Absolute Continuity of Semimartingales

We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov's theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional It\^o-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multi-dimensional It\^o-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.


Introduction
In the 1940s, Cameron and Martin studied the influence of a deterministic drift to Brownian motion. Their result can be seen as the starting point of a sequence of results which are nowadays known as Girsanov's theorems. In the 1970s, mathematicians studied Girsanov's theorem from a semimartingale perspective. The most general results were obtained by Jacod and Mémin [9] and Kabanov, Lipster and Shiryaev [12,13] under a restrictive uniqueness assumption, called local uniqueness in the monograph of Jacod and Shiryaev [10]. In Markovian settings it is wellknown that local uniqueness is implied by uniqueness in law. The question when local uniqueness holds in non-Markovian settings is, however, open in general.
In this article we show that the assumption of local uniqueness can be replaced by the assumption of uniqueness in law. Our tool and also our main result is a generalized Girsanov's theorem, which relates two laws of possibly explosive semimartingales through a candidate density process. The idea of its proof is to work not on the classical Skorokhod space, but on a slightly larger path space, which has the good topological property to allow the extension of many consistent families of probability measures. Since the Skorokhod space is a Borel subset of the larger path space, one can always consider laws of semimartingales as probability measures on the larger space.
The equivalence of two probability measures always implies the martingale property of their density process. Hence, our result can be used to prove that stochastic exponentials are martingales. To illustrate this, we generalize Benes's [1] linear growth condition for the martingale property of stochastic exponentials driven by Brownian motion to quasi-left continuous stochastic drivers.
Let us shortly comment on related literature. Under a local uniqueness assumption, the question when laws of semimartigales are equivalent was studied by Cheridito, Filipovic, and Yor [2], Jacod and Mémin [9] and Kabanov, Lipster, and Shiryaev [12,13]. Kallsen and Muhle-Karbe [14] used local changes of measures and local uniqueness to derive conditions for the martingale property of stochastic exponentials in an affine semimartingale setting. An extension argument related to ours was used by Ruf [20] to prove the martingale property of stochastic exponentials in an Itô-diffusion framework.
We summarize the structure of the article. In Section 2 we introduce our mathematical framework and give our main result. In Section 3 we present our applications.
Let us end the introduction with a remark on notation: All non-explained notations can be found in the monograph of Jacod and Shiryaev [10].

A Generalized Girsanov Theorem
Let ∆ be a point outside of R d and denote by R d ∆ ≡ R d ∪ {∆} the one-point compactification of the locally compact space R d . It is well-known that R d ∆ is a Polish space. We define Ω to be the set of all functions α : We denote by cl(G) the closure of a set In particular, it is easy to see that τ n ↑ n→∞ τ ∆ .
In the spirit of stochastic differential equations up to explosion, we now formulate a semimartingale problem up to explosion. We start by introducing the parameters: (i) Let (B, C, ν) be a so-called candidate triplet consisting of , which we call initial law. The idea is to find a probability measure on (Ω, F) such that the coordinate process is a semimartingale (up to τ ∆ ) with characteristics (B, C, ν) (up to τ ∆ ) and initial law η. In a classical global setting the semimartingale problem was first introduced by Jacod [8].
We stress that, implicitly, we fix a truncation function from the beginning. That means that all following objects should be read w.r.t. the same truncation function.
Clearly, the assumption on the initial value Z 0 may be replaced by a positivity assumption.
For n ∈ N denote by X c,n the continuous local (F, P )-martingale part of X τn , which is unique up to P -indistinguishability. Now, set Finally, we introduce the following modified candidate triplet: Our main result is the following generalized Girsanov's theorem.
In particular, (i) we have (ii) in the case P (τ ∆ = ∞) = 1, we have the following equivalence Proof: For n ∈ N define the probability measure Q n ≡ Z τn · P on (Ω, F). We now construct Q using the classical extension result of Parthasarathy. We recall a definition due to Föllmer [7]. Let T ⊆ [0, ∞) be an index set, and (Ω * , F * t ) t∈T be a sequence of measurable spaces.

an application of Lemma A.2 in Appendix A and
Girsanov's theorem [10, Theorem III. 3.24] implies that the stopped process X τn is . Note that we need the Lemmata in Appendix A to apply Girsanov's theorem [10, Theorem III. 3.24], which is formulated for rightcontinuous filtrations. Moreover, we used the following fact: If two processes are P -indistinguishable, then Q n ≪ P implies that they are also Q n -indistinguishable. The equality Q = Q n on F τn implies that X τn is Q-indistinguishable from an (F, Q)-semimartingale whose (F, Q)-characteristics are Q-indistinguishable from (B ′ ·∧τn , C ·∧τn , 1 [ [0,τn] ] · ν ′ ). Here, we use that for an F-optional process Y and an F-stopping time γ the stopped process Y γ is F γ -measurable. Since P -a.s. Z 0 = 1, it holds that Q = P on F 0 , and we conclude that Q solves the SMP (τ ∆ ; η; B ′ , C, ν ′ ).
We now prove (2.2). Let ξ be an F-stopping time and A ∈ F ξ , then due to the monotone convergence theorem and Doob's stopping theorem.
In particular, this implies for each t ∈ [0, ∞) and A ∈ F t that This equality readily implies all remaining claims.
In the following section we present two applications: First, we study local equivalence of laws of semimartingales. Second, we derive linear growth conditions which imply the martingale property of stochastic exponentials driven by quasi-left continuous semimartingales.

Applications
3.1. Local Equivalence of Laws of Semimartingales. As a first application of Theorem 2.2 we study equivalence of laws of semimartingales possibly defined up to explosion. In non-explosive settings, systematic approaches were given by Kabanov, Lipster and Shiryaev [12,13], Jacod and Mémin [9] and Jacod [8,11] under a restrictive uniqueness assumption. In an explosive Itô-jump-diffusion setting, the question was addressed by Cheridito, Filipović and Yor [2]. We now prove that the results hold under a weaker uniqueness assumption and in non-Markovian settings.
Assume that the probability measure P solves the SMP (τ ∆ ; η; B, C, ν) and that the probability measure Q solves the SMP (τ ∆ ; η; B ′ , C, ν ′ ), where and The main result of this section is given by the following: Corollary 3.1. Assume that Q is the unique solution to the SMP (τ ∆ ; η; B ′ , C, ν ′ ), that for all t ∈ [0, ∞) and all càdlàg functions α :

3)
and that for all n ∈ N the random variable H τn is P -a.s. bounded. Then, for all F-stopping times ξ and all A ∈ F ξ the equality (2.2) holds with Moreover,

5)
Proof: Let us show that we may define a process Z on n∈N [[0, τ n ]] by Then, Z ξ is well-defined on {ξ < τ ∆ }. We verify that Z τn is a positive local (F, P )martingale. Thanks to the assumption that Y maps into (0, ∞) and (3.3), P -a.s.
where we use the convention that 0 0 = 0. Hence, thanks to [10, Theorem II.1.33] and Lemma A.1 in Appendix A, the assumption that H τn is P -a.s. bounded implies that Z τn is a positive local (F, P )-martingale. Moreover, [8,Theorem 8.25] yields that Z τn is even a uniformly integrable (F, P )-martingale. Thus, arguing as in the proof of [10, Lemma III.5.27], the claim follows from Theorem 2.2.

Example 3.2 (Continuous Setting). We set
F-predictable, and we use again the convention that, in the case of divergence, the integrals are set to ∆. Here, σ * denotes the adjoint of σ. We suppose that P solves the SMP (δ 0 ; B, C, 0) and that Q solves the SMP (δ 0 ; B ′ , C, 0). Then, Corollary 3.1 implies that P ∼ loc Q if the SMP (τ ∆ ; δ 0 ; B ′ , C, 0) has a unique solution and τn 0 β(X, s)σ(X, s) 2 ds is P -a.s. bounded. Let us relate this result to the concept of stochastic differential equations. It is well-known that P is the law of a solution process to the stochastic differential equation where W is a d-dimensional Brownian motion, and Q is the law of a solution process to the stochastic differential equation where B is a d-dimensional Brownian motion, cf. [8,Theorem 14.80]. Hence, Corollary 3.1 provides sufficient conditions for the local equivalence of two laws of solution processes of stochastic differential equations. We stress that we do not require any type of strong existence or pathwise uniqueness. For this particular case the result is known in Markovian-type settings, i.e. if µ, σ, and β are homogeneous functions which only depend on X t instead of the path X and are bounded on compact subsets of R d , cf., e.g., [22,Exercise 10.3.2]. If the stochastic differential equation (3.7) satisfies pathwise uniqueness, an alternative way to derive the result is to apply [10,Theorem III.5.34]. In this case, an additional Yamada-Watanabe-type argument is necessary and has to be adapted.
Let us relate our results to the classical results of Jacod and Shiryaev [10, Theorem III. 5.34] and Cheridito, Filipović and Yor [2].
Jacod and Shiryaev show in the case P (τ ∆ = ∞) = Q(τ ∆ = ∞) = 1 together with a stronger uniqueness assumption on the SMP (τ ∆ ; η; B ′ , C, ν ′ ) (i.e. local uniqueness, cf. [10, Definition III.2.37]), that In variations, this result can also be found in [8,12,13]. The proof of Jacod and Shiryaev heavily relies on the assumption of local uniqueness. It is classical that in Markovian settings local uniqueness is implied by global uniqueness, cf. [10, Theorem III.2.40]. The strength of our result is that it does not rely on local uniqueness and hence applies in more general, possibly non-Markovian, frameworks. In contrast to Jacod and Shiryaev, Cheridito, Filipović and Yor work in an explosive Markovian setting. Their main result is very similar to Corollary 3.1. As for the result of Jacod and Shiryaev, the proof of Cheridito, Filipović and Yor heavily relies on the assumption of local uniqueness which is, thanks to their Markovian setting, implied by their global uniqueness assumption. Our result generalizes their observation beyond Markovian settings.
Let us stress that assuming (Ω, F) as underlying measurable space is no restriction, even in non-explosive settings. More precisely, since the space of càdlàg functions [0, ∞) → R d equals the set {τ ∆ = ∞} ∈ F, laws of (global) semimartingales can always be seen as probability measures on (Ω, F). Furthermore, the space of càdlàg functions [0, ∞) → R d ∆ can be shown to be a Borel subset of Ω. This space is used as underlying measurable space in [2].
In summary, the equivalence of laws of semimartingales have been derived in the literature under various different sets of assumptions. In all cases, either structural assumptions or stronger concepts of uniqueness have been evoked. In this regard, it is interesting that Corollary 3.1 shows that for laws of semimartingales to be equivalent it suffices that their characteristics have a local boundedness property and their laws are unique. In particular, it is surprising that no further structural assumptions or stronger uniqueness properties are necessary.

3.2.
Martingale Property of Stochastic Exponentials. As a second application we derive a generalization of the classical linear growth condition of Benes [1]. To recall the result, let B be a d-dimensional Brownian motion and µ be a predictable real time-dependent functional on the Wiener space. Then, the process is a martingale if µ is at most of linear growth. We refer to [15,Corollary 3.5.16] for a precise statement. We generalize Benes's condition to a quasi-left continuous semimartingale setting. Using an analytical method, similar results were obtained by Klebaner and Lipster [16] in a Lévy-diffusion setting.
Let P be a solution to the SMP (η; B, C, ν) with is continuous and increasing with A 0 = 0 and K is a transition kernel from (Ω × [0, ∞), P) to (R d , B(R d )). Here, S d denotes the space of symmetric nonnegative definite d × d matrices. Take an F-predictable β : Ω × [0, ∞) → R d and a P-measurable Y : Ω × [0, ∞) × R d → (0, ∞). In the following, we will denote H as in (3.2).
Proof: By the same arguments as used in the proof of Corollary 3.1, the stopped process Z τn is a positive uniformly integrable (F, P )-martingale. We show the following: For all solutions Q to the SMP (τ ∆ ; η; B ′ , C, ν ′ ) it holds that Q(τ ∆ = ∞) = 1. Here, B ′ and ν ′ are given by (3.1). In this case, Theorem 2.2 implies the claim.
It is obvious that Benes's condition is implied by Corollary 3.3. Let us also mention another typical special case: A t = t, c = σσ * , K(ω, t, G) = 1 G (w(ω, t, x))F (dx), G ∈ B(R d ), 0 ∈ G, where σ : Ω × [0, ∞) → R d ⊗ R d is F-predictable, w : Ω × [0, ∞) × R d → R d is P-measurable and F is a Lévy measure on (R d , B(R d )). In this case, P corresponds to the law of a solution process to the stochastic differential equation where W is a d-dimensional Brownian motion and µ is a Poisson jump measure with intensity measure ν = F ⊗ dt. An univariate version of this setting was studied by Klebaner and Lipster [16].