Characterization of max-continuous local martingales vanishing at infinity

We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by Nikeghbali and Yor [NY06] for continuous filtrations. Our generalization is complementary to the one presented by Kardaras [Kar14], and is obtained by means of similar tools.

It was first noted in [NY06] that honest times can be related to times of maxima of local martingales; see also [NP08], [NP13], [Kar14a]. In this note we shall provide a characterization of the following set of local martingales: M 0 := {L local martingale : L ≥ 0, L 0 = 1, L ∞ = 0, L * is continuous}, where L ∞ := lim t→∞ L t , and L * denotes the running supremum, L * t := sup 0≤s≤t L s for t ≥ 0. Such characterization will be given in terms of the times of maximum of the processes in M 0 , in the sense of the following definition. The specification "last" can be dropped in Definition 1.2 due to Corollary 2.6, where it is proved that ρ L is in fact the only time of maximum of L (i.e., the only time ρ such that L ρ− = L * ∞ ). The maximum is actually attained at ρ L (i.e., L ρ = L * ∞ ) if L has continuous paths, while this is in general not the case if L has jumps. As a consequence of our results, we will show that for L ∈ M 0 it holds that ρ L = sup{t > 0 : L t = L * t }, see Corollary 1.6.
1.2. State of the art and main theorem. Under the assumption of a continuous filtration (meaning that every martingale is continuous), Nikeghbali and Yor [NY06] prove the following characterization of M 0 . Theorem 1.3. Let F be continuous. Then a random time ρ satisfies ρ = ρ L for some local martingale L ∈ M 0 if and only if it is an honest time that avoids all stopping times. In this case, When the filtration is not continuous, this characterization is no more true in general, in the sense that there exist local martingales L in M 0 such that P [ρ L = τ ] > 0 for some stopping time τ ; see Example 1.8. Therefore, in order to obtain a similar characterization, one may either require some additional property on the local martingales, or alternatively consider a larger class of honest times. The first approach is the one adopted by Kardaras in [Kar14a], where the author considers those martingales in M 0 that do no jump at the time of maximum, obtaining the following characterization. On the other hand, in the present paper we look for the set of random times that characterize the times of maxima for the processes in the class M 0 , without any restriction. We shall therefore consider a larger class of honest times; see Remark 1.7. This constitutes the main result of this note.
Theorem 1.5. For a random time ρ the following are equivalent: (I) ρ = ρ L for one (and only one) local martingale L ∈ M 0 ; (II) ρ is the end of a predictable set and it avoids all predictable stopping times.
Corollary 1.6. Define the random time Then Proof. In [NP13] it is proved that, for L ∈ M 0 , the Azèma supermartingale Z ′ associated to ρ ′ L satisfies Z ′ = L/L * . Since clearly ρ L ≤ ρ ′ L , from Theorem 1.5 we get that the optional projection  Remark 1.9. The request that ρ is the end of a predictable set in Theorem 1.5-(II) cannot be weakened by asking ρ to be the end of an optional set, i.e. ρ being an honest time, as Example 1.10 shows.
Example 1.10. Let N be a Poisson process and M its compensated martingale. Define the process S := E(M ) and the random time τ := sup{t : S t = S * t }. Since S t goes to zero as t goes to infinity, τ is a finite honest time. Now, the Azéma supermartingale associated to τ is given by by (2.3). Therefore we have On the other hand, where we use the fact that P Z ρ = 1 = 1 since ρ is an honest time, see [ any nonnegative supermartingale can be written as the product of a local martingale and a nonincreasing process. The decomposition is in general not unique. One can however require conditions on the factorizing processes in order to identify a particular (unique) decomposition. This is what we want to do for the Azéma supermartingale associated to a random time. In [Kar14a] it is noted that the optional multiplicative decomposition given in [Kar14b] is useful for the characterization of honest times. Here we adopt a similar argumentation, using instead the predictable multiplicative decomposition given in [PR13]. (1) L is a non-negative local martingale with L 0 = 1; (2) D is a non-increasing predictable process with 0 ≤ D ≤ 1; (3) a t = − [0,t] L s− dD s for all t ∈ R + , with L 0− := 1; (4) Z t = L t D t , t ∈ R + .

As the continuity of the dual optional projection
so the continuity of the dual predictable projection a will be in proving the main results of this paper. Indeed, note that when a is continuous, then D is continuous as well, which is crucial in the proof of Proposition 2.4. This shows how in the present setting we cannot simply use the multiplicative decomposition considered in [Kar14a], based on the process A, since in our case A is no more continuous. However, in the particular case of ρ avoiding all stopping times, then a ≡ A and the decomposition in Proposition 2.3-(4) coincides with that in [Kar14a].  (ii) ⇒ (i). For u ∈ (0, 1], we have This implies that the function u → E[a ξu ] is continuous, with E[a ξu ] = 1 − u. Therefore, ∆a ξu = 0 for all u ∈ (0, 1], which implies ∆D ξu = 0 for all u ∈ (0, 1]. From this it follows that D is continuous, hence a is continuous, and this concludes the proof by Lemma 2.1-(b).
2.3. Doob's maximal identity. The following characterization of the law of the maximum of a local martingale is shown in [NY06], [Kar14a]. By applying Lemma 2.5 to the local martingale (L τ +t ) t≥0 in the filtration (F τ +t ) t≥0 , for any given stopping time τ , we obtain The equality holds if and only if L ∈ M 0 , in which case we have Corollary 2.6. Let L ∈ M 0 and let ρ be a time of maximum for L, in the sense that P [L ρ− = L * ∞ ] = 1. Then P [ρ = ρ L ] = 1, that is, the time of maximum is unique. Moreover, the Azéma supermartingale corresponding to ρ L is given by Z = L/L * .
Proof. By definition of ρ L we have P [ρ ≤ ρ L ] = 1 and, for any stopping time τ , Now, due to (2.4), all these sets coincide up to evanescence, and it follows that Z = L/L * . This also shows that P [ρ = ρ L ] = 1.
(II) ⇒ (I): Let ρ be the end of a predictable set such that it avoids all predictable stopping times, and let (L, D) be the pair of processes appearing in the multiplicative decomposition of Z in Proposition 2.3-(4). We first prove that ρ is a time of maximum for L, in the sense that P [L ρ− = L * ∞ ] = 1, and that L ∈ M 0 . Since ρ is the end of a predictable set, Z ρ− = 1 holds by [Azé72, Theorem 1.4]. Therefore, Proposition 2.3-(4) gives 1 = Z ρ− = L ρ− D ρ− = L ρ− D ρ . Then we have P [L ρ− > x] = P D ρ < 1 x = 1 x by Proposition 2.4, which implies P L * ρ− > x ≥ 1 x , x ∈ R + . On the other hand, P [L * ∞ > x] ≤ 1/x by Lemma 2.5, and so P [L ρ− = L * ∞ ] = 1, and L ∈ M 0 . Now, ρ = ρ L and Z = L/L * follow from Corollary 2.6. The fact that ρ identifies a unique L ∈ M 0 is shown in Remark 2.7, and this concludes the proof.
Remark 2.7. Note that, as in [NY06], by applying Itô's formula to Z = L/L * we get that a = log(L * ), hence Z has the following Doob-Meyer decomposition In this section we recall the notions of optional (resp. predictable) projection and of dual optional (resp. predictable) projection, which play a fundamental role in this note.
A.1. The optional and predictable projections. Let X be a non-negative measurable process.
The optional projection of X is the unique (up to indistinguishability) optional process Y such that for every stopping time τ .
The predictable projection of X is the unique (up to indistinguishability) predictable process N such that for every predictable stopping time τ .
A.2. The dual optional and predictable projections. Let H be an integrable raw increasing process. The dual optional projection of H is the optional increasing process U defined by for any bounded optional process X.
The dual predictable projection of H is the predictable increasing process V defined by for any bounded predictable process X.
A.3. The additive decompositions in (2.1). Let A and a denote the dual optional and predictable projections of the process H t := I {ρ≤t} , respectively. From the previous section we have that, for every bounded predictable process X, Fix t ∈ R + and F t bounded F t -measurable, then X s := F t 1 {t<s} defines a predictable process, and from (A.1) we have

This in turn yields
which is exactly equation (2.1).