Non-extremal Martingale with Brownian Filtration

Let (Bt)t≥0 be the filtration of a Brownian motion (Bt)t≥0 on (Ω,B,P). An example is given of an non-extremal martingale which generates the filtration (Bt)t≥0. We also discuss a property of pure martingales, we show here that it is a property of a filtration rather than a martingale.


Introduction
Among the series of questions asked at the end of the chap.V of [12]) (or also in [13] and [15]) is the following question: a filtration being given on a probability space, how to recognize if it is generated by a Brownian motion or not? This question is especially of interest for a weakly Brownian filtration (there exists an F-Brownian motion which has the predictable representation property (PRP) with respect to F, see [11] for application of this important property). In all generality, there are weakly Brownian filtrations, which are not Brownian, as it is shown in [6], paper that was followed by other examples of non-Brownian filtrations given in [4], [7], [14]. These works are important progress that raises many new questions, including how to establish the non-Brownian character of a weakly Brownian filtration?
In all the works above, it is the notion of non-cosiness (introduced by Tsirel'son in [14] and that we will not discuss in this paper) of these filtrations which serves as a criterion to show that they are non-Brownian, see [4], [10] for different types of cosiness: I-cosiness, D-cosiness and T-cosiness. One might think that a filtration generated by a non-pure extremal martingale or non-extremal martingale can not be Brownian. In fact we show in Section 3 that this is not true. The non-Brownian character of a weakly Brownian filtration is much more delicate. Section 4 shows that Brownian filtration can be generated by non-pure extremal martingale. In section 5, we discuss the following property denoted by (*) in [1]: If M is a continuous martingale and F = F M , for every, F-stopping time T finite a.s such that P(M T = 0) = 0,then where G T = sup{s ≤ T, M s = 0}, T ∈ [0, ∞[. Authors of [1] have shown that property (*) is satisfied by any pure martingale. It is understood here that (*) is a property of a filtration rather than a martingale.

Preliminaries
We will only consider completed probability spaces and right continuous filtrations. We denote HdX the stochastic integral of H with respect to X and F X the natural filtration of X. An F−continuous local martingale X has the PRP (the predictable representation property) if for every F−continuous local martingale M there exists an F−predictable process H such that (this terminology is justified by the fact that the law of an extremal martingale is an extremal point in the convex set of all probability measures on W = C(R +, R), which make the coordinate process a local martingale). A continuous local martingale is the Brownian motion of Dubins-Schwartz (DDS) associated with X, which is equivalent to say that for all t, X t is F B ∞ −measurable. Every pure martingale is extremal but the opposite is not true. Yor has given in [15] an example of an extremal martingale which is not pure; we will prove here that its natural filtration is Brownian.
Definition 2.1. A filtration F is said to be immersed in a filtration G( defined on the same probability space) if any F-martingale is G-martingale.

Example of non-extremal martingale with Brownian filtration
We have the following characterization of extremal martingales with respect to Brownian filtration: If M is B−extremal, then there exists a B−predictable process K such that B = KdM and dλ = K 2 d M , that is d M is equivalent to λ. If now, d M is equivalent to λ, it is enough to represent B as a stochastic integral with respect to M. We have H = 0, λ ⊗ dP a.s so B = 1 H dM .
Lane [9], gave partial answers to the following question [12]: If B is a Brownian motion, f is borel function and M is the local martingale f (B) dB, under what conditions the filtration F M is Brownian?.An important example is when f ≥ 0 and µ ({f = 0}) > 0 but the set {f = 0} does not contain any interval (µ is the Lebesgue measure on R). This case was studied by knight [8] with F = {f = 0} is a subset of [0, 1], defined by the Cantor method: removing 3 8 , 5 8 then 5 32 , 7 32 and 19 32 , 21 32 and so on. We define the set F n by means of its complementary F c n , 3 Theorem 3.2. Let B be a Brownian motion, B its natural filtration and M the martingale defined by where the numbers (c k n ), n ≥ 1, k ∈ {1, ..., ℓ n }, c ′ and c ′′ are strictly positive and all different. The martingale M is not extremal and we have F M = B. Remark 3.3. In order not to burden the proof of Theorem 1, at the end of this paper (in the appendix) we have gathered some non-detailed points.
We consider the martingales Let us show that the sequence ( So, we know B T 1 and for every r ≥ 1 and t ∈ [S r , T r ] we have Then, if we know M and B T r , we can know B S r (and the inverse is true).
It remains the case where M T r − M S r = 0 so B T r = b (and then B T r = B S r ).Remark that Suppose we know M until time t, since we know B T 1 , then, from (2), we can know B S 2 and B T 3 and so on, we can know the sequence (B T r , B S r ) for T r , S r ≤ t.
To finish the proof, let t 0 ≤ t, the set {B t0 ∈ F c } is F M t0 -measurable (Point 2). If B t0 ∈ F c ,then there exists n and k such that B t0 ∈ A k n and so, there exists r such that t 0 ∈]S r , T r [. We have and equality (1) gives

Examples of extremal non-pure martingales with Brownian filtrations
We will now show that the filtration of the extremal non-pure martingale given in [15] is Brownian.
Theorem 4.1. Brownian filtration is generated by a non-pure extremal martingale.
Proof. Let B be a Brownian motion and B its natural filtration. We start by considering the stochastic equation . We easily check that: The function x 1+|x| is strictly increasing, we apply theorem 3.5(iii), chap.IX of [12] and we get F X = B. We have, X = ϕ 2 (X t )dt, since ϕ 2 is continuous and strictly decreasing 0 sgnγ s dγ s and γ is the DDS Brownian motion associated to X. We have X = M then It remains to show that M is extremal but non-pure. Since ϕ is strictly positive, d M is equivalent to Lebesgue measure and F M is a Brownian filtration, therefore, using Lemma 1, we deduce that M is Here is an other example of non-pure extremal martingale with Brownian filtration :   [12]: is there a strictly positive predictable process H such that the martingale N t = t 0 H s dB s is not pure?

A martingale class that satisfy property (⋆)
In [1], authors discussed a property (⋆) verified by all pure martingales and gave some examples of non-pure extremal martingales and non-extremal martingales that nevertheless satisfy property (⋆) . In [2], we better understand this property that we reset here: Let M be a continuous martingale and F = F M , for every, F-stopping time T finite a.s such that P(M T = 0) = 0, we have where G T = sup{s ≤ T, M s = 0}, T ∈ [0, ∞[. The example given in [1] of non-pure extremal martingale satisfying property (⋆) is in fact the example of Yor [15]. We have shown that its filtration is Brownian and therefore, it is obvious that this martingale satisfies (⋆) using Barlow's property proven in [2]. In the same way, our non-extremal martingale of Theorem 1, satisfies (⋆). In general, the following proposition can be stated: We define the sets C 1 = {P(A | F Gt ) = 0} and C 2 = {P(A c | F Gt ) = 0} which are in F Gt . We have P(A ∩ C 1 ) = 0 and P(A c ∩ C 2 ) = 0.
Here is an example of a filtration with SpM ult ≤ 2.
Definition 5.3. A filtration generated by a pure martingale is called pure filtration.
Proposition 5.4. Let F be a filtration, C = (C t ) time change for F and F = (F Ct ). We have: (a) SpM ult(F) ≤ SpM ult( F). If moreover C is strictly increasing, we have: SpM ult(F) = SpM ult( F).
(b) Let F be the natural filtration of a continuous martingale M and C the inverse of M .we suppose that M is strictly increasing and M ∞ = ∞. If F is Brownian, then M is extremal and F is pure.
Proof. (a) Suppose SpM ult( F) = n ∈ N * . Let M be F-spider martingale of multiplicity n + 1, bounded and M 0 = 0. Then M c = E[M ∞ | F] is F-spider martingale of multiplicity n + 1 vanishing at the origin, Proposition 13 of [2] gives M ∞ = 0 a.s and SpM ult(F) ≤ n. If C is strictly increasing and if τ is its inverse, then by Lemma 5.9 of [13], we have If F is pure, then there exists a time change which we also note C, such that F c is Brownian, then SpM ult( F) = 2 and SpM ult(F) ≤ 2.
(b) Let W be a Brownian motion that generates F and X the martingale W M (by construction, X is pure ).
Let us show that M is extremal: let B be the DDS Brownian motion of M , B is F− Brownian motion that has F − P RP (because F is Brownian ), as F C0 is trivial, F 0 is too, and M is extremal. Notice now that and . Hence X is F-extremal (and since it is extremal), Proposition 7.1 of [13], gives us that F X is immersed in F. So we have F = F X using (4).
The next question naturally arises: The reciprocal of proposition 1 is it true? i.e if all the martingales that generate a filtration F satisfy the property (⋆), do we have SpM ult(F) = 2?
For now, we do not have a general answer to this question. In any case, let us note that the following example given in [1] section 6, does not give a negative answer, let where (X t + iY t ) is a planar Brownian motion starting from z ∈ C * and α ∈] − ∞, 1 2 ]. Let F be the filtration of M , C the inverse of M and F = (F Ct ) t≥0 , F is Brownian, so F is pure and according to proposition 1, M satisfy property (⋆).