Jacod’s criterion and universal supermartingale densities

Let (Ω,F,(F_t)t≥0, P) be a filtered probability space, and let (G_t⁰)t≥0be an initial filtration enlargement of (F_t), by which we mean that there there exists a random variableX such thatG_t⁰ =F_t∨σ(X) for all t≥0. We define the right-continuous regularization of (G_t⁰) by settingG_t:=^_s>tG_s⁰ for all t≥0. Note that we do not require F, (F_t)t≥0, or (G_t)t≥0

to be complete, contrary to Jacod [Jac85] (although we allow them to be complete).

Recall that Hypothèse (H^′) is satisfied if all (F_t)–semimartingales are (G_t )-semimartin-gales.

We now give the classical formulation of Jacod’s criterion, see [Jac85]. For this purpose we need to assume thatX takes its values in a standard Borel space, which we denote by (X,B). For the definition of standard Borel spaces see Parthasarathy [Par67], Definition V.2.2. For a detailed discussion see also Dellacherie [Del69], where standard Borel spaces are referred to as Lusin spaces. Note that (X,B) is a standard Borel space provided that Xis a Polish space and Bits Borel σ–algebra.

If X takes its values in the standard Borel space (X,B), then the regular conditional distribution

Pt(ω,dx) :=P(X ∈dx|F_t)(ω)

exists for all t ≥ 0, see Durrett [Dur10], Theorem 5.1.9 (Durrett calls standard Borel spaces “nice spaces”). We write PX for the distribution of X. Jacod’s criterion states that Hypothèse (H^′) is satisfied provided that for every t≥0 almost surely

Pt(ω,dx)≪PX(dx). (1.27)

Note that this statement only makes sense if the set {ω : Pt(ω,dx) ≪ PX(dx)} is F–

measurable. But since theσ–algebra of a standard Borel space is countably generated (see also [PR13]), it is easily verified that this is indeed the case. Below we give an alternative

proof of Jacod’s result, and we relate it to the existence of a universal supermartingale density.

First observe that Hypothèse (H^′) is satisfied if and only if all nonnegative (F_t)–

martingales are (G_t)–semimartingales: This follows by decomposing every (F_t)–local martingale into a sum of a locally bounded local martingale and a local martingale of finite variation, by observing that every bounded process can be made nonnegative by adding a deterministic constant, and from the fact that local semimartingales are semimartingales (see Protter [Pro04], Theorem II.6).

Definition 1.5.1. Let (G_t) be a filtration enlargement of (F_t). LetZ be a (G_t)–adapted process that is almost surely càdlàg, such that P(Zt > 0) = 1 for all t ≥ 0. Then Z is called universal supermartingale density for (G_t) ifZM is a (G_t)–supermartingale for every nonnegative (F_t)–supermartingale M.

Note that here we do not require Z∞ to be positive, unlike in the previous sections.

This is because here we are interested in the semimartingale property and not primarily in the (NA1) property. Local semimartingales are semimartingales, and therefore it suffices to verify the (G_t)–semimartingale property ofM on [0, t] for everyt≥0. Hence, it suffices ifZ_t>0 for every t≥0.

Also note that we required ZM to be a (G_t)–supermartingale for every nonnegative (F_t)–supermartingaleM, and not just for nonnegative (F_t)–martingales. This has the ad-vantage that now we see immediately that in finite time every process satisfying (NA1) under (F_t) satisfies also (NA1) under (G_t), provided that there exists a universal su-permartingale density Z: if Y is a (F_t)–supermartingale density for S, then ZY is a (G_t)–supermartingale density forS. To extend this result to infinite time, we would have to change the definition of a universal supermartingale density by additionally requiring thatP(Z∞>0) = 1. To obtain a universal supermartingale density under Jacod’s crite-rion, we would then have to assume that also almost surelyP∞(ω,·) ≪P_X(·). Here we do not pursue this further.

The first result of this section shows that Jacod’s criterion is not so much a crite-rion for Hypothèse (H^′) to hold, but rather a criterion for the existence of a universal supermartingale density.

Proposition 1.5.2. Let (G_t)be the right-continuous regularization of an initial enlarge-ment of (F_t) with a random variable X taking its values in a standard Borel space. As-sume Jacod’s criterion (1.27) is satisfied. Then there exists a universal supermartingale density for (G_t).

Proof. 1. Let t ≥ 0. Without loss of generality we may assume that dPt(ω,·) ≪ dP_X(·) for all ω ∈ Ω. This can be achieved by setting P_t(ω,·) := 0 on the mea-surable set {ω : Pt(ω) does not satisfy Pt(ω,·) ≪ PX(·)}. Now we can apply a theorem of Doob, see [YM78], according to which there exists aF_t⊗ B–measurable random variable Y_t : Ω×X→ R+, such that for every ω ∈Ω we have P_X–almost

surely

Y_t(ω, x) = dP_t(ω,·) dP_X (x).

Note that Yor and Meyer [YM78] do not require complete σ–algebras. Let now t, s≥0. We first show thatP ⊗P_X–almost surely

{(ω, x) :Y_t(ω, x) = 0} ⊆ {(ω, x) :Y_t+s(ω, x) = 0}. (1.28) Note that Yt+s ≥ 0, and therefore Fubini’s theorem and the tower property of conditional expectations imply that

This Z is (G_t)–adapted by construction. Let now M be a nonnegative (F_t)–

supermartingale. Let s, t ≥ 0, let A ∈ F_t, and B ∈ B(X). Then we can apply surely nonnegative. Using the (F_t)–supermartingale property of M in conjunction with Fubini’s theorem, we obtain

=



Ω

1_A(ω)



X

1_B(x)Mt(ω)Y_t(ω, x)

Yt(ω, x)1_{Yt(ω,x)>0}PX(dx)P(dω)

=



Ω

1_A(ω)



X

1_B(x)Mt(ω)Z^t(ω, x)Pt(ω,dx)P(dω) =E(1_A1_B(X)MtZt) The monotone class theorem allows to pass from sets of the formA∩X⁻¹(B) to general sets in (G_t⁰), and thereforeM Z is a (G_t⁰)–supermartingale. Taking M ≡1, we see that also Z is a (G⁰_t)–supermartingale.

3. Let us show that Z_t is P–almost surely strictly positive for every t ≥0. For this purpose it suffices to show thatP(ω:Yt(ω, X(ω)) = 0) = 0. By the tower property we have

E(1_{Yt(·,X(·))=0}) =



Ω



X

1_{Yt(ω,x)=0}P_t(ω,dx)P(dω) = 0.

4. Z is not necessarily right-continuous, and also we did not show yet that ZM is a (G_t)–supermartingale and not just a (G_t⁰)–supermartingale. But the construction of a right-continuous universal supermartingale density is now done exactly as in the proof of Theorem 1.3.1. The (G_t)–supermartingale property ofZM follows also in the same way as in the proof of Theorem 1.3.1.

Remark 1.5.3. If we are only interested whether Hypothèse (H^′) holds and not whether there exists a universal supermartingale density, then we can also work with the filtration (G_t⁰) and not with its right-continuous regularization (G_t). Since Hypothèse (H^′) holds for (G_t) and since (G_t⁰) is a filtration shrinkage of (G_t), Stricker’s theorem implies that Hypothèse (H^′) is also satisfied for (G_t⁰).

Remark1.5.4. We could replace assumption (1.27) byPt(ω,dx)≫PX(dx) orPt(ω,dx)∼ P_X(dx). In the first case we could use the same proof as for Proposition 1.5.2 to obtain the existence of a nonnegative martingaleZ, not necessarily strictly positive, such thatZM is a (G_t)–supermartingale for every nonnegative (F_t)–supermartingaleM. In particular, then there exists an absolutely continuous measure Q ≪ P, such that every locally bounded (P,(F_t))–local martingale is a (Q,(G_t))–local martingale. Since (NA) is related to the existence of absolutely continuous local martingale measures, see [DS95b], this indicates that the (NA) property may be stable under initial filtration enlargements that satisfy this “reverse Jacod condition”. Note that it is much harder to satisfy this assumption. For example it will never be satisfied ifX isF_t–measurable for some t≥0.

IfP_t(ω,dx)∼P_X(dx), then the same proof as for Proposition 1.5.2 yields the existence of an equivalent measureQ∼P, such that every nonnegative (P,(F_t))–supermartingale is a nonnegative (Q,(G_t))–supermartingale. In particular, then every locally bounded (P,(F_t))–local martingale is a (Q,(G_t))–local martingale. This condition has been stud-ied by Amendinger, Imkeller and Schweizer [AIS98], as well as Amendinger [Ame00].

Obviously it is harder to satisfy than Jacod’s condition or the reverse Jacod condition.

In financial applications one may however assume that the knowledge of the “insider” is

perturbed by a small Gaussian noise that is independent of F∞ (or more generally by an independent noise with strictly positive density with respect to Lebesgue measure).

Then P_t(ω,dx)∼P_X(dx) is always satisfied.

1.5.2. Universal supermartingale densities and the generalized Jacod