Reflection principle and Ocone martingales.

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Reflection principle and Ocone martingales.

Loïc Chaumont, L. Vostrikova

To cite this version:

Loïc Chaumont, L. Vostrikova. Reflection principle and Ocone martingales.. Stochastic Processes

and their Applications, Elsevier, 2009, 119 (10), pp.3816-3833. �10.1016/j.spa.2009.07.009�. �hal-

00305333v2�

www.elsevier.com/locate/spa

Reflection principle and Ocone martingales

L. Chaumont

, L. Vostrikova

LAREMA, D´epartement de Math´ematiques, Universit´e d’Angers, 2, Bd Lavoisier - 49045, Angers Cedex 01, France Received 12 February 2009; received in revised form 9 July 2009; accepted 16 July 2009

Available online 5 August 2009

Abstract

LetM = (Mt)t≥0be any continuous real-valued stochastic process. We prove that if there exists a sequence(an)n≥1of real numbers which converges to 0 and such thatMsatisfies the reflection property at all levelsanand 2anwithn≥1, thenMis an Ocone local martingale with respect to its natural filtration.

We state the subsequent open question: is this result still true when the property only holds at levelsa_n? We prove that this question is equivalent to the fact that for Brownian motion, theσ-field of the invariant events by all reflections at levelsa_n,n≥1 is trivial. We establish similar results for skip freeZ-valued processes and use them for the proof in continuous time, via a discretization in space.

c

2009 Elsevier B.V. All rights reserved.

MSC:60G44; 60G42; 60J65

Keywords:Ocone martingale; Skip free process; Reflection principle; Quadratic variation; Dambis–Dubins–Schwarz Brownian motion

1. Introduction and main results

Local martingales whose law is invariant under any integral transformations preserving their quadratic variation were first introduced and characterized by Ocone [7]. The motivation for introducing Ocone martingales comes from control theory where often optimal control function is the sign of some observed process. In the Brownian case, this fact is based on the invariance properties of Brownian motion. When Brownian motion is replaced by a martingale, the construction of optimal control can be similar ifMis invariant under integration with respect to processes taking values±1. Namely a continuous real-valued local martingaleM =(M_t)t≥0

∗Corresponding author. Tel.: +33 2 41 73 50 28; fax: +33 2 41 73 54 54.

E-mail addresses:loic.chaumont@univ-angers.fr(L. Chaumont),lioudmila.vostrikova@univ-angers.fr (L. Vostrikova).

0304-4149/$ - see front matter c2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.spa.2009.07.009

with natural filtrationF=(F_t)t≥0is calledOconeif Z t

0

H_sdM_s

t≥0

=L M, (1.1)

for all processesHbelonging to the set

H= {H =(H_t)t≥0|HisF-predictable,|H_t| =1, for allt ≥0}.

In the primary paper [7], the author proved that a local martingale is Ocone whenever it satisfies(1.1)for all processesH belonging to the smaller class of deterministic processes:

H₁= { 1_[0,u](t)−1_]u,+∞[(t)

t≥0, withu≥0}. (1.2)

A natural question for which we sketch out an answer in this paper is to describe minimal sub- classes of Hcharacterizing Ocone local martingales through relation(1.1). For instance, it is readily seen that the subset{(1_[0,u](t)−1_]u,+∞[(t))t≥0, withu∈ E}ofH1characterizes Ocone martingales if and only if E is dense in[0,∞). Let us denote byhMithe quadratic variation of M. In [7] it was shown that for continuous local martingales,(1.1)is equivalent to the fact that conditionally to theσ-algebraσ{hMi_s,s ≥ 0}, M is a gaussian process with independent increments. Hencea continuous Ocone local martingale is a Brownian motion time changed by any independent non-decreasing continuous process.This is actually the definition we will refer to all along this paper.

When the continuous local martingaleM is divergent, i.e.P-a.s.

t→∞limhMi_t = +∞,

we denote byτ the right-continuous inverse ofhMi, i.e. fort≥0, τt =inf{s≥0: hMi_s >t},

and we recall that the Dambis–Dubins–Schwarz Brownian motion associated toM is the(F_τt)- Brownian motion defined by

B^{M (def)}= (M_τt)t≥0.

Then Dubins, Emery and Yor [2] refined Ocone’s characterization by proving that (1.1) is equivalent to each of the following three properties:

(i) The processeshMiandB^M are independent.

(ii) For every F-predictable process H, measurable for the productσ-fieldB(R+)⊗σ (hMi) and such thatR∞

0 H_s²dhMi_s <∞,P-a.s., E

exp

i

Z ∞

0

HsdMs

| hMi

=exp

−1 2

Z ∞

0

H_s²dhMi_s

. (iii) For every deterministic functionhof the formPn

j=1λj1_[0,aj], E

exp

i

Z ∞

0

h(s)dM_s

=E

exp

−1 2

Z ∞

0

h²(s)dhMi_s

.

It can easily be shown that the equivalence between(1.1)and (i), (ii), (iii) also holds in the case when Mis not necessarily divergent. This fact will be used in the proof ofTheorem 1. We also refer to [8] for further results related to Girsanov theorem and different classes of martingales.

In [2], the authors conjectured that the classH1can be reduced to a single process, namely that(1.1)is equivalent to:

Z _t

0

sign(M_s)dM_s

t≥0

=L M. (1.3)

In fact,(1.3)holds if and only ifB^M andhMiare conditionally independent given theσ-field of invariant sets by the L´evy transform of B^M, i.e. B^M 7→ (R·

0sign(B_s^M)dB_s^M), see [2]. Hence, if the L´evy transform of Brownian motion is ergodic, thenB^M andhMiare independent and(1.3) implies thatM is an Ocone local martingale. The converse is also proved in [2], that is if(1.3) implies thatM is an Ocone local martingale, then the L´evy transform of Brownian motionB^M is ergodic.

Different approaches have been proposed to prove ergodicity of the L´evy transform but this problem is still open. Among the most accomplished works in this direction, we may cite papers by Malric [5,6] who recently proved that a.s. the orbits of the L´evy transform of Brownian motion are dense in the set of continuous functions. Let us also mention that in discrete time case this problem has been treated in [3] where the authors proved that an equivalent of the L´evy transform for symmetric Bernoulli random walk is ergodic.

In this paper we exhibit a new sub-class of H characterizing continuous Ocone local martingales which is related to first passage times and the reflection property of stochastic processes. IfMis the standard Brownian motion andTa(M)the first passage time at levela, i.e.

Ta(M)=inf{t≥0:Mt =a}, (1.4)

where here and in all the remainder of this article, we make the convention that inf{∅} = +∞, then for alla∈R:

(M_t)t≥0

=L M_t1_{t≤Ta(M)}+(2a−M_t)1_{t>Ta(M)}

t≥0.

It is readily checked that this identity in law actually holds for any continuous Ocone local mar- tingale. This property is known as thereflection principle at level aand was first observed for symmetric Bernoulli random walks by Andr´e [1]. We will use this terminology for any continu- ous stochastic processMand when no confusion is possible, we will denote byT_a=T_a(M)the first passage time at levelabyM defined as above.

Let(Ω,F,F,P)be the canonical space of continuous functions endowed with its natural right-continuous filtrationF = (F_t)t≥0 completed by negligible sets ofF = W

t≥0F_t. The family of transformationsΘ^a,a ≥0, is defined for all continuous functionsω∈Ωby

Θ^a(ω)=(ωt1_{t≤Ta}+(2a−ωt)1_{t>Ta})t≥0. (1.5) Note thatΘ^a(ω)=ωon the set{ω:T_a(ω)= ∞}. WhenM is a local martingale,Θ^a(M)can by expressed in terms of a stochastic integral, i.e.

Θ^a(M)= Z t

0

1_[0,Ta](s)−1_]Ta,+∞[(s) dMs

t≥0

.

The setH2= {(1_[0,Ta](t)−1_]Ta,+∞[(t))t≥0|a≥0}is a sub-class ofHwhich provides a family of transformations preserving the quadratic variation ofMand we will prove that it characterizes Ocone local martingales. Moreover, the fact that the transformationsω7→Θ^a(ω)are defined for all continuous functionsω∈ Ωallows us to characterize Ocone local martingales in the whole set of continuous stochastic processes as shows our main result.

Theorem 1. Let M = (M_t)t≥0 be a continuous stochastic process defined on the canonical probability space, such that M0=0. If there exists a sequence(an)n≥1of positive real numbers such thatlimn→∞an=0and for all n≥0:

Θ^an(M)=^L Θ^2an(M)=^L M, (1.6)

then M is an Ocone local martingale with respect to its natural filtration. Moreover, if T_a1 <∞ a.s., then M is a divergent local martingale.

Remark 1. It is natural to wonder about the necessity of the hypothesis Θ^2an(M) =^L M in Theorem 1. The discrete time counterpart of this problem which is presented in Section2, shows that it is necessary for a skip free process M to satisfyΘ^a(M) =^L M, fora = 0,1 and 2 in order to be a skip free Ocone local martingale, i.e. the reflection property ata = 0 and 1 is not sufficient, see the counterexamples in Section2.2. This argument seems to confirm that the assumptionΘ^2an(M)=^L M is necessary in continuous time.

In an attempt to identify the sequences(a_n)n≥1which characterize Ocone local martingales, we will prove the following theorem. Leta =(a_n)n≥1be a sequence of positive numbers with lim_n→∞a_n=0 and letI^abe the sub-σ-field of the invariant sets by all the transformationsΘ^an, i.e.

I^a= {F∈F:1_F◦Θ^{an a.s.}= 1_F, for alln≥0}. Theorem 2. The following assertions are equivalent:

(i) Any continuous local martingale M satisfyingΘ^an(M)=^L M for all n≥0is an Ocone local martingale.

(ii) The sub-σ-fieldI^a is trivial for the Wiener measure on the canonical space(Ω,F), i.e. it contains only the sets of measure 0 and 1.

Remark 2. It follows fromTheorems 1and2that if the sequence(a_n)contains a subsequence (2a_n⁰)(this holds, for instance, when (a_n)is the dyadic sequence), then the sub-σ-fieldI^a is trivial for the Wiener measure on(Ω,F). So, our open question is equivalent to: is the sub-σ- fieldI^atrivial for any sequence(an)decreasing to zero?

In the next section, we prove analogous results for skip free processes. We use them as preliminary results to proveTheorem 1in Section3. In Section2.2, we give counterexamples in the discrete time setting, related toTheorem 3. Finally, in Section4, we proveTheorem 2.

2. Reflecting property and skip free processes

2.1. Discrete time skip free processes

A discrete time skip free process M is any measurable stochastic process with M0 =0 and for alln ≥ 1, 1Mn = Mn −Mn−1 ∈ {−1,0,1}. This section is devoted to an analogue of Theorem 1for skip free processes.

To each skip free processM, we associate the increasing process [M]_n=

n−1

X

k=0

(M_k+1−M_k)², n ≥1, [M]₀=0,

which is called thequadratic variation of M. In this section, since no confusion is possible, we will use the same notations for discrete processes as in the continuous time case. For every integera≥0, we denote byTathe first passage time byMto the levela,

Ta=inf{k≥0: Mk =a}.

We also introduce the inverse processτ which is defined byτ0=0 and forn ≥1, τn=inf{k> τn−1: [M]_k=n}.

Then we may define

S^M =(M_τn)n≥0, (2.7)

withM∞ = limn→∞Mn, when this limit exists (note that limn→∞[M]_n = limn→∞τn = ∞, when limn→∞Mndoes not exist). Denote also

T =inf{k≥0: [S^M]_k = [S^M]_∞},

and note that the paths ofS^M are such that1S_k^M ∈ {−1,+1}, for allk≤ T and1S_k^M =0, for allk>T, where1S_k^M =S_k−S_k−1^M .

We recall that skip free martingales are just skip free processes being martingales with respect to some filtration. It is well known that for any divergent skip free martingaleM, that is satisfying lim_n→+∞[M]_n = +∞, a.s., the process S^M is a symmetric Bernoulli random walk onZ. This property is the equivalent of the Dambis–Dubins–Schwartz theorem for continuous martingales.

In discrete time, the proof is quite straightforward and we recall it now.

A first step is the equivalent of L´evy’s characterization for skip free martingales: any skip free martingale S such that S_n+1−S_n 6= 0, for all n ≥ 0 (or equivalently, whose quadratic variation satisfies[S]_n = n) is a symmetric Bernoulli random walk. Indeed forn ≥ 1, S₁, S2−S1, . . . ,S_n−Sn−1are i.i.d. symmetric Bernoulli r.v.’s if and only if for any subsequence 1≤n1≤ · · · ≤n_k ≤n:

E[(Sn₁ −Sn₁−1)(Sn₂−Sn₂−1)· · ·(Sn_k −Sn_k−1)]

=E[S_n1 −S_n1−1]E[S_n2−S_n2−1] · · ·E[S_nk−S_nk−1] =0

and this identity can easily be checked from the martingale property. Finally callF=(Fn)n≥0the natural filtration generated byM. Since[M]_nis anF-adapted process, from the optional stopping theorem,S^Mis a martingale with respect to the filtration(F_τn)n≥0and since its increments cannot be 0, we conclude from L´evy’s characterization.

We also recall the following important property: any skip free process which is a symmetric Bernoulli random walk time changed by an independent non-decreasing skip free process, is a local martingale with respect to its natural filtration.

This leads to the definition:

Definition 1. A discrete Ocone local martingale is a symmetric Bernoulli random walk time changed by any independent non-decreasing skip free process.

We emphasize that in this particular case,Definition 1coincides with the general definition of Ocone [7]. It should also be noticed that the symmetric Bernoulli random walk ofDefinition 1is not necessarily the same as in(2.7). It coincides withS^M ifMis a divergent process. IfMis not divergent, then it can be obtained from S^M by pasting of an independent symmetric Bernoulli random walk (seeLemma 3), otherwise the independence fails.

A counterpart of transformations Θ^a defined in (1.5) for skip free processes is given for all integersa≥0 by

Θ^a(M)n=

n

X

k=1

(1_{k≤Ta}−1_{k>Ta})1M_k, (2.8)

where1M_k =M_k−M_k−1. Again in the following discrete time counterpart ofTheorem 1, we characterize discrete Ocone local martingales in the whole set of skip free processes.

Theorem 3. Let M be any discrete skip free process. Assume that for all a∈ {0,1,2},

Θ^a(M)=^L M, (2.9)

then M is a discrete Ocone local martingale with respect to its natural filtration. If in addition T₁<∞a.s. then M is a divergent local martingale.

The proof ofTheorem 3is based on the following crucial combinatorial lemma concerning the set of sequences of partial sums of elements in{−1,+1}with lengthm≥1:

Λ^m = {(s₀,s₁, . . . ,s_m):s₀=0 and1s_k ∈ {−1,+1}for 1≤k≤m}, where1sk =sk−sk−1.

For each sequences ∈Λ^m, and each integera, we defineTa(s)=inf{k≥ 0:sk =a}. The transformationΘ^a(s)is defined for eachs∈Λ^mby

Θ^a(s)n=

n

X

k=1

(1_{k≤Ta(s)}−1_{k>Ta(s)})1s_k, n≤m.

Lemma 1. Let m ≥ 1 be fixed. For any two elements s and s⁰ of the setΛ^m such that s 6= s⁰, there are integers a₁,a₂, . . . ,a_k ∈ {0,1,2}depending on s and s⁰such that

s⁰=Θ^akΘ^ak−1· · ·Θ^a1(s). (2.10) Moreover, the integers a₁, . . . ,a_k can be chosen so that s ∈ Λ^m_a

1 andΘ^ai−1Θ^ai−2· · ·Θ^a1(s)∈ Λ^m_a

i, for all i =2, . . . ,k where Λ^m_a = {s∈Λ^m,Ta(s)≤m−1}.

Proof. The last property follows from the simple remark that fors∈Λ^mwe have thatΘ^a(s)6=s if and only ifs∈Λ^m_a. So, we suppose that all the transformations involved in the rest of the proof verify the above property.

Lets¯^(m)be the sequence ofΛ_mdefined bys¯⁽₀^m)=0,s¯₁^(m) =1 and1s¯_k^(m) = −1s¯_k−1^(m) for all 2≤k≤m. That iss¯^{(m) (def)}= (0,1,0,1, . . . ,0,1)ifmis odd ands¯^{(m) (def)}= (0,1,0,1, . . . ,1,0) ifmis even.

First we prove that the statement of the lemma is equivalent to the following one: for any sequencesofΛ_msuch thats6= ¯s^(m), there are integersb1,b2, . . . ,b_p∈ {0,1,2}such that

s¯^(m)=Θ^bpΘ^bp−1· · ·Θ^b1(s). (2.11)

Indeed, suppose that the latter property holds and lets⁰ ∈ Λ_m such thats⁰ 6= s. Ifs⁰ = ¯s^(m), then the sequence b1,b2, . . . ,b_p satisfies the statement of the lemma. If s⁰ 6= ¯s^(m), then let c1, . . . ,cl ∈ {0,1,2}such that

s¯^(m) =Θ^clΘ^cl−1· · ·Θ^c1(s⁰).

We notice that the transformationsΘ^aare involutive, i.e. for allx∈Λ_m,

Θ^aΘ^a(x)=x. (2.12)

Then we haveΘ^c1Θ^c2· · ·Θ^cl(s¯^m)=s⁰, so that s⁰=Θ^c1Θ^c2· · ·Θ^clΘ^bpΘ^bp−1· · ·Θ^b1(s),

which implies(2.10). The fact that(2.10)implies(2.11)is obvious.

Now we prove(2.11) by induction inm. It is not difficult to see that the result is true for m=1,2 and 3. Suppose that the result is true up tomand lets ∈Λ^m+1such thats 6= ¯s^(m+1). Forj ≤m, we calls^(j)the truncated sequences^(j) =(s₀,s₁, . . . ,s_j)∈Λ^j. From the hypothesis of induction, there existb₁,b₂, . . . ,b_p∈ {0,1,2}such that

s¯^(m) =Θ^bpΘ^bp−1· · ·Θ^b1(s^(m)), (2.13) where

s^(m) ∈Λ^m_b

1 and Θ^bi−1Θ^bi−2· · ·Θ^b1 s^(m)

∈Λ^m_b

i, for alli =2, . . . ,p. (2.14) Then, let us consider separately the case wheremis even and the case wheremis odd.

Ifmis even and1s_m1s_m+1= −1, then we obtain directly that Θ^bpΘ^bp−1· · ·Θ^b1(s)= ¯s^(m+1).

Indeed, from(2.14), none of the transformationsΘ^bi−1· · ·Θ^b1,i=2, . . . ,paffects the last step ofs, so the identity follows from(2.13).

Ifm is even and1s_m1s_m+1 = 1, then from the hypothesis of induction there existd₁,d₂, . . . ,d_r ∈ {0,1,2}such that

Θ^dr· · ·Θ^d1 s^(m)

=

s¯^(m−1),2

(2.15) which, from the above remark, may be chosen so that

s^(m) ∈Λ^m_d

1 and Θ^di−1Θ^di−2· · ·Θ^d1 s^(m)

∈Λ^m_d

i, for alli =2, . . . ,r. (2.16) Since from(2.16), none of the transformationsΘ^di · · ·Θ^d1,i =1, . . . ,raffects the last step of s, it follows from(2.15)that

Θ^dr· · ·Θ^d1(s)=

s¯^(m−1),2,3

. (2.17)

Then by applying transformationΘ², we obtain:

Θ²

s¯^(m−1),2,3

=

s¯^(m−1),2,1

. (2.18)

Hence, from(2.15) and since none of the transformationsΘ^dr−i· · ·Θ^dr,i = 0,1, . . . ,r −1 affects the last step of(s¯^(m−1),2,1), we have

Θ^d1Θ^d2· · ·Θ^dr

s¯^(m−1),2,1

=

s^(m),sm−1sm+1

.

Finally from(2.13)and(2.14), we have

Θ^bpΘ^bp−1· · ·Θ^b1Θ^d1· · ·Θ^drΘ²Θ^dr· · ·Θ^d1(s)=s^(m+1) and the induction hypothesis is true at the orderm+1, whenmis even.

The proof whenmis odd is very similar and we will pass over some of the arguments in this case. Ifmis odd and1sm1sm+1= −1, then we obtain directly that

Θ^bpΘ^bp−1· · ·Θ^b1(s)= ¯s^(m+1).

Ifmis odd and1sm1sm+1 = 1 then from the hypothesis of induction, there existd1,d2, . . . , dr ∈ {0,1,2}such that

Θ^dr · · ·Θ^d1 s^(m)

=

s¯^(m−1),−1

(2.19) and

s^(m)∈Λ^m_d

1 and Θ^di−1Θ^di−2· · ·Θ^d1 s^(m)

∈Λ^m_d

i, for alli=2, . . . ,r. (2.20) Then it follows from(2.19)and(2.20)that

Θ^dr · · ·Θ^d1(s)=

s¯^(m−1),−1,−2

(2.21) and by performing the transformationΘ⁰Θ¹Θ⁰=Θ⁻¹,

Θ⁰Θ¹Θ⁰

s¯^(m−1),−1,−2

=

s¯^(m−1),−1,0

. (2.22)

From(2.19)and(2.20), it follows that Θ^d1· · ·Θ^dr

s¯^(m−1),−1,0

=

s^(m),s_m−1sm+1

,

which finally gives from(2.13)and(2.14),

Θ^bpΘ^bp−1· · ·Θ^b1Θ^d1· · ·Θ^drΘ⁰Θ¹Θ⁰Θ^dr · · ·Θ^d1(s)= ¯s^(m+1) and ends the proof of the lemma.

In the proof ofTheorem 3, for technical reasons we have to consider two cases:T1<∞a.s.

andP(T1= ∞) >0.Lemma 2proves that in the first caseMis a divergent process.

Lemma 2. Any skip free process such that T₁ < ∞ a.s. and Θ^a(M) =^L M for a = 0 and 1satisfies:

n→+∞lim [M]_n= +∞, a.s.

Proof. Let us introduce the first exit time from the interval[−a,a]: σa(M)=inf{n: |M_n| =a},

whereais any integer. Let us put Ψ^a(M)=

n

X

k=1

(1_{k≤σa}−1_{k>σa})1M_k

!

n≥0

,

where1M_k = M_k − M_k−1. First we observe that if Θ^a(M) =^L M for a = 0 and 1, then Ψ^a(M)=^L M, fora =0 and 1. This assertion is obvious fora=0 sinceσ0=T₀. Fora =1, it follows from the almost sure identity:

Ψ^a(M)=Θ^a(M)1_{Ta<T−a}+Θ^−a(M)1_{T−a<Ta}

and the equalities:

{T_a(M) <T_−a(M)} = {T_a(Θ^a(M)) <T_−a(Θ^a(M))}, {T_−a(M) <T_a(M)} = {T_−a(Θ^−a(M)) <T_a(Θ^−a(M))}. Then from the almost sure inequality

σ3(Ψ¹(M))≤max{T₁(M),T₋₁(M)},

the fact thatT₁(M) <∞,T₋₁(M) <∞a.s. and the identity in lawΨ¹(M)=^L M, we deduce thatσ3(M) <+∞, a.s. Generalizing the above inequality, we obtain

σa+2(Ψ¹(M))≤max{Ta(M),T−a(M)}.

This gives in the same manner as before, that for eacha ≥ 0,σa <∞a.s. From this it is not difficult to see that lim_n→∞[M]_n= +∞,P-a.s.

The next lemma shows that in the caseP(T₁ = ∞) > 0 we can modify our process M by pasting to it an independent symmetric Bernoulli random walk S and reduce the caseP(T₁ =

∞) >0 to the caseT₁<∞a.s.

We denote by[M]_∞ = limk→∞[M]_k which always exists since it is an increasing process and we put

T =inf{k≥0: [M]_k = [M]_∞},

with inf{∅} = +∞. We denote the extension of the process M byX. It is defined for allk ≥0 by

X_k =M_k1_{k<T}+(M_T +S_k−T)1_{k≥T}. Note thatX =M, on the set{T = ∞}.

Lemma 3. Let M be a discrete skip free process which satisfiesΘ^a(M)=^L M for some a ∈ Z. Then X also satisfies Θ^a(X) =^L X . Moreover, the σ-algebras generated by the respective quadratic variations coincide, i.e. σ ([M]) = σ ([X]), X is a divergent process P-a.s. and M =S_[M]^X .

Proof. We show that reflection property holds forX. In this aim, we consider the two processes Y andZ such that for allk≥0,

Yk =Θ^a(M)k1_{k<T}+(Θ^a(M)T −Sk−T)1_{k≥T}, Zk =Mk1_{k<T}+(MT +Θ^a−MT(S)k−T)1_{k≥T}. We remark that

Θ^a(X)=Y1_{Ta(Y)≤T}+Z1_{Ta(Z)>T} (2.23)

and we write the same kind of decomposition forX:

X =X1_{Ta(X)≤T}+X1_{Ta(X)>T}. (2.24)

In view of(2.23)and(2.24), to obtain X =^L Θ^a(X)it is sufficient to show that for all bounded and measurable functionalF,

E[F(X)] =E[F(Y)1_{Ta(Y)≤T}] +E[F(Z)1_{Ta(Z)>T}].

Since reflection is a transformation which preserves the quadratic variation of the process, the random timeT can be defined as a functional ofY as well as a functional of Z. So we see that the last equality is equivalent to X =^L Y andX =^L Z. The first equality in law follows from the fact that

(M,S)=^L(Θ^a(M),−S)

which holds due to the reflection property ofM andS, and the independency of M andS. The second one holds since it can be reduced to the reflection property of Sitself, by conditioning with respect toM.

Finally, the identityM =S_[M^X _]just follows from the construction ofX.

Proof of Theorem 3. The quadratic variations ofM andΘ^a(M)are measurable functionals of MandΘ^a(M), which together with(2.9)gives for alla=0,1,2:

(M,[M])=^L (Θ^a(M),[Θ^a(M)]).

Since both processesMandΘ^a(M)have the same quadratic variation, the identity in law of the statement is equivalent to: for alla =0,1,2

(M,[M])=^L (Θ^a(M),[M]).

Then we remark that the above equalities are equivalent to: for alla =0,1,2 (S^M,[M])=^L(S^Θa(M),[M]).

Now it is crucial to observe the path by path equality: for eacha=0,1,2 S^Θa(M)=Θ^a(S^M),

from which we obtain

(S^M,[M])=^L(Θ^a(S^M),[M]). (2.25)

From the last identity we conclude that the conditional laws ofS^M andΘ^a(S^M)with respect to theσ-algebra generated by[M]are equal:

L(S^M|[M])=L(Θ^a(S^M)|[M]). (2.26)

Fix m ≥ 1 and let s,s⁰ ∈ Λ^m with s 6= s⁰ be fixed. Consider the sequence of integersa1, a2, . . . ,ak ∈ {0,1,2}given inLemma 1such that

s=Θ^akΘ^ak−1· · ·Θ^a1(s⁰). (2.27) Denote by S^M,m the restricted path (S0,S1, . . . ,Sm). Iterating(2.26), we may write for allu

∈Λ^m: P

S^M,m =u| [M]

=P

Θ^a1Θ^a2· · ·Θ^ak(S^M,m)=u| [M] .

Applying(2.12), we see that the right-hand side is equal to P

S^M,m =Θ^akΘ^ak−1· · ·Θ^a1(u)| [M] . Take nowu=s⁰and use(2.27), to obtain

P

S^M,m =s⁰| [M]

=P

S^M,m =s| [M]

. (2.28)

Consider now two cases:T1<∞a.s. andP(T1= ∞) >0. IfT1<∞a.s. then fromLemma 2 we can see thatM is divergent and for allm≥0

P(S^M,m ∈Λ^m)=1.

Then from(2.28)and theP-a.s. identity X

s∈Λ^m

P

S^M,m =s| [M]

=1, we have,

P

S^M,m =s| [M]

= 1

card(Λ^m).

It follows that the law ofS^M,m is uniform overΛ^m and that it coincides with the conditional law ofS^M,m given[M]. Hence,S^M,mis symmetric Bernoulli random walk on[0,m]independent of [M]. Since this holds for allm≥0, we conclude thatS^M is a symmetric Bernoulli random walk which is independent of[M]. So, fromDefinition 1,M is a divergent Ocone local martingale.

IfP(T₁= ∞) >0, we consider the extensionXof the processMdefined inLemma 3. From this lemma, X satisfies the hypotheses ofTheorem 3andP(T1(X) <∞)= 1. From what has just been provedS^X is a symmetric Bernoulli random walk which is independent of[X]. Since theσ-algebra generated by[X]is the same as theσ-algebra generated by[M],S^X and[M]are independent. FromLemma 3we have M =S_[M]^X , and, hence, the processM is itself an Ocone martingale according toDefinition 1.

2.2. Counterexamples

In this part, we give two examples of a discrete skip free processM which satisfy M₀ =0, Θ⁰(M)=^L MandΘ¹(M)=^L M, but which are not discrete Ocone martingales.

Counterexample 1. Let (k)k≥1 be a sequence of independent symmetric Bernoulli random variables. We put M₀ = 0, 1M₁ = 1, 1M₂ = 2, 1M₃ = 2and for k > 3, 1M_k = k. We also introduce

M_n=

n

X

k=1

1M_k.

Since[M]_n =nfor alln ≥1 and sinceMis not Bernoulli random walk, it cannot be an Ocone martingale.

Let us verify thatΘ^a(M)=^L Mfora∈N\ {2}. Fora=0, the reflection property holds since thek’s are symmetric and independent. Fora =1 we consider four possible cases related with the values of(M1,M2,M3). Let us putRn=Pn

k=4k forn≥4.

In fact, ifM1=1,M2=2,M3=3, we haveΘ¹(M)=(0,1,0,−1, (−1−R_n)n≥4). IfM₁=1,M₂=0,M₃= −1, thenΘ¹(M)=(0,1,2,3, (3−R_n)n≥4).

IfM1= −1,M2=0,M3=1, thenΘ¹(M)=(0,−1,0,1, (1−R_n)n≥4).

IfM1= −1,M2= −2,M3= −3, thenΘ¹(M)=(0,−1,−2,−3,Θ¹(−3−Rn)n≥4). Similar presentation is valid forM:

ifM₁=1,M₂=2,M₃=3, thenM =(0,1,2,3, (3+R_n)n≥4), ifM1=1,M2=0,M3= −1, thenM =(0,1,0,−1, (−1+R_n)n≥4), ifM1= −1,M2=0,M3=1, thenM =(0,−1,0,1, (1+Rn)n≥4),

ifM1= −1,M2= −2,M3= −3, thenM =(0,−1,−2,−3,Θ¹(−3+Rn)n≥4).

To see that the laws ofΘ¹(M)and M are equal it is convenient to pass to increments of the corresponding processes.

If we take a pass with M1 = 1,M2 = 2,M3 = 3, then Θ²(M)of such trajectory has a probability zero which is not the case for the corresponding trajectory ofM. So,Θ²(M)6=^L M. Fora ≥3 we can write

Θ³(M)=

M1,M2,M3,Θ³((Mk)k≥4)

and we conclude from symmetry of Bernoulli random walk.

Counterexample 2. Let (εk)k≥0 be a sequence of independent {−1,+1}-valued symmetric Bernoulli random variables. Set k_n = j

ln(n+1) ln 2

k

−1, where bxcis the lower integer part of xand let us consider the following skip free process:

M₀=0 and for n≥1, M_n=

kn

X

k=0

2^kεk+(n−2^kn)εn.

Actually,Mis constructed as follows:M0=0,M1=ε0and for allk≥1 andn ∈ [2^k,2^k+1−1], the incrementsM_n−Mn−1have the sign ofεk. In particular, the increments of(M_n)are−1 or 1 and since, from the discussion at the beginning of Section2, the only skip free local martin- gale with such increments is the Bernoulli random walk, it is clear thatM is not an Ocone local martingale.

The equalityΘ⁰(M)=^L M only means thatM is a symmetric process, which is straightforward from its construction. Now let us check thatT1<∞, a.s. andΘ¹(M)=^L M. Almost surely on the set{M1 = −1}, there existsk ≥0 such thatεi = −1 for alli ≤ kandεk+1 =1. The later assertion is equivalent to say that for all integern∈(0,2^k+1−1],Mn−Mn−1= −1 and for all n∈ [2^k+1,2^k+2−1],Mn−Mn−1=1. It is then easy to check that

M₂k+2−1=1.

So we have proved that{M₁= −1} ⊆ {T₁<∞}, but since we also have{M₁=1} ⊆ {T₁<∞}, it follows thatP(T₁<∞)=1.

Then we see from the construction of(M_n)that almost surely,T₁belongs to the set{2^j−1: j ≥1}and that for j ≥1, conditionally toT1=2^j−1,(M_n,n ≤T1)and(M_T1+n,n ≥0)are independent. Moreover,

(M_T1+n,n≥0)=^L(2−M_T1+n,n≥0), so this proves thatΘ¹(M)=^L M.

Finally note that 0 and 1 are the only non-negative levels at which the reflection principle holds for the processM, i.e.Θ^a(M)=^L M impliesa =0 or 1. Indeed, at least it is clear from the construction of M that the only times and levels at which the sign of its increments can change belong to the set{2^j−1,j ≥0}, i.e. ifa≥0 is such thatΘ^a(M)=^L M, then necessarily a∈ {2^j−1,j≥0}andT_a∈ {2^j−1,j ≥0}. But suppose that fori≥2 we haveT₁=2ⁱ−1 and recall that all the incrementsM_T1+k+1−M_T1+kfor allk=0,1, . . .2ⁱ−1 have the same sign. If these increments are 1, then the processMreaches the level 2ⁱ−1 at timeT1+2ⁱ−2=2ⁱ⁺¹−3 which does not belong to the set{2^j −1,j ≥ 0}. So the sign of the increments ofM cannot change at this time and the level 2ⁱ−1 cannot satisfy the identity in lawΘ²ⁱ⁻¹(M)=^L M. 2.3. Continuous time lattice processes

As a preliminary result for the proof ofTheorem 1, we state an analogue ofTheorem 3for continuous time lattice processes. We say thatM =(M_t)t≥0is acontinuous time lattice process if M₀ = 0 and if it is a pure jump c`adl`ag process whose jumps1M_t = M_t −M_t− verify :

|1M_t| =η, for some fixed realη >0. If we denote by(τk)k≥1the jump times ofM, i.e.τ0=0 and fork≥1,

τk =inf{t> τk−1: |M_t−M_τk−1| =η}, then for allt ≥0,P-a.s.

M_t =

∞

X

k=1

1M_τk1{τk≤t}.

The quadratic variation ofMis given by:

[M]_t =

∞

X

k=1

(1M_τk)²1{τk≤t}=η²

∞

X

k=1

1{τk≤t}.

Note thatτkadmits the equivalent definitionτk =inf{t ≥0: [M]_t =kη²}. We define the time changed discrete processS^MbyS^M =(M_τk)k≥0which has values in the latticeηZ. In particular, we have:

M_t =S_η^M₋₂

[M]t, t ≥0. (2.29)

We say that M is a continuous time lattice Ocone local martingale if it can be written as M_t = S_At, where S is a symmetric Bernoulli random walk with values in the latticeηZand Ais a non-decreasing continuous time lattice process with values inNwhich is independent of S. In the case whereM is divergent,S coincides with S^M given in formula(2.29). WhenM is not divergent,S is different fromS^M, namely ifT =inf{k ≥0 : [S^M]_k = [S^M]_∞}thenS can be taken as:

Sk=S_k^M1_{k≤T}+(S_T^M + ˜ST−k)1_{k>T},

whereS˜ is a symmetric Bernoulli random walk which is independent fromS^M. In this caseS is independent from[M]. Therefore, when considering a continuous time lattice Ocone local martingale M, in identity (2.29) we can and will suppose that S^M is a symmetric Bernoulli random walk with values in the latticeηZand which is independent of[M].

Recall the definitions(1.4)and(1.5)of the hitting timeT_a and transformationsΘ^a, respec- tively.

Proposition 1. Let M be any continuous time lattice process such that for all k=0,1,2, Θ^kη(M)=^L M,

then M is a continuous time lattice Ocone local martingale with respect to its own filtration. If in addition T_η<∞a.s., then M is a divergent local martingale.

Proof. SetN =η⁻¹M. We remark that fork=1,2,3, Θ^k(N)=^L N.

Then following the proof of Theorem 3along the same lines for the continuous time process N, we obtain thatS^N conditionally to[N]is Bernoulli random walk. Hence S^N is a Bernoulli random walk which is independent of[N]. SinceS^N =η⁻¹S^M andη⁻²[N] = [M], we obtain thatS^M is a symmetric Bernoulli random walk on the latticeηZwhich is independent of[M]. It means that it is a local martingale with respect to its own filtration. Finally, whenT_η <∞a.s., Mis a divergent local martingale since Nis so.

3. Proof ofTheorem 1

Let(Ω,F,F,P)be the canonical space of continuous functions with filtrationFsatisfying usual conditions. LetM be a continuous stochastic process which is defined on this space and satisfying the assumptions ofTheorem 1. Without loss of generality we suppose that the sequence (a_n)is decreasing.

Proof of Theorem 1. First of all we note that since the map (x, ω) → Θ^x(ω) from R×C(R⁺,R)toC(R⁺,R)is continuous, the hypothesis of this theorem imply thatΘ⁰(M)=^L M, i.e.M is symmetric process.

Now, fix a positive integern. We define the continuous lattice valued process Mⁿ by using discretization with respect to the space variable. In this aim, we introduce the sequence of stopping times(τ_kⁿ)k≥0i.e.τ₀ⁿ =0 and for allk≥1

τ_kⁿ=inf{t > τ_k−1ⁿ : |Mt −M_τⁿ

k−1| =an}. ThenMⁿ=(M_tⁿ)t≥0is defined by:

M_tⁿ=

∞

X

k=0

M_τⁿ

k1_{τⁿ

k≤t<τk+1ⁿ }.

We can easily check that Mⁿis a continuous time lattice process verifying the assumptions of Proposition 1forη=a_n. Therefore according to this proposition,Mⁿis a continuous time lattice Ocone local martingale.

From the construction ofMⁿwe have the almost sure inequality sup

t≥0

|M_t −M_tⁿ| ≤a_n. (3.30)

Hence the sequence(Mⁿ)converges a.s. uniformly on[0,∞)towardM. The condition sup

n≥1

sup

t≥0

|1M_tⁿ| ≤a₁

and(3.30)imply (cf. [4], Corollary IX.1.19, Corollary VI.6.6) thatM is a local martingale and that

(Mⁿ,[Mⁿ])→^L (M,hMi). (3.31)

Since the properties (i) and (iii) given in the introduction are equivalent, it is sufficient to verify that for every deterministic functionh of the formPk

j=1λj1_]tj−1,tj]witht₀ =0 <t₁ <· · ·t_k we have:

E

exp

i Z ∞

0

h(s)dM_s

=E

exp

−1 2

Z ∞

0

h²(s)dhMi_s

. (3.32)

From(3.31)we see that

n→∞lim E

exp

i Z ∞

0

h(s)dM_sⁿ

=E

exp

i Z ∞

0

h(s)dM_s

.

Then in order to obtain(3.32), we will show by straightforward calculations that

n→∞lim E

exp

i Z ∞

0

h(s)dM_sⁿ

=E

exp

−1 2

Z ∞

0

h²(s)dhMi_s

. (3.33)

To prove(3.33)we first write E

exp

i

Z ∞

0

h(s)dM_sⁿ

= Z

E

exp

i Z ∞

0

h(s)dM_sⁿ

| [Mⁿ] =ω

dP[Mⁿ](ω), whereP_[Mⁿ]is the law of[Mⁿ]. Then fromProposition 1we have that

Mⁿ=^LanS_a−2 n [Mⁿ],

whereSis symmetric Bernoulli random walk independent from[Mⁿ]. Moreover, Z ∞

0

h(s)dM_sⁿ=

k

X

j=1

λj1M_tⁿ_j =^Lan k

X

j=1

λj1Sr_j, where1M_tⁿ_j =M_tⁿ

j −M_tⁿ

j−1,1Sr_j =Sr_j −Sr_j−1 andrj =a_n⁻²[Mⁿ]_t

j, 1≤ j ≤k.

SinceSand[Mⁿ]are independent andE

exp(ia1S_k)

=cos(a)for alla ∈R, we have:

E

exp

i Z ∞

0

h(s)dM_sⁿ

| [Mⁿ] =ω

=

k

Y

j=1

[cos(λjan)]^{(unj−unj−1)}, (3.34) whereuⁿ_j = ba_n⁻²ωt_jc, j =0,1, . . . ,kandbxcis the lower integer part ofx. Moreover, it is not difficult to see that

n→∞lim

k

Y

j=1

[cos(λja_n)]^{(unj−unj−1)}=exp −1 2

k

X

j=1

λ²_j(ωtj −ωtj−1)

!

, (3.35)

uniformly on compact sets ofR^k+. Then, the expression(3.34)and the convergence relations (3.31)and(3.35)imply(3.33).

4. Proof ofTheorem 2

In what follows, we assume without loss of generality, that the process M is divergent. We begin with the following classical result of ergodic theory, a proof of which may be found in [2, Lemma 1].

Recall that (Ω,F,F,P) is the canonical space of continuous functions endowed with its natural right-continuous filtrationF=(F_t)t≥0completed by negligible sets ofF=W

t≥0F_t. LetΘ be a measurable transformation fromΩ to Ω which preserves P. We say that Z is invariant a.s. byΘ if

Z◦Θ ^a=^.s. Z.

Lemma 4. LetΘ be a measurable transformation fromΩ toΩ which preservesP. A random variable Z ∈L²(Ω,F,P)is a.s. invariant byΘ if and only if

E(Z·(Y ◦Θ))=E(Z·Y), for all Y ∈ L²(Ω,F,P).

LetΘ_n,n≥1 be a family of transformations defined on canonical space of continuous func- tions(Ω,F,F,P). LetIbe the sub-σ-algebra of the invariant events by all the transformations Θ_n,n≥1, i.e.

I= {F∈F:1_F◦Θ_n^a=^.s.1_F, for alln ≥1}.

The following lemma extends Theorem 1 in [2]. For simplicity of writing both notations X◦Θ_nandΘ_n(X)are used forΘ_n-transformation ofX.

Lemma 5. Let M be a continuous divergent local martingale defined on the filtered probability space(Ω,F,F,P). Assume that the transformationsΘ_npreserve the Wiener measure, i.e. if B is the standard Brownian motion then for all n ≥1, B◦Θ_n =^L B. The following assertions are equivalent:

(j)For all n≥1,(B^M,hMi)and(Θ_n(B^M),hMi)have the same law.

(jj) B^M andhMiare conditionally independent given theσ-fieldI^M =(B^M)⁻¹(I).

Proof. The proof almost follows from that of Theorem 1 in [2] along the same lines. We first prove that (j) implies (jj).

Leth,gtwo measurable functions fromΩtoΩ. Then (j) implies:

E

h(hMi)g(B^M)

=E

h(hMi)g(B^M ◦Θ_n)

=E

h(hMi)(g(B^M)◦Θ_n)

. (4.36) We take conditional expectation with respect to B^M. For this we denote by f the following function:

E

h(hMi)|B^M_a.s.

= f(B^M).

Then(4.36)implies that E

f(B^M)g(B^M)

=E

f(B^M)(g(B^M)◦Θ_n)

. (4.37)