A discrete approach to the chaotic representation property.

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To cite this version:

Michel Emery. A discrete approach to the chaotic representation property.. 2000. �hal-00129658�

A DISCRETE APPROACH

TO THE CHAOTIC REPRESENTATION PROPERTY

M. ´ Emery

Abstract. — In continuous time, let (X t ) _{t > 0} be a normal martingale (i.e. a process such that both X _t and X _t ² − t are martingales). One says that X has the chaotic representation property if L ² ¡

σ(X) ¢

is the (direct) Hilbert sum L

p∈N

χ _p (X), where χ _p (X) is the space of all p-fold iterated stochastic integrals

Z

0<t 1 <...<t p

f (t ₁ , . . . , t _p ) dX _{t 1} . . . dX _{t p}

with f square-integrable (χ _p (X) is called the p ^th chaotic space; by convention, χ ₀ (X) is the one-dimensional space of deterministic random variables). An open problem is to characterize those processes X.

Instead of working in continuous time, we shall address an analogue of this problem where the time-axis is the set Z of signed integers; in this setting, we shall give a sufficient (but probably far from necessary) condition for the chaotic representation property to hold.

Notation and preliminaries

We shall use the set Z of all signed integers as our time-axis; the set of all finite subsets of Z will be denoted by P. For m and n in Z , we shall have to do with the following “intervals”:

ce m, n ce = { k ∈ Z : m < k 6 n } ; ce n, ∞bd = { k ∈ Z : n < k } ; ce−∞ , n ce = { k ∈ Z : k 6 n } . ce−∞ , n bd = { k ∈ Z : k < n } .

Given a filtration F = (F _n ) _{n ∈Z} , a process X = (X n ) _{n ∈Z} is adapted (respectively predictable ) if for each n the random variable X _n is F _n -measurable (respectively F _n−1 -measurable); a stopping time is an F _∞ -measurable random variable T with values in Z ∪ { + ∞} , such that for each n ∈ Z the event { T = n } (or, for that matter, { T 6 n } ) belongs to F _n ; notice that the value −∞ is not allowed to stopping times.

An empty sum P

i∈∅ x _i is always null, an empty product Q

i∈∅ x _i is always 1.

With Z as the time-axis, the analogue of a normal martingale is no longer a martingale, but a sort of normalized martingale increment:

Definition. — On a probability space (Ω, A, P ), let F = (F _n ) _{n ∈Z} be a filtration.

A process X = (X _n ) _n∈Z is a novation (more precisely: an F-novation) if, for each time n ∈ Z , X _n belongs to L ² (F _n ) and verifies

Ebd X _n | F _{n − 1} ce = 0 ; (N1)

Ebd X _n ² | F _n−1 ce = 1 .

(N2)

The name ‘novation’ aims at suggesting that X plays the rˆole of an innovation, but the prefix ‘in’ has been dropped to stress that no independence is required. ¹ Condition (N1) says that X n should be understood as a martingale increment; and (N2) is a normalization hypothesis. The simplest example of a novation is a sequence of independent random variables with mean 0 and variance 1.

Proposition 1 and definitions. — Let X be a novation on (Ω, A, P , F); for each A ∈ P, denote by X _A the product Q

n ∈ A

X _n . The set of random variables { X _A , A ∈ P } is orthonormal in L ² (Ω, A, P ).

So this set is the orthonormal basis of some closed subset of L ² (Ω, A, P ), called the chaotic space associated to X, and denoted by χ(X ).

If { X _A , A ∈ P } is total in L ² (Ω, F _∞ , P ), or equivalently if the chaotic space is equal to L ² (Ω, F _∞ , P ), one says that X has the chaotic representation property.

The simplest example of a novation with the chaotic representation property is the fair coin-tossing: the X _n are independent and uniformly distributed on {− 1, 1 } , and F is the filtration generated by X.

Proof of Proposition 1. — Fix A and B in P. For n ∈ Z , the formula Ebd X _A X _B | F _n ce =

½ 0 if A ∩ce n, ∞bd 6 = B ∩ce n, ∞bd ;

X _{A∩ce−∞,nce} X _{B∩ce−∞,nce} if A ∩ce n, ∞bd = B ∩ce n, ∞bd .

is true if n is large enough for ce−∞ , n ce to contain A and B; and if it holds for some n, it holds for n − 1 too because X is a novation. So it holds for every n ∈ Z , and in particular when n is small enough for A and B to be included in ce n, ∞bd . Thus, for such an n,

Ebd X _A X _B | F _n ce =

½ 0 if A 6 = B 1 if A = B,

and the proposition is proved by taking expectations on both sides.

Here are five necessary conditions for a novation X to have the chaotic represen- tation property.

Proposition 2 and definition. — Let X be a novation defined on some filtered probability space (Ω, A, P , F). If X has the chaotic representation property, then (i) for each n ∈ Z , the set ©

X _A , A ∈ P, A ⊂ ce−∞ , n ce ª

is an orthonormal basis of L ² (Ω, F _n , P );

(ii) the filtration F is generated by X;

(iii) the σ-field F _−∞ ( = T

n F _n ) is degenerate;

(iv) for all n ∈ Z and U ∈ L ² (F _n ), there exist Q and R in L ² (F _n−1 ) such that U = Q + RX _n ;

(v) for each U ∈ L ² (F _∞ ), there exists an F-predictable process H = (H _n ) _n∈Z such that

E £ P

n ∈Z H _n ² ¤

< ∞ and U = Ebd U ce + X

n ∈Z

H _n X _n .

When (v) holds, one says that the novation X has the predictable representation property (with respect to the filtration F).

1. Ne crains doncques, Po¨ ete futur, d’innover quelques termes.

(J. du Bellay, La deffence et

illustration de la langue fran¸coyse.)

Discrete chaotic representation 3

The analogy between this definition and the predictable representation property in continuous time (see for instance bd 3 ce ) is plain: X _n replaces dX _t and P

replaces R . Proof. — (i) Fix n ∈ Z . For each A ∈ P, one has X _A ⊥ L ² (F _n ) if A meets ce n, ∞bd and X _A ∈ L ² (F _n ) if A is included in ce−∞ , n ce . Thus ©

X _A , A ∈ P; A ⊂ ce−∞ , n ce ª is an orthonormal basis of L ² (F _n ) (and ©

X _A , A ∈ P; A meets ce n, ∞bd ª

is an orthonormal basis of its orthogonal supplement).

(ii) is a consequence of (i).

(iii) For A ∈ P and A 6 = ∅ , X _A is orthogonal to L ² (F _n ) for every n < sup A, and a fortiori to L ² (F _−∞ ). So L ² (F _−∞ ) is included in the orthogonal supplement to { X _A , A ∈ P, A 6 = ∅} ; as this supplement consists of deterministic random variables, F _−∞ is degenerate.

(iv) We know from (i) that every random variable U ∈ L ² (F _n ) admits an L ² -expansion as

U = X

A ∈ P A⊂ce−∞,nce

u _A X _A

with P

u ² _A < ∞ . Setting

Q = X

A ∈ P A⊂ce−∞,nbd

u _A X _A and R = X

A ∈ P A⊂ce−∞,nbd

u _{A ∪{ n }} X _A ,

one has

RX _n = X

A∈ P sup A=n

u _A X _A

(approximate both sides by finite sums and take limits in L ² ), whence (iv).

(v) By the chaotic representation property, each U ∈ L ² (F _∞ ) has an expansion U = X

A ∈ P

u _A X _A = u _∅ + X

n∈Z

X

A ∈ P A ⊂ce−∞ ,n bd

u _{A ∪{ n }} X _A X _n .

Now, the random variable H _n = X

A ∈ P A⊂ce−∞,nbd

u _{A ∪{ n }} X _A = X

A ∈ P sup A=n

u _A X _{A −{ n }}

belongs to L ² (F _{n − 1} ) by (i), with squared norm Ebd H _n ² ce = P

A ∈ P sup A=n

u _A ² ; summing in n gives

E £P

n

H _n ² ¤

= X

A∈ P A 6 = ∅

u _A ² < ∞ ,

and, as u _∅ = Ebd U X _∅ ce = Ebd U ce , the formula for U becomes Ebd U ce + P

n

H _n X _n .

The well-known equivalence between extremality and the predictable representa-

tion property (see Theorem (V.4.6) of bd 3 ce ) becomes completely elementary in our

discrete setting; it is recalled in the next proposition. We shall call L the set of all

probability laws on the real line that are carried by two points, and have mean 0 and

variance 1. In other words, an element of L is a probability of the form pδ _a + qδ _b ,

with p > 0, q > 0, p + q = 1, pa + qb = 0, and pa ² + qb ² = 1.

Lemma 1. — For each h ∈ R , there is a unique law `(h) ∈ L supported by the two roots of the quadratic equation x <sup>2</sup> = 1 + hx; it gives mass 1/(1 + x <sup>2</sup> ) to each root x of the equation. Moreover, this map ` : R → L is a bijection.

Proof. — The roots of x ² = 1 + hx are two real numbers y and z with product

− 1, so one is strictly positive and the other strictly negative. There is a unique probability law λ carried by y and z and having mean 0; it weights y with mass z/(z − y) = 1/(1+y ² ) and z with y/(y − z) = 1/(1+z ² ). And any random variable X with law λ verifies X ² = 1 + hX and Ebd X ce = 0, whence Ebd X ² ce = 1, so λ ∈ L.

Conversely, any λ ∈ L is supported by two points, so a random variable X with law λ verifies a quadratic equation X ² = hX + k; taking expectations gives k = 1, so ` is surjective.

The elements of L can also be characterized as the centered laws with unit variance that are extremal in the set of all centered laws. They are a fortiori extremal in the smaller set of all centered laws with unit variance, but this is no longer a characterization: the extreme points of the set of all centered laws with unit variance are the centered laws with unit variance carried by two or three points.

Proposition 3. — Let X be a novation on a filtered probability space (Ω, A, P , F).

a) The following three conditions are equivalent:

(i) there exists a predictable process H such that X ² = 1 + HX ;

(ii) there exists a predictable process L with values in L such that, for all Borel f and all n ∈ Z , Ebd f (X _n+1 ) | F _n ce = L _n+1 (f );

(iii) there exists a predictable process L with values in L such that, for all Borel f and all stopping times T , Ebd f (X _{T +1} ) | F _T ce = L _{T +1} (f ) on the event { T < ∞} .

If X has the predictable representation property, then F _−∞ is degenerate, and the three conditions (i) – (iii) hold.

b) If F _−∞ is degenerate, the following five conditions are equivalent:

(iv) X has the predictable representation property;

(v) for each n ∈ Z , there exists an event Γ such that F _n+1 = σ(F _n , Γ);

(vi) for all n ∈ Z and all random variables U ∈ L ² (F _n+1 ), one has U = Ebd U | F _n ce + Ebd U X n+1 | F _n ce X n+1 .

(vii) for all n ∈ Z and all F _n+1 -measurable random variables U , there are two F _n -measurable random variables Q and R such that U = Q + RX _n+1 ;

(viii) for all stopping times T and all F _{T +1} -measurable random variables U , there are two F _T -measurable random variables Q and R such that U = Q + RX T +1 on the event { T < ∞} .

c) If F _−∞ is degenerate and if F is the filtration generated by X, all eight conditions (i) – (viii) are equivalent to each other, and to the following further two conditions:

(ix) for all n ∈ Z and all U ∈ L ² (F _∞ ), one has U = P

A∈ P A ⊂ce n, ∞bd

Ebd U X _A | F _n ce X _A ;

(x) for all stopping times S and T such that S 6 T and all U ∈ L ² (F _T ), one has U = P

A ∈ P

1l _{{ A ⊂ce S,T ce}} Ebd U X _A | F _S ce X _A .

Discrete chaotic representation 5

Condition (i) is called a structure equation ; its analogue in continuous time has the form d bd X, X ce _t = dt+H t dX t . Conditions (i) to (iii) say that the natural filtration of X is dyadic. Conditions (iv) to (viii) say that the filtration F itself is dyadic (given the past F _n−1 , the innovation consists in choosing among two possible values only for X n , or equivalently in choosing the sign of X n ). But these conditions (iv) to (viii) do not imply that F is generated by X; they do not even imply that F is generated by any novation whatsoever (see Vershik’s Example 2 in bd 5 ce ). Conditions (ix) and (x) are the conditional chaotic representation property at times n and S; it is essential here that n and S are not allowed to take the value −∞ : when n and S are −∞ , these conditions become the (unconditional) chaotic representation property, which is in general strictly stronger than the predictable representation property. The rest of this work will precisely be concerned with the gap between these properties: which hypothesis should be added to the predictable representation property to imply the chaotic representation property? We shall only give a very partial answer.

Proof of Proposition 3. — (i) ⇒ (ii). Assuming (i), define an L-valued predictable process L by L n = `(H n ), where ` is the map defined in Lemma 1.

As X _n+1 ² = 1 + H _n+1 X _n+1 , X _n+1 is a.s. one of the two points of the support of

`(H _n+1 ); as Ebd X _n+1 | F _n ce = 0 and Ebd X _n+1 ² | F _n ce = 1, the conditional law of X _n+1 given F _n must be L _n+1 . This gives (ii).

(ii) ⇒ (iii). Assuming (ii), for each n ∈ Z the conclusion holds on the event { T = n } , so it holds on { T < ∞} .

(iii) ⇒ (ii) is trivial, and to obtain (ii) ⇒ (i) it suffices to define the predictable process H by `(H) = L.

Assuming X has the predictable representation property, for every U ∈ L ² (F _∞ ), there is a predictable H such that P

n Ebd H _n ² ce < ∞ and U = Ebd U ce + P

n H _n X _n ; this implies that the (square-integrable) martingale M n = Ebd U | F _n ce is given by M _n = Ebd U ce + P

m6n H _m X _m . For U ∈ L ² (F _−∞ ), one has U = Ebd U ce , showing that F _−∞ is degenerate.

We now pass to the equivalence of (iv) – (viii); the end of a), that is, (iv) ⇒ (i), will be established later.

(iv) ⇒ (vi). We suppose X has the predictable representation property. The martingale argument a few lines above implies that, for every U ∈ L ² (F _n+1 ), one has U = Ebd U | F _n ce + H n+1 X n+1 . As U and Ebd U | F _n ce are in L ² , so is H n+1 X n+1 too.

Multiplying both sides by X _n+1 and conditioning by F _n gives H _n+1 = Ebd U X _n+1 | F _n ce , whence (vi).

(vi) ⇒ (v). Choosing U = 1l _{{ X n+1 =0 }} in (vi) yields 1l _{{ X n+1 =0 }} = Pbd X _n+1 = 0 | F _n ce , and shows that { X _n+1 = 0 } is in F _n . Using (N2), this implies X _n+1 6 = 0 a.s., and, using (N1), Ebd X _n+1 ⁺ | F _n ce = Ebd X _n+1 ⁻ | F _n ce > 0 a.s.

Choosing now U = 1l _{{X n+1 >0}} in (vi) gives 1l _{{X n+1 >0}} = Q + RX _n+1 , with Q and R measurable for F _n and R = Ebd X _n+1 ⁺ | F _n ce > 0. So X n+1 = (1l _{{ X n+1 >0 }} − Q)/R, and (vi) becomes

∀ U ∈ L ² (F _n+1 ) U = Ebd U | F _n ce + Ebd U X n+1 | F _n ce (1l _{{ X n+1 >0 }} − Q)/R ,

showing that F _n+1 is generated by F _n and the event { X _n+1 > 0 } .

(v) ⇒ (vii). Hypothesis (v) implies for each n the existence of two F _n -measurable random variables F and G such that X _n+1 = F 1l _Γ + G 1l _Γ ^c . Observing that X _n+1 ² − (F + G)X _n+1 + F G = 0 and conditioning on F _n , one obtains F G = − 1.

Consequently, F 6 = G a.s. and 1l _Γ = (X _n+1 − G)/(F − G). Using (v) again, every F _n+1 -measurable U has the form V 1l _Γ + W , with V and W measurable for F _n ; replacing 1l _Γ by (X _n+1 − G)/(F − G) shows (vii).

(vii) ⇔ (viii). Given an F _{T +1} -measurable U , apply (vii) to each U _n = U 1l _{{ T =n }} . (vii) and (F _−∞ degenerate) ⇒ (iv). Let U be any random variable in L ² (F _∞ ) and M be the martingale M _n = Ebd U | F _n ce . When applied to M _n+1 − M _n , (vii) gives M n+1 − M n = H n+1 X n+1 for some F _n -measurable H n+1 (Q vanishes by conditioning on F _n ); so one has M _n − M _m = H _m+1 X _m+1 + . . . + H _n X _n for m < n.

Since F _−∞ is degenerate, M _n tends to Ebd U ce a.s. and in L ² when n → −∞ ; it also tends to M _∞ = U when n → + ∞ , so U = Ebd U ce + P

n ∈Z H _n X _n . Writing Ebd H _n+1 ² ce = E £

H _n+1 ² Ebd X _n+1 ² | F _n ce ¤

= Ebd H _n+1 ² X _n+1 ² ce = E £

(M _n+1 − M _n ) ² ¤ , one obtains Ebd H _m+1 ² + . . . + H _n ² ce = E £

(M _n − M _m ) ² ¤

for m < n, giving in the limit E £P

n

H _n ² ¤

= Ebd U ² ce − Ebd U ce ² < ∞ .

Proposition 3 b) is completely proved; to end proving a), that is, proving (iv) ⇒ (i), it suffices to establish (vii) ⇒ (i). That is quite easy: Hypothesis (vii) gives X _n+1 ² = Q + RX _n+1 , and Q = 1 is obtained by conditioning on F _n .

We now start showing c); from here on, we assume F to be generated by X and F _−∞ to be degenerate.

(i) ⇒ (v). From X _n ² = 1 + H _n X _n it follows that X _n = f(H _n , 1l _{{X n >0}} ), where f (h, 0) = ¹ ₂ (h − √

h ² + 4) and f(h, 1) = ¹ ₂ (h + √

h ² + 4). This formula shows that F _n , which we know is generated by F _n−1 and X _n , is also generated by F _n−1 and the event { X _n > 0 } .

(vi) ⇒ (ix). Iterating (vi), one obtains for all m and n in Z such that m 6 n

∀ U ∈ L ² (F _∞ ) Ebd U | F _n ce = X

A ⊂ce m,n ce

Ebd U X _A | F _m ce X _A .

Indeed, fixing n, it is true when m = n (for the right-hand side consists in one term only, Ebd U X _∅ | F _m ce X _∅ ); and if it holds for some m 6 n, one sees that it also holds for m − 1 by applying (vi) to replace each Ebd U X _A | F _m ce by Ebd U X _A | F _m−1 ce + Ebd U X _m X _A | F _m−1 ce X _m .

To obtain (ix), it suffices to let n tend to infinity in this formula; convergence takes place in L ² owing to the following estimate:

X

A ⊂ce m,n ce

Ebd U X _A | F _m ce ² = X

A,B ⊂ce m,n ce

Ebd U X _A | F _m ce Ebd U X _B | F _m ce Ebd X _A X _B | F _m ce

= E £

Ebd U | F _n ce ² ¯ ¯ F _m ¤

6 Ebd U ² | F _m ce .

(ix) ⇒ (x). For m 6 n and U ∈ L ² (F _n ), Ebd U X _A | F _n ce = 0 if sup A > n, so (ix) implies U = P

A⊂cem,nce Ebd U X _A | F _m ce X _A . Thus,

∀ U ∈ L ² (F _∞ ) Ebd U | F _n ce = X

A⊂cem,nce

Ebd U X _A | F _m ce X _A .

Discrete chaotic representation 7

If S is a stopping time and if m 6 n,

1l _{S=m} Ebd U | F _n ce = 1l _{S=m} X

A⊂cem,nce

Ebd U X _A | F _m ce X _A

= X

A ∈ P

1l _{{ A ⊂ce S,n ce}} 1l _{{ S=m }} Ebd U X _A | F _S ce X _A ; summing in m gives

( ∗ ) Ebd U | F _n ce = X

A∈ P

1l _{{ A ⊂ce S,n ce}} Ebd U X _A | F _S ce X _A on the event { S 6 n } .

On the complementary event { S > n } , the right-hand side is just Ebd U | F _S ce ; taken together, these two results can be rewritten

Ebd U | F _{S ∨ n} ce = X

A ∈ P

1l {A⊂ceS,nce} Ebd U X _A | F _S ce X _A . Letting now n tend to ∞ , this yields, for all U ∈ L ² (F _∞ ),

U = X

A∈ P

1l _{{ A ⊂ce S, ∞bd }} Ebd U X _A | F _S ce X _A .

Given a stopping time T > S and a U ∈ L ² (F _T ), we have to show that U = X

A∈ P

1l _{{ A ⊂ce S,T ce}} Ebd U X _A | F _S ce X _A ;

it suffices to verify that the difference between the right-hand sides of these two formulas vanishes:

¿ X

A ∈ P

1l _{{ A ⊂ce S, ∞bd }} 1l _{{ A meets ce T, ∞bd }} Ebd U X _A | F _S ce X _A = 0 ?

Saying that A meets ce T, ∞bd amounts to saying that sup A > T (with the convention sup ∅ = −∞ ). This sum can be rewritten

X

n∈Z

1l _{{T 6n}} X

A ∈ P sup A=n+1

1l {A⊂ceS,∞bd } Ebd U X _A | F _S ce X _A

= X

n ∈Z

1l _{{ T 6n }} X

B∈ P

1l _{{ B ⊂ce S,n ce}} Ebd U X _B∪{n+1} | F _S ce X _B∪{n+1}

= X

n∈Z

1l _{{T 6n}} X _n+1 X

B ∈ P

1l {B⊂ceS,nce} Ebd (U X _n+1 )X _B | F _S ce X _B . Now, on the event { T 6 n } , one has a fortiori S 6 n, so we may use ( ∗ ) to transform the sum over B into Ebd U X _n+1 | F _n ce , yielding P

n 1l _{{T 6n}} X _n+1 Ebd U X _n+1 | F _n ce . Taking into account that { T 6 n } belongs to F _n and that U 1l _{{ T 6n }} is F _n -measurable (because U is F _T -measurable), the conclusion is obtained by writing

1l _{{T 6n}} Ebd U X _n+1 | F _n ce = Ebd U 1l _{{T 6n}} X _n+1 | F _n ce = U 1l _{{T 6n}} Ebd X _n+1 | F _n ce = 0 . (x) ⇒ (vi) is trivial by taking S = n and T = n + 1.

From now on, we suppose given a novation X and its natural filtration F, and we

assume that X enjoys the predictable representation property with respect to F; so

all ten conditions (i) – (x) of Proposition 3 hold. They do not imply the chaotic

representation property (see bd 2 ce for a counterexample); the question is to find

additional conditions that are sufficient for the chaotic representation property to

hold. Observe that the problem depends only on the law of the process X; so the

conditions we are looking for are conditions on the law of X .

Two simple instances of chaotic representation property

The simplest case is when the X _n are independent; by Proposition 3 (ii), the law of each X _n belongs to L, and an easy dimension argument gives the chaotic representation property:

Proposition 4. — If the novation X consists of independent random variables X _n , each with law in L, the chaotic representation property holds.

Proof. — Let (A _p ) _{p ∈N} be an increasing sequence in P with limit S

p A _p = Z . By martingale convergence, any random variable in L ² (F _∞ ) can be approximated by its projection on L ² ¡

σ(X n , n ∈ A p ) ¢

; so it suffices to establish that for fixed A ∈ P, the space S _A = L ² ¡

σ(X _n , n ∈ A) ¢

is included in the chaotic space χ(X). Each X _n takes two values, hence the random vector (X _n ) _n∈A takes 2 ^|A| values, and S _A has dimension 2 ^{| A |} . But the subspace S _A ⁰ of S _A with orthonormal basis { X _B , B ⊂ A } also has dimension 2 ^{| A |} ; thus S _A = S _A ⁰ , whence S _A ⊂ χ(X).

Another case is when the novation X is deterministic in some neighbourhood of −∞ ; before giving a precise statement (Proposition 5), we prove an auxiliary lemma saying that the chaotic representation property needs to be checked near

−∞ only. Recall that stopping times are not allowed to assume the value −∞ . Lemma 2. — Let T be a stopping time. If L ² (F _T ) ⊂ χ(X), the chaotic representation property holds.

Proof. — By replacing T with T ∧ 0, we may suppose T < + ∞ a.s. To prove the lemma, it suffices to show

( ∗ ) L ² (F _T ) ⊂ χ(X) ⇒ L ² (F _{T +1} ) ⊂ χ(X ) ;

for this implies first L ² (F _{T +p} ) ⊂ χ(X) for each p ∈ N , and then L ² (F _∞ ) ⊂ χ(X) since S

p ∈N L ² (F _{T +p} ) is dense in L ² (F _∞ ) by martingale convergence.

To show ( ∗ ), take any U ∈ L ² (F _{T +1} ) and apply Condition (viii) of Proposition 3 to U − Ebd U | F _T ce ; this yields U = Ebd U | F _T ce + KX _{T +1} for some F _T -measurable K. As

Ebd U ² | F _T ce = Ebd U | F _T ce ² + K ² Ebd X _T ² ₊₁ | F _T ce = Ebd U | F _T ce ² + K ² , K belongs to L ² (F _T ). Observing that U = Ebd U | F _T ce + P

n K 1l _{{ T =n }} X _n+1 and that Ebd U | F _T ce ∈ χ(X ), it suffices to verify that K1l _{{T =n}} X _n+1 is in χ(X). But we know K1l _{T P _=n} to be in L ² (F _T ) ∩ L ² (F _n ) ⊂ χ(X) ∩ L ² (F _n ); so it has an expansion of the form

A ∈ P A⊂ce−∞,nce

u _A X _A , and its product with X _n+1 is in χ(X ) too, with chaotic expansion X

A ∈ P sup A=n+1

u _A\{n+1} X _A .

Proposition 5. — The following five conditions are equivalent:

(i) there exist a random variable S with values in Z ∪ { + ∞} and a predictable process Y = (Y n ) _{n ∈Z} such that X = Y on the random interval cece−∞ , S bdbd ;

(ii) there exist a stopping time T such that Pbd T > n ce > 0 for all n ∈ Z and a deterministic process y = (y _n ) _{n ∈Z} such that X = y on cece−∞ , T bdbd , X _T = − 1/y _T on { T < ∞} , and

P £

X _n = y _n ¯ ¯ T > n ¤

= 1

1 + y _n ² P £

X _n = − 1/y _n ¯ ¯ T > n ¤

= y _n ²

1 + y ² _n ;

Discrete chaotic representation 9

(iii) the predictable process H of Proposition 3 (i) verifies X

n 6 0

1

1 + H _n ² < ∞ a.s.;

(iv) the series P

n60 X _n ² converges a.s.;

(v) with probability 1, | X _n | < 1 for all n small enough.

When these conditions are met, the σ-field F _T is generated by T , and the chaotic representation property holds.

For an analogue of this statement in continuous time, see Th´eor`eme 5 of bd 1 ce and Th´eor`eme 5.3.6 of Taviot bd 4 ce .

Convergence of the series in (iv) holds a.s. but not in L ¹ , for Ebd X _n ² ce = 1.

Proof of Proposition 5. — (i) ⇒ (ii). Fix n such that Pbd S > n ce > 0. We shall first see by induction that for every m 6 n there exists an F _m -measurable random variable Z m such that X n = Z m on the event { S > n } . This is true for m = n with Z n = X n . Supposing it to hold for some m 6 n, there exists a Borel function f such that Z m = f(. . . , X m − 2 , X m − 1 , X m ); so on { S > n } , X m = f(. . . , X m − 2 , X m − 1 , Y m ), and it holds for m − 1 too, with Z _m−1 = f(. . . , X _m−2 , X _m−1 , Y _m ) being F _m−1 -measurable because Y is predictable.

Consequently, Ebd X n 1l _{S>n} | F _m ce = Ebd Z m 1l _{S>n} | F _m ce = Z m Pbd S > n | F _m ce , and 1l _{{ S>n }} Ebd X n 1l _{{ S>n }} | F _m ce = 1l _{{ S>n }} X n Pbd S > n | F _m ce .

Letting m tend to −∞ , we get 1l _{{ S>n }} Ebd X _n 1l _{{ S>n }} ce = 1l _{{ S>n }} X _n Pbd S > n ce , showing that X _n is the constant x _n = Ebd X _n | S > n ce on the event { S > n } . Unfixing n, we obtain that X agrees with some deterministic process x on the interval cece−∞ , S bdbd .

By Condition (i) of Proposition 3, there is a predictable process H such that X ² = 1 + HX ; so Φ = ¹ ₂ (H + √

H ² +4) and Ψ = ¹ ₂ (H − √

H ² +4) are two predictable processes such that ΦΨ = − 1 and that for each m, X m is a.s. equal to Φ m or Ψ m . There are two Borel functions φ _m and ψ _m such that Φ _m = φ _m (. . . , X _m−2 , X _m−1 ) and Ψ m = ψ m (. . . , X m − 2 , X m − 1 ).

Fix again n such that Pbd S > n ce > 0. Define a deterministic process y = (y m ) _{m ∈Z} by y _m = x _m if m 6 n and (inductively) by y _m = φ _m (. . . , y _m−2 , y _m−1 ) if m > n.

For m 6 n, Pbd X k = y k ∀ k 6 m ce > Pbd S > n ce > 0; putting E m = { X k = y k ∀ k < m } , one has Pbd X _m = y _m | E _m ce > 0. But the conditional law of X _m given E _m is the law in L supported by the two points φ _m (. . . , y _{m − 2} , y _{m − 1} ) and ψ _m (. . . , y _{m − 2} , y _{m − 1} ). So y _m is one of these two points and − 1/y _m is the other one; this holds for m > n too by the very definition of y _m .

Put T = inf { m : X _m 6 = y _m } . As T is minorated by S ∧ n, it does not take the value −∞ , and T is a stopping time. On the event { T > m } , X and y agree up to time m − 1, and X _m takes the two values y _m and − 1/y _m with respective probabilities given by Lemma 1:

Pbd X _m = y _m | T > m ce = 1/(1+y ² _m ) and Pbd X _m = − 1/y _m | T > m ce = y ² _m /(1+y _m ² ) .

On { T = m } , one has furthermore X _m 6 = y _m , whence X _m = − 1/y _m , and

X _T = − 1/y _T on { T < ∞} . Last, for each m ∈ Z , the essential supremum of T

cannot be m since Pbd T > m | T > m ce = Pbd X _m = y _m | T > m ce = 1/(1+y _m ² ) > 0; thus

T is not bounded above.

(ii) ⇒ (iv). If (ii) holds, iterating the relation Pbd T > n | T > n − 1 ce = 1

1 + y _n ² gives for n < 0

Pbd T > 0 | T > n ce = Y

m ∈ce n,0 ce

1 1 + y _m ² and, taking the limit when n → −∞ ,

Pbd T > 0 ce = Y

m 6 0

1 1 + y ² _m .

The left-hand side being strictly positive by hypothesis, the infinite product must converge, and P

n 6 0

y _n ² < ∞ . As X _n = y _n for all n < T , P

n 6 0

X _n ² < ∞ a.s.

(iv) ⇒ (iii). According to the structure equation X ² = 1 + HX , the process X never vanishes and H = X − 1/X. Hence 1/(1+H ² ) = X ² /(1 − X ² +X ⁴ ) 6 ⁴ ₃ X ² , and if the series P

n60 X _n ² is a.s. convergent, so is also P

n60 1/(1+H _n ² ).

(iii) ⇒ (i). One of the roots of the structure equation x ² = 1 + H _n x satisfied by X _n is Φ _n = ¹ ₂ (sgn H _n ) ¡

| H _n | + p

H _n ² + 4 ¢

, with the convention sgn 0 = 1. Notice that the predictable process Φ verifies | Φ _n | > | H _n | , so the series P

n60 1/(1+Φ ² _n ) is a.s. convergent. Set

T = inf n

n ∈ Z : X

m6n

1

1+Φ ² _m > 1 o .

Because the series is convergent, T > −∞ a.s. As Φ is predictable, T is a predictable stopping time (i.e. T − 1 is a stopping time) and the event { n < T } is in F _{n − 1} . By Proposition 3 (ii) and Lemma 1, Pbd X _n =Φ _n | F _{n − 1} ce = 1/(1+Φ ² _n ), so one can write

E h X

n<T

1l _{{ X n =Φ n }} i

= E hX

n

1l _{{ n<T }} Pbd X n = Φ n | F _{n − 1} ce i

= E h X

n<T

1 1 + Φ ² _n

i 6 1 .

Consequently, the sum P

n<T 1l _{{X n =Φ n }} is a.s. finite; so, with probability 1, for all but finitely many n 6 0, X n is the other root − 1/Φ n of the equation, and (i) holds with Y = − 1/Φ.

(iv) ⇒ (v) is trivial.

(v) ⇒ (i). The two roots of the structure equation x ² = 1 + H _n x satisfied by X n are Φ n = ¹ ₂ (sgn H n ) ¡

| H _n | + p

H _n ² + 4 ¢

and − 1/Φ n ; they verify | Φ n | > 1 and

|− 1/Φ n | 6 1. So X is equal to the predictable process Y = − 1/Φ on the random set {| X | < 1 } . By hypothesis, this random set contains a random interval cece−∞ , S bdbd with S > −∞ a.s., so (i) holds.

(ii) ⇒ (σ(T ) = F _T ). Supposing (ii) to hold, let U be any F _T -measurable random variable. There is for each n ∈ Z∪{ + ∞} a Borel function u _n such that

U = U 1l _{{T =∞}} + X

n ∈Z

U 1l _{{T =n}}

= u _∞ (X _m , m ∈ Z )1l _{{T =∞}} + X

n∈Z

u _n (. . . , X _{n − 2} , X _{n − 1} , X _n )1l _{{T =n}}

= u _∞ (y m , m ∈ Z )1l _{{T =∞}} + X

n ∈Z

u _n (. . . , y n − 2 , y n − 1 , − 1/y n )1l _{{T =n}} ;

since y is deterministic, U is σ(T )-measurable.

Discrete chaotic representation 11

(ii) ⇒ (chaotic representation property). Fix n ∈ Z . For any A ∈ P, one has Ebd X _A 1l _{{T =n}} ce =

 

 

 



y _A Pbd T = n ce if A ⊂ ce−∞ , n bd ; y _A\{n} ( − 1)

y _n Pbd T = n ce if sup A = n;

0 if sup A > n.

Squaring and summing over A yields X

A ∈ P

¡ Ebd X _A 1l _{{T =n}} ce ¢ 2

= Pbd T = n ce ² X

A ∈ P A⊂ce−∞,nbd

³

y _A ² + y _A ² 1 y ² _n

´

= Pbd T = n ce ² ³ 1 + 1

y _n ²

´ X

A ∈ P A⊂ce−∞,nbd

y _A ²

= Pbd T = n ce ² ³ 1 + 1

y _n ²

´ Y

m<n

¡ 1 + y ² _m ¢ .

Now, by induction on k < n, Pbd T = n | T > k ce = y ² _n 1+y _n ²

Y

k<m<n

1

1+y ² _m , so, in the limit when k → −∞ ,

y _n ² 1+y ² _n

Y

m<n

1

1+y _m ² = Pbd T =n ce , and the above sum becomes

X

A ∈ P

¡ Ebd X _A 1l _{{T =n}} ce ¢ 2

= Pbd T = n ce ² 1

Pbd T = n ce = Pbd T = n ce = Ebd (1l _{T =n} ) ² ce . The left-hand side is the squared L ² -norm of the orthogonal projection of 1l T =n on the chaotic space χ(X); the right-hand side is the squared L ² -norm of 1l T =n itself.

Their being equal shows that 1l _{T =n} belongs to χ(X), and, n being arbitrary, that L ² (σ(T )) ⊂ χ(X). We have seen above that σ(T ) = F _T ; so L ² (F _T ) ⊂ χ(X), and the chaotic representation property holds by Lemma 2.

Another, less simple, case of chaotic representation property

Recall the context: X is a novation, F is its natural filtration, and all ten conditions of Proposition 3 are in force; in particular, by condition (v), F is dyadic and by condition (iv) F _−∞ is degenerate. In this section, we shall work in a narrower setting: we shall further suppose that F is generated by a process taking values in a two-point space (the set {− 1, 1 } will be convenient). Example 2 of Vershik bd 5 ce shows that this additional hypothesis is not a consequence of the other assumptions.

Lemma 3. — a) Let ε be a process with values in {− 1, 1 } ; call E the natural filtration of ε and suppose Var bd ε n | E _{n − 1} ce > 0 a.s. for each n ∈ Z . There exists a unique E-novation X ^ε such that sgn X ^ε = ε; moreover, X ^ε has the same natural filtration E as ε.

b) Suppose given a filtration F and an F-novation X; put ε = sgn X (with for instance sgn 0 = 1). The following two conditions are equivalent:

(i) both processes X and ε generate the same filtration;

(ii) the novation X ^ε defined in a) is equal to X.

Remarks. — a) In Lemma 3 b), since X is an F-novation, Var bd ε _n | F _n−1 ce > 0, and a fortiori Var bd ε n | E _{n − 1} ce > 0 where E is the filtration generated by ε. Hence the process X ^ε in condition (ii) is well defined.

b) If the time-axis is { n ∈ Z : n 6 0 } instead of Z , conditions (i) and (ii) in b) are also equivalent to the seemingly weaker condition:

(i ⁰ ) the processes X and ε generate the same σ-field.

To check this, calling F (respectively E) the natural filtration of X (respectively ε), it suffices to verify that if F _n = E _n , then F _n−1 = E _n−1 . Supposing F _n = E _n , X n = U 1l _{{ε n =1}} + V 1l _{{ε n =−1}} for some E _{n − 1} -measurable U and V . This implies (X _n − U )(X _n − V ) = 0; expanding and conditioning on F _n−1 gives U V = − 1;

in particular, U 6 = V a.s., and { X _n = U } = { ε _n = 1 } , { X _n = V } = { ε _n = − 1 } . Now if W is any F _{n − 1} -measurable random variable, it is also E _n -measurable, so W = Q1l _{{ε n =1}} + R1l _{{ε n =−1}} with Q and R measurable for E _{n − 1} . This can be rewritten as W = AX n + B, where Ax + B is the E _{n − 1} -measurable affine function mapping U to Q and V to R. Conditioning on F _n−1 kills the term AX _n , so W = B, and W is E _{n − 1} -measurable. This proves F _{n − 1} = E _{n − 1} .

Proof of Lemma 3. — a) If X is any E-novation, then, owing to the filtration E being dyadic, X n = Aε n + B, where A and B are E _{n − 1} -measurable. Condition (N1) implies X _n = A ¡

ε _n − Ebd ε _n | E _n−1 ce ¢

, and (N2) then yields 1 = A ² Var bd ε _n | E _n−1 ce . If furthermore sgn X = ε, A cannot be negative, and one gets

X _n = ε n − Ebd ε n | E _{n − 1} ce

√ Var bd ε _n | E _n−1 ce .

Conversely, X defined by this formula is an E-novation; and as ¯ ¯ Ebd ε _n | E _n−1 ce ¯ ¯ < 1 and | ε _n | = 1, one has sgn X _n = ε _n . This proves existence and uniqueness.

Since sgn X = ε, the natural filtration E of ε is included in that of X ; but the explicit formula for X n shows that X is adapted to E; so X generates E.

b) (ii) ⇒ (i) is an immediate consequence of a). Conversely, if an F-novation X and its sign ε have the same natural filtration E, E is included in F, so X is also an E-novation, and X = X ^ε by uniqueness in a).

Proposition 6. — Let ε = (ε _n ) _{n ∈Z} be a process with values in {− 1, 1 } and call F its natural filtration. Suppose

(i) the process ε is Markov (but not necessarily homogeneous);

(ii) the σ-field F _−∞ is degenerate;

(iii) Var bd ε n | E _{n − 1} ce > 0 a.s. for each n ∈ Z .

Under these assumptions, the F-novation X ^ε (defined in the previous lemma) has the chaotic representation property.

Proof. — We shall simply write X instead of X ^ε . Notice that all ten conditions of Proposition 3 hold. If (Z _n ) _{n ∈Z} is any process, we shall set Z _A = Q

n ∈ A

Z _n for A ∈ P.

As ε is a Markov process, the conditional expectation Ebd ε n | F _{n − 1} ce is a function

of ε _n−1 ; it takes values in bd− 1, 1 ce , and more precisely in the open interval ( − 1, 1)

since Var bd ε _n | F _n−1 ce > 0. Hence we may put Ebd ε _n | F _n−1 ce = sin Θ _n , for some random

Discrete chaotic representation 13

variable Θ _n = θ _n (ε _{n − 1} ), depending on ε _{n − 1} only, and with values in ( − ^π ₂ , ^π ₂ ). And as ε _{n − 1} takes only the values − 1 and 1, Θ _n = α _n ε _{n − 1} + β _n for two real numbers

α _n = θ _n (1) − θ _n ( − 1)

2 and β n = θ _n (1) + θ _n ( − 1) 2

that are both in the open interval ( − ^π ₂ , ^π ₂ ). With these notations, the formula giving X n in the proof of Lemma 3 becomes X n = (ε n − sin Θ n )/ cos Θ n . This implies ε _n = X _n cos Θ _n + sin Θ _n ; squaring both sides gives X _n ² = 1 − 2 X _n tan Θ _n and shows that the predictable process H appearing in the structure equation satisfied by the novation X is H = − 2 tan Θ.

For m 6 n in Z , call χ ⁿ (respectively χ ⁿ _m ) the closed subspace of χ(X ) with orthonormal basis ©

X _A , A ∈ P, A ⊂ ce−∞ , n ce ª

(respectively { X _A , A ⊂ ce m, n ce} );

notice that χ ⁿ _m has finite dimension 2 ^{n − m} and that for U ∈ χ ^m and V ∈ χ ⁿ _m the product U V belongs to χ ⁿ .

As F _∞ = σ(ε _n , n ∈ Z ), the chaotic representation property will be established if we show that the chaotic space χ(X) contains every random variable of the form f (ε m+1 , . . . , ε n ). For fixed m and n, those random variables form a finite-dimensional vector space, with basis { ε _A , A ⊂ ce m, n ce} ; so it suffices to show that each ε _A belongs to χ(X).

The first step of the proof will consist in establishing that for every A ⊂ ce m, n ce , there exist Q and R in χ ⁿ _m such that ε _A = Q + Rε _m . For fixed n, this will be shown by induction on m 6 n. If m = n, the only possible A is A = ∅ , and the property holds trivially with Q = 1 = X _∅ and R = 0. Suppose now it holds for some m 6 n.

Replacing Θ _m by α _m ε _{m − 1} + β _m in the formula ε _m = X _m cos Θ _m + sin Θ _m gives an expression of the form ε _m = (aX _m +b) + (cX _m +d)ε _{m − 1} . Now every subset A of ce m − 1, n ce is either of the form B, or of the form { m }∪ B, for some B ⊂ ce m, n ce . By induction hypothesis, ε _B = Q + Rε _m with Q and R in χ ⁿ _m ; so ε _A is either Q + Rε _m or Qε _m + R, and replacing ε _m by (aX _m +b) + (cX _m +d)ε _m−1 establishes the claim.

Owing to this property, to show that ε _A is in χ(X) for A ⊂ ce m, n ce , it suffices to show that ε m is in χ ^m . Without loss of generality, we shall do it for m = 0 only: the rest of the proof will consist in establishing that ε ₀ belongs to the chaotic space χ ⁰ generated by (X _n ) _{n 6 0} .

Set µ

Q _n R n

¶

=

µ cos β _n sin β _n

− sin β n cos β n

¶ µ X _n 1

¶ . Rewriting (N1) and (N2) as

E ^{F n−1}

·µ X n

1

¶

( X _n 1 )

¸

= Id , one has

E ^{F n−1}

·µ Q _n R n

¶

( Q n R n )

¸

=

µ cos β _n sin β _n

− sin β n cos β n

¶ Id

µ cos β _n − sin β _n sin β n cos β n

¶

= Id , whence Ebd Q ² _n | F _n−1 ce = Ebd R _n ² | F _n−1 ce = 1 and Ebd Q _n R _n | F _n−1 ce = 0. Consequently, by induction on n 6 0, if A and B (respectively A ⁰ and B ⁰ ) are two disjoint sets with union A ∪ B = A ⁰ ∪ B ⁰ = ce n, 0 ce ,

Ebd (Q _A R _B )(Q _A 0 R _B 0 ) | F _n ce =

½ 1 if A = A ⁰ and B = B ⁰

0 else;

and when (A, B) ranges over all pairs of complementary subsets of ce n, 0 ce , the r.v.’s Q _A R _B form an orthonormal basis of the subspace χ ⁰ _n . The orthogonal projection of ε ₀ on this subspace is

Proj _χ ⁰

n ε ₀ = X

A ∪ B= ce n,0 ce A ∩ B= ∅

Ebd ε ₀ Q _A R _B ce Q _A R _B .

To show that ε ₀ is in the chaotic space, it suffices to show that it is the L ² -limit of Proj _χ 0

n ε ₀ when n → −∞ ; as k ε ₀ k ² = 1, this reduces to proving that k Proj _χ 0 n ε ₀ k ² tends to 1, or equivalently that

X

A∪B=cen,0ce A∩B= ∅

¡ Ebd ε ₀ Q _A R _B ce ¢ 2

−→ 1 when n → −∞ .

Set U _n = Q _n cos α _n and V _n = R _n sin α _n . One has ε _n = X _n cos Θ _n + sin Θ _n

= ¡

cos(α _n ε _n−1 + β _n ) sin(α _n ε _n−1 + β _n ) ¢ µ X _n

1

¶

= ¡

cos(α _n ε _{n − 1} ) sin(α _n ε _{n − 1} ) ¢ µ

cos β _n sin β _n

− sin β n cos β n

¶ µ X _n 1

¶

= ( cos α _n ε _{n − 1} sin α _n ) µ Q n

R _n

¶

= U _n + ε _{n − 1} V _n Iterating this formula, one obtains

ε ₀ = U ₀ + U ₋₁ V ₀ + U ₋₂ V ₋₁ V ₀ + . . . + U _n+1 V _n+2 . . . V ₀ + ε _n V _n+1 . . . V ₀

= cos α ₀ Q ₀ + cos α _{− 1} sin α ₀ Q _{− 1} R ₀ + . . .

+ cos α _n+1 sin α _n+2 . . . sin α ₀ Q _n+1 R _n+2 . . . R ₀ + ε n sin α _n+1 . . . sin α ₀ R n+1 . . . R 0 . Multiplying by Q _A R _B (where A ∪ B = ce n, 0 ce and A ∩ B = ∅ ) and conditioning by F _n , all terms cancel but one, and there only remains

• if A = ∅ and B = ce n, 0 ce , E £

ε ₀ Q _A R _B ¯ ¯ F _n ¤

= Ebd ε _n sin α _n+1 . . . sin α ₀ R ² _n+1 . . . R ² ₀ | F _n ce

= ε _n sin α _n+1 . . . sin α ₀ ;

• if A 6 = ∅ and sup A = m ∈ ce n, 0 ce , Ebd ε ₀ Q _A R _B | F _n ce

= E £

cos α _m sin α _m+1 . . . sin α ₀ Q A∩cen,m−1ce R B∩cen,m−1ce Q ² _m R ² _m+1 . . . R ² ₀ ¯ ¯ F _n ¤

= cos α _m sin α _m+1 . . . sin α ₀ E £

Q _{A ∩ce n,m − 1 ce} R _{B ∩ce n,m − 1 ce} ¯ ¯ F _n ¤

= cos α _m sin α _m+1 . . . sin α ₀ Y

a<m a∈A

sin β _a Y

b<m b∈B

cos β _b

(the latter follows from Ebd Q ` | F <sub>` − 1</sub> ce = sin β ` and Ebd R ` | F <sub>` − 1</sub> ce = cos β ` ). Taking expectations, squaring and summing gives k Proj _χ ⁰

n ε ₀ k ² as the sum of two terms.

The first term, corresponding to A = ∅ , is

¡ Ebd ε _n ce ¢ 2

sin ² α _n+1 . . . sin ² α ₀ ;

Discrete chaotic representation 15

the second term is the sum X

m ∈ce n,0 ce

cos ² α _m sin ² α _m+1 . . . sin ² α ₀ X

A ∪ B= ce n,m − 1 ce A ∩ B= ∅

Y

a ∈ A

sin ² β _a Y

b∈B

cos ² β _b

= X

m∈cen,0ce

cos ² α _m sin ² α _m+1 . . . sin ² α ₀ Y

a∈cen,m−1ce

(sin ² β _a + cos ² β _a )

= X

m ∈ce n,0 ce

cos ² α _m sin ² α _m+1 . . . sin ² α ₀

= X

m ∈ce n,0 ce

¡ sin ² α _m+1 . . . sin ² α ₀ − sin ² α _m sin ² α _m+1 . . . sin ² α ₀ ¢

= 1 − sin ² α _n+1 . . . sin ² α ₀ . Putting both terms together gives

k Proj _χ 0

n ε ₀ k ² = 1 − ¡

1 − Ebd ε _n ce ² ¢

sin ² α _n+1 . . . sin ² α ₀

= 1 − sin ² α _n+1 . . . sin ² α ₀ Var ε _n ,

and to establish the chaotic representation property it suffices to verify that sin ² α _n+1 . . . sin ² α ₀ Var ε _n −→ 0 when n tends to −∞ .

Clearly, this holds if the product sin ² α _n+1 . . . sin ² α ₀ tends to 0 (for Var ε _n 6 1);

hence we only have to consider the case when the infinite product Q

n60 sin ² α _n is convergent.

We shall show that in that case, condition (iii) of Proposition 5 is fulfilled; as the chaotic representation property always holds in the degenerate situation considered in that proposition, our Proposition 6 will thus be established in full generality.

That condition says that the series P

n60 (1+H _n ² ) ⁻¹ converges a.s., where H is the predictable process featuring in the structure equation for the novation X. At the beginning of the proof, we saw that H _n = − 2 tan Θ _n ; consequently

1

1 + H _n ² = 1

1 + 4 tan ² Θ _n 6 1

1 + tan ² Θ _n = cos ² Θ _n , and it only remains to establish that the sum P

n60 cos ² Θ n is a.s. finite.

Put α _n = ^π ₂ − | α _n | . Since the values assumed by Θ _n are β _n − α _n and β _n + α _n , one has − ^π ₂ < β _n − | α _n | < β _n + | α _n | < ^π ₂ , whence

½ π

2 − 2α _n = − ^π ₂ + 2 | α _n | < β _n + | α _n | < ^π ₂

− ^π ₂ + 2α _n = ^π ₂ − 2 | α _n | > β _n − | α _n | > − ^π ₂ and Θ _n = β _n ± α _n ∈ ¡

− ^π ₂ , − ^π ₂ +2α _n ¢

∪ ¡ _π

2 − 2α _n , ^π ₂ ¢

. As the infinite product is convergent, sin ² α _n → 1, so α _n → 0, and 2α _n < ^π ₂ for all n small enough. For these n one has | cos Θ _n | < | cos( ^π ₂ − 2α _n ) | = sin(2 | α _n | ) = 2 sin | α _n | cos α _n 6 2 cos α _n , hence also cos Q ² Θ n < 4 cos ² α _n = 4 (1 − sin ² α _n ); so convergence of the infinite product

n60 sin ² α _n implies convergence of the series P

n60 cos ² Θ _n .

The Markov hypothesis (i) in Proposition 6 has been used to perform explicit

computations on the process ε; it is not clear whether the result remains true or not

when this hypothesis is dropped.

References

bd 1 ce M. ´ Emery. Quelques cas de repr´esentation chaotique. S´eminaire de Probabilit´es XXV, Lecture Notes in Mathematics 1485, Springer 1991.

bd 2 ce . M. ´ Emery. On the chaotic representation property for martingales. Probability theory and Mathematical Statistics. Lectures presented at the semester held in St-Petersburg, Russia, March 2 – April 23, 1993. Gordon and Breach, 1996.

bd 3 ce D. Revuz & M. Yor. Continuous Martingales and Brownian Motion. Springer- Verlag, 1999.

bd 4 ce G. Taviot. Martingales et ´equations de structure : ´etude g´eom´etrique. Th`ese, Universit´e de Strasbourg I, 1999.

bd 5 ce A. M. Vershik. The theory of decreasing sequences of measurable partitions.

St. Petersburg Math. J. 6, 705–761, 1995.

Universit´ e Louis Pasteur et C.N.R.S.

I.R.M.A.

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emery@math.u-strasbg.fr