Filtration shrinkage, the structure of deflators, and failure of market completeness

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.


Introduction
Optional projections of martingales to smaller filtrations retain the martingale property; for the class of local martingales, the latter preservation may fail. For instance, the projection of a nonnegative local martingale can only be guaranteed to be a supermartingale in the smaller filtration, but might fail to be a local martingale; see [Str77] and [FP11].
Positive local martingales appear naturally as deflators in arbitrage theory. (See Section 1 for definitions and a review of classical concepts in the theory of no-arbitrage.) Consider two nested, right-continuous filtrations F · ⊆ G · and a continuous and F · -adapted process S, having the interpretation of the discounted price of a financial asset. Then, the existence of a strictly positive G · -local martingale Y such that Y S is also a G · -local martingale is equivalent to the so-called absence of arbitrage of the first kind. If no such arbitrage opportunities are possible under G · , then the same is true under the smaller filtration F · ; we refer to Section 1 for a rigorous argument of this assertion. Hence, there must exist an F · -local martingale L such that LS is an F · -local martingale. Let now Y G· and Y F· denote the set of all G · -adapted and F · -adapted local martingale deflators, respectively. The above no-arbitrage considerations yield the implication It is natural to ask at this point if there is a direct way to construct an element of Y F· from a given Y ∈ Y G· . The optional projection o Y of Y on F · is not necessarily an F · -local martingale, as discussed above; hence it cannot be expected to be in Y F· . However, as our first main result, Theorem 2.1, implies, the local martingale part L of the multiplicative Doob- is an element of Y F· . Section 2 contains the proof of Theorem 2.1 and related results.
The previous motivates another natural question: when does the projection of Y lose the local martingale property, that is, under which circumstances is it the case that K ∞ > 0? In Section 3, we investigate this question from a Bayesian viewpoint. As it turns out, whenever certain models (which were possible under the Bayesian prior) become impossible under the observed data (the stock price path, in this case), the projection of the deflator Y loses the local martingale property, and K increases. In Section 4, we generalise the Bayesian viewpoint, under the assumption that a certain dominating probability measure exists.
Markets admitting local martingale deflators are complete if, and only if, such a deflator is unique. Since different local martingale deflators in Y G· might have the same projection, it is easy to find an example such that a market is incomplete under G · but complete under F · . Indeed, consider a complete market under F · and add an independent Brownian motion to get to a filtration G · ; then, the market is automatically incomplete under G · .
The reverse question is of more interest: given that the market is complete under G · , is it also complete under F · ? As it turns out, this is not always true; it is possible that certain F ·local martingale deflators do not result from the local martingale component of a projections of G · -local martingale deflators, and completeness in financial markets may be lost when we pass to smaller filtrations. We provide an explicit example in Section 5. This counterexample uses the Lévy transformation B of a standard Brownian motion W , namely where Λ is the local time of W at zero; see [RY99, Theorem VI.1.2]. In fact, we provide a rather general class of counterexamples, whose construction is of independent interest. To wit, let F W · and F B · denote the smallest right-continuous filtrations making W and B measurable, respectively. Both W and B are standard Brownian motions with the predictable representation property on F W · . Furthermore, B is a standard Brownian motion with the predictable representation property on F B · , and it holds that F B · = F
The information lost when passing from W to B consists of the signs of the excursions of W ; in view of [Blu92,page 114], conditional on F B ∞ = F |W | ∞ , these signs are independent and identically distributed. Given that there are countably many excursions of W , there clearly exist non-deterministic F W ∞ -measurable random variables, which are independent of F B ∞ . As we argue in Theorem 5.1 in Subsection 5.2, one may construct such random variables in an F W · -adapted way: there exist F W · -stopping times with any prescribed probability law on the positive real line, independent of F B ∞ . The last result provides an interesting corollary: the existence of two nested filtrations F · ⊆ G · and a one-dimensional continuous stock price process S, adapted to F · , such that the market is complete under G · and under F · , but not under some "intermediate information" model.

Notation, Definitions, and Review of Some Classical Results
In this section, we introduce the framework, and recall certain classical results which shall find use later on.
Fix a probability space (Ω, G ∞ , P), equipped with two right-continuous filtrations F · ≡ (F t ) t≥0 and G · ≡ (G t ) t≥0 , which are nested in the sense that F · ⊆ G · , i.e., F t ⊆ G t holds for all t ≥ 0. For a given process X = (X t ) t≥0 , we use F X · to denote the smallest right-continuous filtration that makes X adapted. If X is additionally nonnegative, let o X denote its F ·optional projection, which always exists but could take the value ∞; see, for example, [Nik06,Theorem 4.1]. If X is a semimartingale, we use E(X) to denote its stochastic exponential.
We consider an F · -adapted continuous-path G · -semimartingale S, representing the price of a financial asset expressed in terms of a certain denomination. Everything that follows carries over to the case of a multi-asset case S = (S 1 , · · · , S d ) for d ∈ N, at the expense of more complicated notation; we refrain from considering multi-asset models as notation is already a bit heavy. However, we stress that continuity of the paths of S will be important, as we shall explain at places.
The financial notions below can be considered under different filtrations, so we use H · to generically denote either the "small" F · or the "large" G · filtration.
For given x ≥ 0, let X H· (x) denote the set of all nonnegative wealth processes, i.e., all nonnegative processes V x,θ of the form where θ is H · -predictable and S-integrable. We set X H· := x≥0 X H· (x).
to be the hedging capital associated with ξ.
The concept of viability (for a specific filtration) is also known as absence of arbitrage of the first kind, or as the condition of no unbounded profit with bounded risk.
The class of all H · -local martingale deflators will be denoted by Y H· . [Kar10]). The following statements are equivalent: (1) The market is H · -viable.
(3) There exists an H · -supermartingale deflator. Note that the structural condition (4) in Theorem 1.3 above implies that S-integrability of an H · -predictable process θ amounts to since the validity of (1.1) and the Cauchy-Schwartz inequality already imply that, P-a.e., In particular, under the structural condition (4), H is S-integrable, and we may define the specific H · -local martingale deflator Y = 1/ V , where The above V is a special wealth process in X H· (1) called the H · -numéraire.
Remark 1.4. The nesting property F · ⊆ G · seems to yield directly that G · -viability implies F · -viability. (1.2) However, the implication in (1.2) is a bit more subtle, the reason being that the inclusion X F· ⊆ X G· is not in general true when F · ⊆ G · . Indeed, an F · -predictable process θ might be S-integrable under F · , but not under G · . For example, assume that F · is the natural filtration of a Brownian motion W and that G · is the smallest right-continuous filtration that makes W adapted and W 1 a G 0 -measurable random variable; the process θ given by but not under G · ; hence, in this example, X F· ⊆ X G· .
The previous remarks notwithstanding, G · -viability implies that X F· ⊆ X G· . Indeed, if an F · -predictable R d -valued process θ is S-integrable, then it satisfies (1.1) a fortiori, which then implies that it is also S-integrable in the filtration G · in view of G · -viability and the discussion right before this Remark. Therefore, since G · -viability implies X F· ⊆ X G· , (1.2) follows.
A wealth process X ∈ X H· is called H · -maximal if, whenever X ′ ∈ X H· is such that Definition 1.5. The market S is called H · -complete if, for any T > 0 and ξ ∈ L 0 + (H T ) with x = x H· (T, ξ) < ∞, there exists a maximal X ∈ X H· (x) such that P[X T = ξ] = 1.
and K a nondecreasing F · -predictable [0, 1)-valued process with K 0 = 0. Then, L is an F ·local martingale deflator for S. Proposition 2.2. Let Y be a G · -supermartingale deflator for S. Then its F · -optional pro- Proof. For any 0 ≤ s ≤ t < ∞ and X ∈ X F· ⊆ X G· , thanks to Remark 1.4, it holds that It remains to show that o Y is strictly positive. For this, fix t ≥ 0, and note that The filtration in the statement of Proposition 2.3 below is implicit. Proof. Since a supermartingale deflator for S exists, the market is viable by Theorem 1.3; therefore, upon performing a change of numéraire, we may assume that S is a continuous local martingale; see [DS95] and [KK07,Theorem 4.12]. The Kunita-Watanabe decomposition yields the representation L = E( · 0 θ u dS u )N , where θ is predictable and S-integrable, and N is a strictly positive local martingale such that [N, S] = 0 holds. In order to prove the statement, it now suffices to show that L = N , i.e., · 0 θ u dS u = 0. For each n ∈ N, consider the strictly positive wealth process E(n · 0 θ u dS u ) ∈ X . Since is a supermartingale, N is a strictly positive local martingale strongly orthogonal to the where C is nondecreasing and predictable, with ∆C < 1. Then, S] u has to be a non-increasing process. Since this has to hold for all n ∈ N, we obtain · 0 θ 2 u d[S, S] u = 0, which is the same as · 0 θ u dS u = 0.
2.2. Ramifications. As mentioned after Theorem 1.3, a special G · -local martingale deflator is the one corresponding to the reciprocal of the G · -numéraire in X G· . It is natural to ask whether the F · -optional projection of the reciprocal of the G · -numéraire is the reciprocal of the F · -numéraire. The following example shows that this is not necessarily the case, even if the reciprocal of the G · -numéraire is a G · -martingale. For a positive result in this direction, under additional assumptions, we refer to Proposition 4.10 later on. for q ∈ (0, 1). The filtration G · is given by Define the wealth process V ∈ X G· by where we have used the fact that Θ = Θ 2 since Θ is {0, 1}-valued. It is straightforward to check that Y := 1/ V = E(−ΘW ) is a G · -local martingale deflator, which is obviously a G · -martingale. Moreover, Θ is F 1 -measurable, yielding F t = G t for all t ≥ 1 and Straightforward computations give Hence, o Y has a jump at time t = 1. Since the F · -numéraire has continuous paths, its reciprocal clearly cannot equal o Y .
We now provide a result concerning the dynamics of S in the smaller F · -filtration. To make headway, note that Theorem 1.3 yields some G · -predictable process G and some G ·local martingale M such that Then, the F · -predictable projection F of G exists and satisfies Proof. Without loss of generality, and upon using F · -localisation, we may assume that the F · - An appropriate modification of [Mey73, Theoreme 1 ′ ] then yields that the dual optional F · - As the next example illustrates, although (2.3) always holds for some F · -predictable process F , the predictable F · -projection of G does not need exist in general. (We are grateful to Walter Schachermayer, who proposed the idea for this example.) Example 2.6. Let Ω = N × C([0, ∞); R). Define Θ(θ, w) = θ and W t (θ, w) = w t for all (θ, w) ∈ Ω and t ∈ [0, ∞). Let G · denote the smallest right-continuous filtration making Θ a G 0 -measurable random variable and W adapted. Consider any probability measure µ on 2 N with θ∈N θµ[{θ}] = ∞, and let P denote the product probability on G ∞ of µ and Wiener measure. Note that E[Θ] = ∞ and that Θ and W are independent under P.
Theorem 1 in [PS10] and a simple conditioning argument yield the existence of a G ·predictable process H taking values in {−1, 1}, such that the process Note also that Θ and X are independent under P, again by a conditioning argument. Thus we have Defining now S := E(X), (2.1) holds with M := · 0 S u H u dW and G := HΘ/S. In particular, · 0 G 2 u du = Θ 2 · 0 S −2 u du < ∞ holds. As shown above, (2.2) fails. Nevertheless, (2.3) holds with F = 0 and N = X; here, F is not the predictable F · -projection of G, as the latter does not exist.
To see this, we may assume by localization that the F · -measurable processes [S, S] and are uniformly bounded, say by a constant κ > 0. It now suffices to observe that here the last inequality uses the fact that the stochastic integrals have bounded quadratic variation. This then yields (2.2).
For example, assume that G · supports a Brownian motion W and a G 0 -measurable R-valued Consider now a filtration F · with F · ⊆ G · such that S and sign(H) are F · -adapted. With

A Bayesian Framework
3.1. Set-up. Consider some parameter space R, equipped with sigma algebra R and probability measure µ, which will be the "prior" law of a parameter. Let Ω = R × C([0, ∞); R).
Define Θ(θ, x) = θ and X t (θ, x) = x t for all (θ, x) ∈ Ω and t ∈ [0, ∞). Define F · as the smallest right-continuous filtration making X adapted; i.e., F · = F X · . Moreover, let G · be the smallest right-continuous filtration containing F · and further making Θ a G 0 -measurable random variable. Next, define Q as the product probability measure on G ∞ of µ and Wiener measure; under Q, X is a G · -Brownian motion independent of Θ, the latter random variable having law µ. Also, under Q, X is an F · -Brownian motion. Let W = Q| F∞ denote Wiener measure.
Consider a functional G : Ω×[0, ∞) → [−∞, ∞], assumed to be G · -optional, which will serve as the drift functional for the stock returns in the filtration G · . We allow G to take the values ±∞, although such values will not be "seen" by the solutions of the martingale problems we consider later. Define also A : For µ-a.e. θ ∈ R, we assume the existence of a probability P θ on F ∞ such that P θ ≪ Ft W for all t ≥ 0, · 0 |G(θ, X, u)|du is P θ -a.e. finitely valued and X − · 0 G(θ, X, u)du is an F · -local P θ -martingale.
Some remarks are in order. First of all, under the previous assumptions, the process W θ := X − · 0 G(θ, X, u)du is actually an (F · , P θ )-Brownian motion, as follows from Lévy's characterisation theorem. Secondly, defining the set-valued process Σ : Girsanov's theorem implies that which, in particular, implies that P θ is necessarily unique. Thanks to [SY78, Proposition 5] and a localisation argument, ζ : may be chosen to be jointly measurable by taking an appropriate version.
, and note that W := X − · 0 G(Θ, X, u)du is a standard (G · , P)-Brownian motion; in particular, W and Θ are independent under P.
In order to connect with the financial setting of the previous sections, one may define the asset price S to equal to X or, if one insists on positive asset prices, one may set S = E(X). Choosing one or the other is plainly a matter of interpretation, and will not affect the mathematical content of the discussion here. The fact that P θ [θ ∈ Σ t ] = 0 for all t ≥ 0, equivalent to finiteness of the process A(θ, x, ·) = · 0 G(θ, x, u) 2 du, implies by Theorem 1.3 the F · -viability of the P θ -model for all θ ∈ R.
We shall be interested in the dynamics of X on F · under P. For this, we shall make one final assumption (recall also the discussion in Remark 2.7): Under the force of all the previous assumptions, Bayes' formula yields In fact, upon defining the random measure-valued process (µ t ) t≥0 via . Therefore, defining the functional , whenever The G · -local martingale deflators for S = X (or S = E(X)) are of the form Note that K h is a nondecreasing, F · -predictable process. In particular, there is "loss of mass" exactly when certain models become impossible. If the conditional law of Θ under P given F t maintains the same support as Θ has for all t ≥ 0, it follows that K h = 0. By Theorem 2.1, 1/ζ is an F · -local martingale deflator. This uses the fact that 1/ζ is indeed an (F · , P)-local martingale since ζ is an (F · , Q)-Brownian martingale, hence continuous, not jumping to zero.
Proof. Simply note that Given that ζ is the density process of P with respect to W on F · , we have which immediately gives the result.
As a corollary of the above, we obtain that the "default" (in the terminology of [ELY99]) of the (F · , P)-local martingale 1/ζ equals This is clear: 1/ζ will be an (F · , P)-martingale if and only if P and W are locally equivalent, which will happen exactly when, under W, a strictly positive µ-measure of densities are still strictly positive.
Remark 3.2. The (F · , P)-market is complete. Indeed, fix some T ≥ 0 and some nonnegative ] < ∞ and the martingale representation theorem gives the existence of an Example 3.3. Let µ be an arbitrary law on R := R, and set G(·, ·, θ) : holds; hence Σ = ∅ in this case. Moreover, we have Hence, all the assumptions of the present section, including (3.1), are satisfied and for some (G · , P)-Brownian motion W and some (F · , P)-Brownian motion W F· . However heavytailed the law of Θ may be (and even if it does not have any moments), its generalised conditional expectations given F · exist. This example with H = 1 is discussed in [Kai71]; see also Remark 2.7. and, correspondingly, lim t↑∞ X t = ∞ on {Θ < 0}, P-a.e.
Let us also note that the distribution of the overall maximum X * ∞ := max t≥0 X t can be computed in this setup. To this end, fix y > 0 and recall from (3. Here we used the facts that the the (F · , P θ )-local martingale 1/ζ θ satisfies 1/ζ θ ∞ = 0 for each θ > 0. Hence we get P[X * ∞ > y] = µ −∞, Similar computations hold also for the overall minimum of X.
Remark 3.5. Explicit formulas for the quantities in Example 3.4 may be obtained for nice laws µ. For example, if µ[dθ] = 1 θ>0 θ −3 e −1/θ dθ for all θ ∈ R (inverse Gamma distribution), one This then yields Θ = 1/(1 + X * ), where X * := max u∈[0,·] X u is the running maximum of X, We thus obtain giving, in conjunction with (3.3), that the limiting conditional law for Θ is in other words, 1/Θ − X * ∞ given F ∞ has the standard exponential law under P. Moreover, (3.4) yields that X * ∞ has also the standard exponential law under P. Hence 1/Θ is the sum of the two independent standard exponentially distributed random variables 1/Θ − X * ∞ and X * ∞ . Note also that the overall maximum X * ∞ of X has the same distribution as the overall maximum of Brownian motion with drift rate −1/2; see, for example, [KS91, Exercise 3.5.9].
Remark 3.6. Fix t > 0 and an F t -measurable nonnegative random variable ξ, representing the payoff of a contingent claim. As already observed in Remark 3.2, the (F · , P)-market is complete. Indeed, the price p of ξ in the (F · , P)-market equals by Lemma 3.1. Similarly, in the (G · , P)-market, one has the G 0 -measurable price p Θ , where It is clear, both by economic and by mathematical reasoning, that p Θ ≤ p, P-a.e.
Let us now consider the question how p and ess P sup p Θ relate (that is, how does the hedging cost of an "uninformed" agent relate to the worst-case hedging cost of an "informed" agent) in the context of Example 3.4. Using the fact that Σ = R \ (θ, θ), we have (This is for example the case when ξ = 1.) Then we have Therefore, even the worst-case hedging cost of the informed agent is strictly smaller than the uninformed agent's hedging cost. Note that, in all cases, the replication strategy for the informed agent starting from p Θ depends on Θ. In the case ess P sup p Θ < p, the superreplication strategy of the informed agent starting from deterministic amount ess P sup p Θ also depends on Θ; however, when ess P sup p Θ = p, no knowledge of Θ is required in order to (super)replicate starting from p.

Under the Presence of a Dominating Measure
We now consider a more general setup than in Section 3. We start from the existence of a G · -deflator Y ; that is, Y is a strictly positive G · -local martingale. Moreover, we shall make the following assumption.
Assumption 4.1. There exists a probability measure Q and a (G · , Q)-martingale Z such that (dP/dQ)| Gt = Z t for all t ≥ 0 and Z = 1/Y , P-a.e.
We refer to [Föl72] and [PR15] for sufficient conditions for the existence of such a probability measure Q and process Z. Note, in particular, that P ≪ Ft Q holds for all t ≥ 0.
In the sequel, we shall need to consider optional projections under both probabilities P and Q; therefore, for the purposes of this section, we shall denote explicitly, via a superscript, the probability under which the projection is considered.
Under Assumption 4.1, Bayes' rule yields where we have introduced the G · -stopping time To preserve uniqueness of the multiplicative decomposition we assume that M = M ρ and K = K ρ , where ρ is the first time that Q [ Z · > 0| F · ] hits zero, and additionally that ∆M ρ = 0 on the event {K ρ = 1}.
Note that the Bayesian setup of Section 3 leads to Z = ζ Θ and M = 1.
Let us collect some properties on these processes that we have introduced so far.
Proposition 4.2. In the notation of this section, and under Assumption 4.1, the following statements hold.
(1) The process 1/ o Z Q is an (F · , P)-supermartingale, and satisfies (2) The process M/ o Z Q is an (F · , P)-local martingale. Hence, the right-hand-side of (4.1) also leads to the multiplicative Doob-Meyer decomposition of the (F · , P)-supermartingale o Y P .
Proof. Thanks to (dP/dQ)| Ft = o Z Q t we have (4.5), which then yields the statement in (1). Fix now s, t ≥ 0 with s < t and A ∈ F s and let τ denote any bounded F · -stopping time such that M τ is an (F · , Q)-martingale and o Z Q is uniformly bounded away from zero on Hence M τ /( o Z Q ) τ is an (F · , P)-martingale.
Let now (τ ′ n ) n∈N denote an (F · , Q)-localization sequence of M and let τ ′′ n denote the first time that o Z Q crosses the level 1/n, for each n ∈ N. Defining now τ n = τ ′ n ∧ τ ′′ n for each n ∈ N we get lim n↑∞ τ n = ∞, P-a.s, M τn is an (F · , Q)-martingale, and o Z Q is uniformly bounded away from zero on [[0, τ n [[. This then yields statement (2).
Under any of the above equivalent conditions, it holds that M = 1, Q-a.e.; hence also • Proof. Let us first assume that statement (1) holds, i.e., Next, since M is also an (F · , Q)-local martingale we obtain that M = 1, Q-a.e. Then again recalling (4.6) yields that K is {0, 1}-valued, Q-a.e.
Example 4.4. Assume that the underlying probability space, equipped with the probability measure Q, supports a Q-Brownian motion W and an independent R-valued random variable Θ. Consider the filtration G · to be the smallest that makes W adapted, and such that Θ is G 0 -measurable. Moreover, consider the G · -stopping time Consider also the nonnegative (G · , Q)-supermartingale Since Z is continuous it is indeed a (G · , Q)-local martingale. We shall assume from now on that Q[Θ ∈ {0} ∪ [1/2, ∞)] = 1, as this is a necessary and sufficient condition for Z to be a (G · , Q)-martingale, by the arguments in [Ruf15].
Set now S := E(W ) and F · := F W · and define the F · -predictable time Then In the later case, • Y P = 1 is an (F · , P)-martingale.

Proof. Note that
This yields the equivalence of statements (1) and (2). For the equivalence of statements (2) and (3) Example 4.7. Assume that the underlying probability space, equipped with the probability measure Q, supports a Q-Brownian motion W and an independent random variable Θ with 1]. Consider the filtration G · to be the smallest right-continuous one that makes W adapted, and such that Θ is G 0 -measurable. Moreover, consider the G · -stopping times Note that Q[0 < ν < ρ < ∞] = 1, and that τ 0 equals Q-a.e. either ν, ρ or ∞ (the latter infinite value if, and only if, Θ = 0). Consider also the nonnegative (G · , Q)-martingale Z := 1−ΘW τ 0 .
Let F · be the smallest filtration that makes W adapted, and such that Θ is F ρ -measurable.
Under F · , Θ is only revealed at ρ, as opposed to G · where Θ is known from the beginning of time. We set S := E(W ), which is both a (G · , Q)-and an (F · , Q)-martingale. We also note that ν and ρ are F · -predictable times. Observe that, despite the different filtration structure, this example resembles Example 3.4. In both cases, W is Brownian motion conditioned to never hit level 1/Θ.
With the above set-up, we compute Hence, we have with the understanding that M = 1 if q = 1. Note that M is a bounded (F · , Q)-martingale.

Moreover, straightforward computations give
Hence, when q ∈ (0, 1), P-a.e., which is a bounded (F · , P)-martingale; however, when q = 1, then P-a.e., which can be seen to be a strict local (F · , P)-martingale. These observations are consistent with the result of Proposition 4.6.
We next modify Example 4.7 to illustrate that it is also possible that the local martingale part M in the multiplicative decomposition of (Q[τ > t|F t ]) t≥0 is continuous. and let G · denote the smallest right-continuous filtration that makes W adapted, and contains all the information of B already at time 0. Consider the process ψ := √ 2 · 0 exp(−u)dB u , and note that Θ := ψ ∞ is G 0 -measurable with standard normal distribution, and that the conditional law of Θ given F t is Gaussian with mean ψ t and standard deviation exp(−t) for each t ≥ 0. Set, as before, τ 0 := inf{t ≥ 0 | ΘW t = 1}, and consider the nonnegative (G · , Q)-martingale Z := 1 − ΘW τ 0 .
With θ := 1/ inf u∈[0,·] W u and θ := 1/ sup u∈[0,·] W u , note that {τ 0 > t} = {θ t < Θ < θ t }. It follows that where Φ denotes the standard normal distribution function. Writing the dynamics of the above, we see that the local martingale part in the additive decomposition of A has non-zero quadratic variation everywhere. The same properties carry over to the multiplicative decomposition, yielding that M is a Brownian local martingale with strictly increasing quadratic variation.
The next example has similar features as the setup of Section 3, in the sense that he projection of the local martingale deflator loses mass whenever in the small filtration one learns about the sign of an excursion of a Brownian motion. This example also relates to the framework of the following section. . Let us write where Λ denotes local time of W at zero. We now set S := E(B) and consider the process We claim that Z has continuous paths and is a (G · , Q)-martingale. To see path-continuity, note that just before τ 0 the process 1 + W − B = 1 + W − |W | + Λ behaves like twice a Brownian motion hitting level zero, given that Λ will be flat (since W is away from zero); then, it suffices to note that · 0 (1 + β u ) −2 du explodes at the first time that a Brownian motion β hits −1. Path-continuity of Z, coupled with its definition, implies that it is a (G · , Q)-local martingale. To see the actual martingale property of Z, we follow the arguments in [Ruf15], where we send the reader for further details. To this end, consider the Föllmer measure P, given by the extension of the measures defined via the Radon-Nikodym derivatives (Z τn∧n ) n∈N on the increasing sequence (F τn∧n ) n∈N , where (τ n ) n∈N is a localization sequence of Z. Here we assume, without loss of generality, that the underlying probability space allows for such an extension. For some P-Brownian motion U we then have Hence, whenever 1+W −B becomes small then W moves like a two-dimensional (G · , P)-Bessel process. In particular, 1 − W − B never hits zero and · 0 (1 + W u − B u ) −2 du < ∞, P-a.e., yielding that Z is indeed a martingale.
Let us now consider the F · -predictable times (ρ i ) i∈N 0 and (τ i ) i∈N , defined inductively by ρ 0 := 0 and Then we have Hence, by (4.3), we get For the the F · -optional P-projection • Y P of the G · -local martingale deflator Y = 1/Z, we then have • Y P = (1 − K)/ o Z Q , where 1/ o Z Q is an (F · , P)-martingale by Proposition 4.6.
In Example 2.4, it was shown that projections of reciprocals of G · -numéraires are not necessarily reciprocals of F · -numéraires. However, we have the following result.
Proof. For statement (1), it suffices to argue that Q[τ 0 ≤ ·|F ∞ ] is the F · -predictable projection of the process 1 {τ 0 ≤·} . That is, for a given F · -predictable time ρ we need to argue that We shall argue this assertion by showing that the right-hand side is indeed the F ∞ -conditional expectation of 1 {τ 0 ≤ρ<∞} . To this end, fix A ∈ F ∞ and note that Q[A|F ρ− ] = Q[A|G t− ] since each (F · , Q)-martingale is also a (G · , Q)-martingale by assumption. This now yields where the last equality uses the fact that τ 0 is Q-a.e. equal to a G · -predictable time since Z does not jump to zero; see [LR18,Lemma 3.5]. This now yields (4.8).
For statement (2), observe that we have o Z Q = E( · 0 θ u dS u )N for some nonnegative F ·predictable process θ and some (F · , Q)-local martingale N with [N, S] = 0. By assumption, N is also a (G · , Q)-local martingale. By the product rule, so is N Z. Hence, N is a nonnegative (G · , P)-local martingale, thus an (F · , P)-local martingale, as it is F · -adapted. Moreover, o Z Q /N = E( · 0 θ u dS u ) is an (F · , Q)-local martingale; hence 1/N is also an (F · , P)-local martingale. This implies that N = 1.
Remark 4.11. From a modelling point of view, it is convenient to observe that Jacod's hypothesis (H) holds, for example, if G · is of the form where H · is a filtration such that F ∞ and H ∞ are independent under Q. Indeed, fix any (F · , Q)-martingale N , some s, t ≥ 0 with s < t, and some A ∈ H s . Then where we have used repeatedly the independence of F ∞ and H ∞ under Q. In particular, the Bayesian setup of Section 3 satisfies the assumptions of Proposition 4.10. As a corollary, Assumption 4.1 and Jacod's hypothesis (H) holding under Q does not imply that each (F · , P)martingale is also a (G · , P)-martingale. Consider the F B · -stopping times (ρ i ) i∈N 0 and (τ i ) i∈N , defined inductively by ρ 0 = 0 and

Completeness and
These stopping times allow to define the F · -adapted process which is piecewise constant and jumps only at the times before τ when |W | hits one. More precisely, N jumps up or down by 1/2 with probability 1/2, depending on whether W hits 1 or −1; hence, it is an F · -martingale, but not a G · -local martingale. The discontinuous process N a fortiori cannot be expressed as a stochastic integral with respect to the geometric Brownian motion S; hence, the market is indeed incomplete under F · .

5.2.
A more general construction. The following result is of independent interest. Corollary 5.2. There exist two nested filtrations F · ⊆ G · and a one-dimensional continuous stock price process S, adapted to F · , such that the market is complete under G · and under F S · , but not under the "intermediate information" model F · .
Proof. Using the above notation, set G · := F W · and S := E(B), where we recall that E(·) denotes the stochastic exponential operator, and note that F S · = F B · . Next, take µ to be the law of a non-deterministic distribution, and consider an F W · -stopping time τ as in Theorem 5.1, with distribution µ. Define now F · to be the right-continuous modification of the progressive enlargement of the filtration F B · with the random time τ . Clearly, F S · = F B · ⊆ F · ⊆ F W · = G · , and B is a Brownian motion on all three considered filtrations. However, although B (hence S) has the predictable representation property on both F B · and F W · , it loses the predictable representation property on F · . This can be readily seen by considering the (non-continuous) In the context of the proof of Corollary 5. functions w such that w(0) = 0, R(w) > 0, and w(t) = 0 for all t ≥ R(w). We denote by U + (respectively, U − ) the subset of U with the extra property that w(t) > 0 (respectively, w(t) < 0) holds for all t ∈ (0, R(w)), in which case we speak of positive (respectively, negative) excursions. With δ : [0, ∞) → R denoting the function that is identically equal to zero (which in particular implies that δ / ∈ U ), consider the state space U δ := U ∪ {δ}.
Recalling that Λ is the local time of the Brownian motion W at zero, define defined via e s (t) = W (σ s− + t) for t ∈ [0, σ s − σ s− ], and e s (t) = 0 for t > σ s − σ s− . We shall use |e| := (|e| s ) s∈[0,∞) to denote the process such that |e| s (t) = |e s (t)| holds for all s, t ≥ 0, and note that |e| is also a Poisson point process, with state space U + ∪ {δ}; in effect, |e| forgets the excursion signs.
With the above notation set, we need the following facts: (1) The sigma-algebra generated by the Poisson point process |e| (2) Conditional on the process |e| (i.e., conditional on F |W | ∞ ), the signs of the excursions are (a countable number of) independent and identically distributed random variables taking the values −1 and +1 with probability 1/2. The first statement above is a consequence of [RY99, Chapter XII, Proposition 2.5]. Indeed, since Λ is F |W | ∞ -measurable, the process (σ s ) s≥0 is also F |W | ∞ -measurable, and then it is straightforward that (|e| s ) s≥0 is F |W | ∞ -measurable. On the other hand, one may reconstruct |W | from |e| as follows: first, for s ≥ 0 one defines σ s := v∈(0,s] R(|e| v ), then one obtains Λ as the right-continuous inverse of σ, and then one defines |W t | := |e| Λt (t − σ(Λ t −)) for all t ≥ 0.
The second statement above comes, for example, as a consequence of the discussion in [Blu92, Chapter IV, mostly page 114]; see also [Pro09].
The following result is the main tool in establishing the validity of Theorem 5.1.
Lemma 5.3. Fix a strictly decreasing sequence (s n ) n∈N in (0, ∞) with lim n↑∞ s n = 0. Then, there exists a countable collection (U n ) n∈N of random variables such that: • for each n ∈ N, U n is F W σs n -measurable; and • (U n ) n∈N consists of independent and identically distributed random variables with the standard uniform law, and is further independent from F |W | Proof. For each n ∈ N, write σ n := σ sn for typographical simplicity. Consider the intervals I n := (s n+1 , s n ]; since the sequence (s n ) n∈N is strictly decreasing, (I n ) n∈N consists of disjoint intervals.
Define the random, P-a.e. countable set where excursions actually happen in the local time clock. The set D ∩ I n corresponds to excursion times that happen in I n in the local time clock, and is clearly countable and infinite.
Furthermore, (e s ) s∈In is F W σn -measurable, for all n ∈ N. It is straightforward (for example, by ordering the excursion sizes) to see that one may find an F W σn -measurable enumeration (v n,k ) k∈N of D ∩ I n , for each n ∈ N. Then, for all n ∈ N, the F W σn -measurable random variables X n,k := 1 {ev n,k ∈U + } are {0, 1}-valued. Moreover, using the fact that the intervals (I n ) n∈N are disjoint, we obtain that, conditional on F B ∞ = F |W | ∞ , the doubly-indexed collection (X n,k ) (n,k)∈N×N consists of independent and identically distributed random variables with P [X n,k = 0] = 1/2 = P [X n,k = 1]. Therefore, upon defining U n := ∞ k=1 2 −k X n,k for all n ∈ N, we obtain a sequence (U n ) n∈N of independent and identically distributed random variables with the standard uniform law, that are further independent of F |W | ∞ . Finally, since (X n,k ) k∈N are F W σn -measurable for each n ∈ N, we obtain that U n is F W σn -measurable for each n ∈ N, which completes the argument. Given Lemma 5.3 above, we may now proceed to prove Theorem 5.1. Note that (µ s ) s≥0 is increasing in first-order stochastic dominance and that, as s ↓ 0, µ s converges (actually, in total variation) to µ 0 = µ.
Pick (s n ) n∈N to be any strictly decreasing sequence of positive numbers with lim n↑∞ s n = 0.
Define now τ := lim n↑∞ τ n = n∈N τ n ; since τ n is an F W · -stopping time for all n ∈ N, τ is also an F W · -stopping time. Given that the conditional law of τ n given F B ∞ is µ σn for each n ∈ N and that (σ n ) n∈N decreases P-a.e. to zero, it follows that the conditional law of τ given F B ∞ is µ 0 = µ. This implies both that τ is independent of F B ∞ , and that its probability law equals µ, which concludes the proof of Theorem 5.1.

5.4.
A further example where incompleteness arises though filtration shrinkage.
The markets described in Corollary 5.2 have an interesting "quasi-completeness" property: for each T ≥ 0, any F S T -measurable contingent claim can by replicated; see [Ruf11] for a discussion of this weaker notion of completeness. Below, we shall provide an example where the market is complete under the large filtration G · , but not even quasi-complete under the smaller filtration F · .
Let B be a one-dimensional Brownian motion, and set G · the filtration generated by B; i.e., G · := F B · . Set S := E( · 0 θ u dB u ), where Since θ is nonzero, (P × [B, B])-a.e., the market is complete under G · . Now, let F · := F S · be the filtration generated by S. Since S t = E((B 2 t − t)/2) holds for all t ≤ 1, it follows that F t = F |B| t holds for all t < 1. On the other hand, for all t > 1, since {B 1 > 0} = {[S, S] t − [S, S] 1 = t − 1} ∈ F t , it follows that B 1 ∈ F t . Furthermore, (B s − B 1 ) s∈[1,t] is also F t -measurable, which then implies that (B s ) s∈[1,t] is F t -measurable. Using right-continuity of the filtration, it then easily follows that, for all t ≥ 1, F t is generated by F Here, there is a jump of information at the filtration F · happening exactly at the (deterministic) time 1: the sign of B at t = 1 is suddenly revealed, while the only previous information was on the absolute value of B. In particular, any process of the form for all α ∈ (−1, 1) is a strictly positive F · -martingale, and it is clearly purely discontinuous.
Therefore, S cannot have the predictable representation property in F · .
Under the filtration F · , its hedging cost is at least sup α∈(−1,1) in fact, this is the actual hedging cost since one may trivially hedge starting from this amount.
This can also be argued in a "dual" way, observing that the probabilities Q α constructed from Z α for all α ∈ (−1, 1) are exactly the class of equivalent local martingale measures under the filtration F · .