t0−δ/2
Then, for every n>n ,
0
P[|∆Xn|>ε]
tn
= P[|∆Xn|1 >ε]+P[|∆Xn|1 >ε] (13)
tn |tn−t0|>δ/2 tn |tn−t0|<δ/2
6 P[|tn−t |>δ/2]+P[|Xn −Xn |1 >ε/3]
0 tn t0+δ/2 |tn−t0|<δ/2
+P[|Xn −Xn |>ε/3]+P[|Xn −Xn |1 >ε/3]
t0+δ/2 t0−δ/2 t0−δ/2 tn− |tn−t0|<δ/2
8
(Xn) is tight. So, we can find δ >0 and n ∈N such that for everyδ 6δ , for every n>n ,
n 0 1 0 1
P[w′(Xn,δ)>ε/6]6η/6.
Then, we can find a finite subdivision {t } such that
k
∀k,t −t >δ and P[maxw(Xn,[t ,t [)>ε/3]6η/4.
k+1 k k k+1
k
We know that for every n>max(n ,n ),
0 1
2η 6 P[|∆X |>2ε]
t0
6 P[|∆Xn −∆X |>ε]+P[|∆Xn|>ε]
tn t0 tn
6 η/2+P[|∆Xn|>ε].
tn
In particular, for every n>max(n ,n ),
0 1
P[|∆Xn|>ε]>3η/2. (14)
tn
So, for every n>max(n ,n ), tn ∈{t }.
0 1 k
Then, for every δ 6δ , for every n>max(n ,n ),
0 0 1
P[|Xn −Xn |1 >ε/3]6P[maxw(Xn,[t ,t [)>ε/3]6η/4. (15)
tn t0+δ/2 |tn−t0|<δ/2 k k k+1
On the same way,
P[|Xn −Xn |1 >ε/3]6η/4. (16)
t0−δ/2 tn− |tn−t0|<δ/2
Finally, using (13) and inequalities (12), (14), (15) and (16), for every δ 6 δ , for every n >
0
max(n ,n ),
0 1
3η/26P[|∆Xn|>ε]6η/2+η/4+P[|Xn −Xn |>ε/3]+η/4.
tn t0+δ/2 t0−δ/2
So, for every δ 6δ , for every n>max(n ,n ),
0 0 1
η/2 6 P[|Xn −Xn |>ε/3]
t0+δ/2 t0−δ/2
6 sup P[|Xn −Xn |>ε/3].
Tn+δ Tn
S,T∈Tn,S6T6S+δ
L
Taking the limsup when n tends to infinity and the limit when δ decreases to 0, we have :
η/26lim limsup sup P[|Xn −Xn |>ε/3],
Tn+δ Tn
δ↓0 n→+∞ S,T∈Tn,S6T6S+δ
L
which is in contradiction with Aldous’ criterion. The result follows. (cid:3)
5 Proof of the inequality Γ(L) > limsupΓ (L) when for ev-
n
ery n, Fn ⊂ F
5.1 Randomized stopping times
The notion of randomized stopping times has been introduced in (Baxter and Chacon, 1977)
and this notion has been used in (Meyer, 1978) under the french name ”temps d’arrˆetflous”.
We are given a filtration F. Let us denote by B the Borel σ-field on [0,1]. Then, we define
thefiltrationG onΩ×[0,1]suchthat∀t,G =F ×B. Amapτ :Ω×[0,1]→[0,+∞]iscalleda
t t
randomizedF stopping time if τ is a G stopping time. We denote by T∗ the set ofrandomized
stopping times and by T∗ the set of randomized stopping times bounded by L. T is included
L
in T∗ and the application τ 7→τ∗, where τ∗(ω,t)=τ(ω) for every ω and every t, maps T into
9