Martingale spaces and representations under absolutely continuous changes of probability

In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.


Introduction
Martingale representation results have fundamental applications in stochastic control, filtering, backward stochastic differential equations and mathematical finance, notably in connection with the property of market completeness. In all these fields, absolutely continuous changes of probability play an equally important role, often leading to a substantial simplification of the problem under consideration. This motivates the interest of studying how spaces of martingales under two absolutely continuous probabilities are connected and, more specifically, the behavior of the martingale representation property (MRP) under absolutely continuous (not necessarily equivalent) changes of probability. In this paper, we aim at developing a general theory for these issues under minimal assumptions. This enables us to simplify and extend previous results to full generality, covering situations that cannot be addressed by the existing theory.
To the best of our knowledge, the most general result available in the literature on the behavior of the MRP under absolutely continuous changes of probability can be found in [HWY92,Theorem 13.12] and can be stated as follows (see also [JS15,Lemma 2.5]): Let P and Q be two probability measures on (Ω, F, F) such that Q ≪ P, with density process Z, and let X = (X t ) t≥0 be a real-valued P-local martingale having the MRP under P. Suppose that the process [X, Z] has locally integrable variation under P. Then, the process X ′ := X − (Z − ) −1 · X, Z P is a Q-local martingale and has the MRP under Q. A multi-dimensional version of this result, under the additional assumption of local boundedness of 1/Z under P, was first obtained in [Duf85].
The crucial assumption in the above result is the requirement that [X, Z] has locally integrable variation under P (or, equivalently, that X is a special semimartingale under Q). This leaves open the question of whether, in the absence of such a condition, the MRP is preserved or not under an absolutely continuous change of probability. We provide a positive answer in full generality, without any further assumption beyond local absolute continuity (Theorem 2.3). One of the key steps in our approach consists in replacing the usual version of Girsanov's theorem (see e.g. [JS03,Theorem III.3.11]) with its most general version as originally proven in [Len77]. Besides the greater generality, our proofs are more elementary and constructive than those in [Duf85,HWY92] and yield an explicit description of the stochastic integral representation (Remark 2.4). As shown by means of an explicit example (Section 3), there exist simple situations that are not covered by the existing theory and for which our results yield an explicit MRP.
From a more abstract standpoint, we obtain a new and general characterization of the set of Q-martingales as the smallest stable subset generated by suitable transformations of P-martingales (Theorem 2.2). By relying on our main results, we then address further issues, including the practically relevant case of locally equivalent probabilities and the dimension of martingale spaces under locally absolutely continuous probabilities (Section 2.3). In particular, these results enable us to provide a general solution to an open problem formulated in [Žit06]. We want to point out that, even though the present paper focuses on theoretical aspects, our results have relevant applications, notably in mathematical finance in the context of equilibrium models (see e.g. [Žit06,KP17]). 1 The paper is structured as follows. Section 1.1 introduces all necessary notations and terminology. In Section 2.1, we recall the setting and a crucial preliminary result due to [Len77]. Section 2.2 contains our main results, while further properties and ramifications are presented in Section 2.3. In Section 3, we give a simple example which falls beyond the scope of the existing results and to which our theory applies. The proofs of all results are collected in Section 4.
1.1. Notation. Throughout the paper, we shall make use of the following notation, referring to [JS03] for all unexplained notions. Let (Ω, F, P) be a probability space endowed with a rightcontinuous (not necessarily complete) filtration F = (F t ) t≥0 . On (Ω, F, P), we denote by M(P) (M loc (P), resp.) the set of all real-valued martingales (local martingales, resp.) and by A loc (P) the set of all real-valued adapted processes of locally integrable variation. For A ∈ A loc (P), we denote by

Results
2.1. Setting and preliminaries. We consider a probability space (Ω, F, P) endowed with a rightcontinuous (not necessarily P-complete) filtration F = (F t ) t≥0 and a probability measure Q on (Ω, F) such that Q ≪ loc P (i.e., Q is locally absolutely continuous with respect to P). This means that Q| Ft ≪ P| Ft , for all t ≥ 0, while Q ≪ P does not necessarily hold. In view of [JS03, Theorem III.3.4], the density process of Q relative to P is the unique non-negative process Z ∈ M(P) such that dQ| Ft = Z t dP| Ft , for all t ≥ 0. Let us define the stopping times ζ := inf{t ∈ R + : Z t = 0} and η := ζ1 Λ + ∞1 Ω\Λ , with Λ := {ζ < +∞, Z ζ− > 0}.
The behavior of local martingales under locally absolutely continuous, but not necessarily equivalent, changes of probability has been studied in [Len77], form which we recall the following fundamental result (compare also with [HWY92, Theorems 12.12 and 12.20]).
Proposition 2.1. For an adapted real-valued càdlàg process X, the following hold: (iii) if X ∈ M loc (P), then As mentioned in the introduction, part (iii) of the above proposition represents the most general formulation of Girsanov's theorem. In particular, unlike the usual version of Girsanov's theorem The above theorem shows that all Q-local martingales are generated by stochastic integrals of elements M , with M ∈ M(P). Loosely speaking, we can say that Q-martingales correspond to Lenglart transformations as in (2.1) of P-martingales. We want to emphasize that, despite the generality of the statement, the proof relies on rather basic facts of stochastic calculus, notably integration by parts and Itô's formula (see Section 4.1).
Theorem 2.2 does not assume any structure on the filtered probability space (Ω, F, P). An especially important case is when all P-local martingales can be represented as stochastic integrals of some fixed P-local martingale. More precisely, we say that an R d -valued P-local martingale X has the martingale representation property Our second main result asserts the stability of the MRP under locally absolutely continuous changes of probability in its most general form, without any further assumption. times (τ n ) n∈N increasing Q-a.s. to infinity such that (ZN ) τn ∈ M 0,loc (P), for each n ∈ N. Since X has the MRP under P, there exist H ∈ L m (X, P) and K n ∈ L m (X, P), for each n ∈ N, such that As shown in Section 4.2, the integrand φ ∈ L m ( X, Q) appearing in the stochastic integral representation N = φ · X is explicitly given by where τ 0 := 0. Note that each process φ n is well-defined under Q, since Q(ζ < +∞) = 0. In In particular, the assumption that ∆X η ac = 0 P-a.s. on {η ac < +∞} always holds in the following cases: (ii) if Q ∼ loc P, in which case P(η = η ac = +∞) = 1; (iii) if the process X = (X t ) t≥0 is P-a.s. quasi-left-continuous.  Under the slightly stronger assumption that Q ∼ P, a version of Proposition 2.8 has been recently established in [KP17]. Note also that, in the case Q ∼ P, the decomposition corresponds to the general version of Girsanov's theorem presented in [Mey76].

Dimension of H 1 -martingale spaces.
In this subsection, we study how the dimension of the martingale space H 1 behaves under locally absolutely continuous changes of probability. In particular, Proposition 2.8 enables us to prove the invariance of the dimension with respect to locally equivalent changes of probability. In line with [Jac79, Definition 4.38], let us recall that an R d -valued local martingale X on (Ω, F, P) is said to be a basis for H 1 (P) if L 1 (X, P) = H 1 0 (P) and there exists no R m -valued local martingale Y on (Ω, F, P), with m < d, such that L 1 (Y, P) = H 1 0 (P). In this case, d is said to be the dimension of H 1 (P), denoted as dim H 1 (P). This is also closely related to the notion of martingale multiplicity introduced in [DV74]. Under Q, the notions of basis and dimension are defined in an analogous way. 2 Proposition 2.9. If Q ≪ loc P, it holds that dim H 1 (Q) ≤ dim H 1 (P). If furthermore Q ∼ loc P, then dim H 1 (P) = dim H 1 (Q) and an R d -valued local martingale X on (Ω, F, P) is a basis for H 1 (P) if and only if X is a basis for H 1 (Q).
This last result generalizes [Duf85, Theorem 3.2 and its Corollary] by removing all restrictive boundedness assumptions on the density process Z.

An example
In this section, we present an example of a simple situation where classical results on the stability of the MRP under absolutely continuous changes of probability cannot be applied, while on the contrary our Theorem 2.3 yields the existence of a process having the MRP. By [HWY92, Lemma 13.8], the stopped martingale M τ 1 has the MRP on (Ω, F τ 1 , P), where F τ 1 denotes the stopped filtration (F t∧τ 1 ) t≥0 . We then define the process X = (X t ) t≥0 by It holds that X ∈ H 1 0 (P), as follows from the fact that Moreover, since the integrand 1/ √ u is strictly positive, it is immediate to verify that the martingale X inherits the MRP of M τ 1 under P in the filtration F τ 1 .
However, as a consequence of Theorem 2.3, the process X has the MRP under Q and, in view of Proposition 2.1-(iii), can be explicitly computed as follows. Note that η = +∞ for all T ∈ (0, 1/4], so that ∆X η 1 {η<+∞} = 0. Therefore, for all t ≥ 0, it holds that

Proofs
In this section, we give the proofs of our results, together with some auxiliary technical results.

Proofs of the results stated in Section 2.2.
Proof of Theorem 2.2. In view of [Jac79, Corollary 4.12], in order to prove H 1 0 (Q) = L 1 ( M(P), Q), it suffices to show that every bounded N ∈ M 0 (Q) such that N M ∈ M loc (Q), for all M ∈ M(P), is null. Recalling that Q(ζ < +∞) = 0, we can apply integration by parts under Q and compute where the last equality makes use of the identities as can be readily verified by applying Itô's formula (under Q), see also (4.4) below. Furthermore, In turn, making use of representation (2.1), this enables us to rewrite (4.1) under Q as follows: Since N is bounded, [N ] ∈ A loc (Q) and therefore the predictable quadratic variation N Q of N under Q is well-defined and can be explicitly computed as We proceed to proving our second main result (Theorem 2.3). This makes use of the following two technical lemmata, which concern the behavior of continuous and purely discontinuous local martingales and stochastic integrals under locally absolutely continuous changes of probability.
As a preliminary, let us recall that, for two semimartingales X and Y on (Ω, F, P), the quadratic We have thus shown that [ M ] = s≤· (∆ M s ) 2 , which means that M ∈ M d loc (Q). Since Q ≪ loc P, the density process Z is a strictly positive semimartingale under Q (see [HWY92, Theorem 12.14]). Therefore, as in (4.2), an application of Itô's formula yields that By MRP under P, there exists a process H ∈ L m (X, P) such that Z = Z 0 + H · X (see equation (2.2)). In view of [JS03, Proposition III.6.24], the process H is integrable with respect to X under Q in the semimartingale sense. Hence, by the associativity of the stochastic integral, we have that Recall from (2.2) that there exists a sequence of stopping times (τ n ) n∈N increasing Q-a.s. to infinity such that (ZN ) τn = K n · X, with K n ∈ L m (X, P), for each n ∈ N. Similarly as above, K n · X also makes sense as a semimartingale stochastic integral under Q. Therefore, using similar arguments as in the proof of Theorem 2.2, we can apply integration by parts and equation (4.5) under Q, thus where, for each n ∈ N, Moreover, on {η ≤ τ n } ∩ {η < +∞} (under the measure P) it holds that In turn, this implies that 0 = (φ n η ∆X τn ,P up to an evanescent set. Therefore, by (4.6) together with (2.1), it follows that N τn = φ n · X τn , for each n ∈ N.

4.3.
Proof of the results stated in Section 2.3. In this section, we present the proof of the remaining results of the paper, starting with Proposition 2.5.
Proof of Proposition 2.5. It suffices to show that, if ∆X η ac = 0 P-a.s. on {η ac < +∞} and [X i , X j ] ≡ 0, for all i, j = 1, . . . , d with i = j, then [ X i , X j ] ≡ 0 (up to a Q-evanescent set), for all i, j = 1, . . . , d with i = j. We first compute where the first equality follows from Lemma 4.1 and the uniqueness of the decomposition of a local martingale into a continuous part and a purely discontinuous part (see [JS03,Theorem I.4.18]).
The assumption that [X i , X j ] ≡ 0 implies that [(X i ) c , (X j ) c ] ≡ 0, so that [( X i ) c , ( X j ) c ] ≡ 0 up to a Q-evanescent set. It remains to show that ∆ X i t ∆ X j t = 0 Q-a.s. for all t ≥ 0 and i, j = 1, . . . , d with i = j.
Recall that [X i , X j ] ≡ 0 implies that ∆X i t ∆X j t = 0 P-a.s. for all t ≥ 0. Since by assumption Q ≪ loc P, we deduce that ∆ X i t ∆ X j t = 0.
We proceed with the proof of Proposition 2.8, which relies on the symmetric role of the two probabilities Q and P under the assumption Q ∼ loc P.
Proof of Proposition 2.8. By Theorem 2.3, it suffices to show that, if X has the MRP under Q, then X has the MRP under P. Since Q ∼ loc P, the density process of P relative to Q is given by 1/Z. We can then apply Proposition 2.1-(iii) on (Ω, F, Q) to the process X, yelding As a consequence of Theorem 2.3 applied to the local martingale X on (Ω, F, Q), the process X has the MRP under P, thus proving the claim.
Finally, we conclude with the short proof of Proposition 2.9.
Proof of Proposition 2.9. It suffices to consider the case dim H 1 (P) < +∞. Let X be an R dvalued local martingale on (Ω, F, P) that is a basis for H 1 (P) and let X be defined as in (2.1). By Theorem 2.3, it holds that L 1 ( X, Q) = H 1 0 (Q). In view of [Duf85, Lemma 2.2], this implies that dim H 1 (Q) ≤ d = dim H 1 (P). If Q ∼ loc P, using the result of Proposition 2.8 and reversing the role