On Absolute Continuity and Singularity of Multidimensional Diffusions

Consider two laws \(P\) and \(Q\) of multidimensional possibly explosive diffusions with common diffusion coefficient \(\mathfrak{a}\) and drift coefficients \(\mathfrak{b}\) and \(\mathfrak{b} + \mathfrak{a} \mathfrak{c}\), respectively, and the law \(P^\circ\) of an auxiliary diffusion with diffusion coefficient \(\langle \mathfrak{c},\mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{a}\) and drift coefficient \(\langle \mathfrak{c}, \mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{b}\). We show that \(P \ll Q\) if and only if the auxiliary diffusion \(P^\circ\) explodes almost surely and that \(P\perp Q\) if and only if the auxiliary diffusion \(P^\circ\) almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.


Introduction
Consider two laws P and Q of multidimensional possibly explosive diffusions with common diffusion coefficient a and drift coefficients b and b + ac, respectively. We are interested in finding analytic conditions for the absolute continuity P ≪ Q and the singularity P ⊥ Q. Such conditions are of interest in many branches of probability theory. In mathematical finance, for instance, mutual absolute continuity is of importance in the study of the absence of arbitrage, see [11,34].
For one-dimensional diffusions precise integral test were proven in [6] under the Engelbert-Schmidt conditions. For multidimensional diffusions the situation is less well-understood and only a few analytic conditions are known, see [1] for an integral test for Fuchsian diffusions.
The starting point for our research is the following probabilistic characterization of absolute continuity and singularity: Let X be the coordinate process and set A t t∧θ 0 c(X s ), a(X s )c(X s ) ds, t ∈ R + , where θ is the explosion time. It has been proven in [1,9] that P ≪ Q is equivalent to P (A θ < ∞) = 1 and that P ⊥ Q is equivalent to P (A θ = ∞) = 1. In other words, P ≪ Q and P ⊥ Q are characterized by P -a.s. divergence and convergence of the perpetual integral A θ . Again, these properties are well-understood for one-dimensional diffusions, see [3,10,18,23,27], and it seems that less work has been done for the multidimensional case, see [12,38] for results concerning Bessel processes, [9] for some conditions in radial cases, and [19] for results on divergence in case X is a conservative Feller process possibly with jumps.
In [3,10,18,19] the perpetual integral A θ was related to the hitting time of a time-changed process. In this article we pick up this idea and prove the following: Let f : R d → (0, ∞) be a Borel function which is locally bounded away from zero and infinity. Under the assumptions that the diffusion P exists and b and a are locally bounded, we show existence of a diffusion P • with diffusion coefficient f −1 a and drift coefficient f −1 b such that the law of the perpetual integral T θ θ 0 f(X s )ds under P coincides with the law of the explosion time θ under P • . Furthermore, we show that P • is unique whenever P is unique.
Returning to our initial problem, we note that in case f = c, ac the absolute continuity P ≪ Q is equivalent to P • (θ < ∞) = 1 and the singularity P ⊥ Q is equivalent to P • (θ = ∞) = 1. This observation is very useful, because the literature contains many conditions for explosion and non-explosion of multidimensional diffusions, see [22,28,36]. For illustration, we formulate a Khasminskii-type integral test for absolute continuity and singularity.
The result can also be applied in the converse direction: In case we have criteria for absolute continuity and singularity, these can be used to deduce explosion criteria for time-changed diffusions. To illustrate this, we derive an integral test for almost sure explosion and non-explosion of time-changed Brownian motion, using results on singularity of Fuchsian diffusions proven in [1]. The integral test improves several conditions known in the literature, see [22].
The absolute continuity P ≪ Q is intrinsically connected to the uniform integrable (UI) Q-martingale property of a certain stochastic exponential (see Eq. 2.2 below), which has been studied for one-dimensional diffusions in [24]. Independent of the dimension, it is known that for the conservative case the loss of the martingale property has a one-to-one relation to the explosion of an auxiliary diffusion, see, e.g., [5,35]. This turned out to be wrong in the non-conservative setting of [24]. Our result explains that for the UI martingale property the statement is true irrespective whether the diffusions are conservative or non-conservative.
As a third application, we use the relation of the UI martingale property and absolute continuity to study a problem in mathematical finance: We derive an explosion criterion for a local martingale measure to be a true martingale measure on the infinite time horizon.
Let us close the introduction with comments on related literature. To the best of our knowledge, the relation of absolute continuity/singularity and explosion of a timechanged process has not been reported before. We think that our new integral tests for absolute continuity/singularity and explosion/non-explosion illustrate that working out this connection is fruitful. The integral tests in [6,24] for absolute continuity, singularity and the UI martingale property in one-dimensional frameworks can be deduced from our result and Feller's test for explosion under additional assumptions on the coefficients. For general one-dimensional diffusion models with finite and infinite time horizon, analytic conditions for a local martingale measure to be a martingale measure were given in [25]. Conditions for one-and multidimensional diffusion models with finite time horizon were proven in [7,8,35]. We extend part of these results to a multidimensional setting with infinite time horizon. Beginning with [37], existence and uniqueness results for time-changed Markov processes have a long history, see, e.g., [4,36] for more information. In most of the classical work, the function f is assumed to be uniformly bounded away from zero, which implies that the time-changed process is conservative in case the original process is conservative. In combination with conditions for non-explosion, general positive continuous f are considered in the recent article [19].
The novelty of our existence and uniqueness result is that we work without additional assumptions for non-explosion. This is crucial for the question of absolute continuity and singularity. Moreover, we work under sort of minimal assumptions on the original diffusion P by assuming only existence and locally bounded coefficients.
The article is structured as follows: In Section 2 we present our main results, in Section 3 we discuss applications and in Section 4 we prove our main theorem.

Main Results
Let R d ∆ R d ∪ {∆} be the one-point compactification of R d . We define Ω be the space of all continuous functions R + → R d ∆ which are absorbed in ∆. Let X be the coordinate process on Ω, i.e. X t (ω) = ω(t) for all ω ∈ Ω and t ∈ R + , and define Except stated otherwise, all terms such as martingale, local martingale, stopping time etc. correspond to F (F t ) t≥0 as underlying filtration. Let S d×d be the space of symmetric non-negative definite real-valued d × d matrices.
For n ∈ N we set It is well-known that θ n and θ are stopping times. We fix two Borel functions b : R d → R d and a : R d → S d×d and impose the following: (S1) b and a are locally bounded. Here, S is an acronym for standing, which indicates that the assumption is in force for the remainder of the section.
The following definition of a martingale problem is taken from [28], where it is called generalized martingale problem due to the possibility of explosion. For simplicity we drop the term generalized.
Definition 2.1. We say that a probability measure P on (Ω, F ) solves the martingale problem MP (a, b, x 0 ), where x 0 ∈ R d , if P (X 0 = x 0 ) = 1 and for all n ∈ N and f ∈ C 2 (R d ) the process is a P -martingale. A solution P is called conservative (or non-explosive), if P (θ = ∞) = 1, and almost surely explosive, if P (θ < ∞) = 1.
The following theorem is the key observation in this article. It explains that perpetual integrals are distributed as the explosion time of a time-changed diffusion. The proof is given in Section 4 below.
Theorem 2.2. Let f : R d → (0, ∞) be Borel and locally bounded away from zero and infinity. There exist R d ∆ -valued right-continuous measurable processes Y and U such that for every x 0 ∈ R d and every solution P x 0 to the MP (a, b, x 0 ) the following hold: (i) Y has P x 0 -a.s. continuous paths.
In particular, existence and uniqueness hold simultaneously for the MPs (a, b, x 0 ) and Remark 2.3. While our main interest lies in the equality (2.1), also the existence and uniqueness parts of Theorem 2.2 are useful, because they lead to localizations of known existence and uniqueness theorems. For example, suppose that det(a) is locally bounded away from zero. This is a weak ellipticity assumption without continuity. There exists a continuous function f : R d → (0, ∞) such that fb and fa are bounded. Thus, by an existence result from [31], the MP (fa, fb, x 0 ) has a (conservative) solution. Now, Theorem 2.2 implies that the MP (a, b, x 0 ) has a (not necessarily conservative) solution, too. We give more details on this in Appendix A.
Fix a third Borel function c : R d → R d with the following property: (S2) ac is locally bounded.
Before we turn to our main application, we report a simple observation which might be of independent interest: Absolute continuity and singularity are invariant under time-changes. For an application of a related result to mathematical finance see [21].
Corollary 2.4. Let f : R d → (0, ∞) be a Borel function locally bounded away from zero and infinity, x 0 ∈ R d , P x 0 be the unique solution to the MP (a, b, x 0 ), Q x 0 be the unique solution to the MP (a, b + ac, x 0 ), P * x 0 be the unique solution to the MP (f −1 a, f −1 b, x 0 ) and let Q * x 0 be the unique solution to the MP (f −1 a, f −1 (b + ac), x 0 ). The following hold: where Y is as in Theorem 2.2. Then, Theorem 2.2 yields that Q x 0 (B) = Q * x 0 (A) = 0 and P x 0 (B) = P * x 0 (A) > 0. This is a contradiction and we conclude that P * x 0 ≪ Q * x 0 . The converse implication in part (i) follows by symmetry. Part (ii) can be shown in the same manner.
We assume the following: (S3) For every x 0 ∈ R d there exists a unique solution P x 0 to the MP (a, b, x 0 ) and a unique solution Q x 0 to the MP (a, b + ac, x 0 ).
Analytic conditions for (S3) are given by Proposition A.1 in Appendix A. Next, we introduce a non-negative local Q x 0 -martingale which relates Q x 0 and P x 0 . For this, we assume the following: (S4) c, ac is locally bounded.
By assumption (S4), the integral process Y ·∧θn ·∧θn 0 c(X s ), dX s is well-defined as a continuous Q x 0 -martingale starting at zero with quadratic variation process is a non-negative local Q x 0 -martingale and Q x 0 -a.s. the terminal value Z ∞ lim t→∞ Z t exists and is finite.
Proof: This follows similar to the proof of [15,Lemma 12.43].
As in the introduction, we set Part (i) of the next proposition is a version of [9, Corollary 5.1] (see also [32,Theorem 3.3]) and part (ii) is an extension of [1, Theorem 1 (i)] to a non-conservative setting. The proofs are similar and omitted.
(i) The following are equivalent: The following are equivalent: From now on we extend (S4) and assume the following: (S5) c, ac is locally bounded away from zero. Due to Theorem 2.2 there exists a unique solution P • x 0 to the time-changed MP ( c, ac −1 a, c, ac −1 b, x 0 ) and . This observation relates Proposition 2.6 to P • x 0 : Corollary 2.7.
(i) (i.a) -(i.c) in Proposition 2.6 are equivalent to the following: in Proposition 2.6 are equivalent to the following: (c) P • x 0 (θ = ∞) = 1. Remark 2.8. Due to [9, Corollary 5.1], the following are equivalent: In comparison, the absolute continuity P x 0 ≪ Q x 0 and the UI Q x 0 -martingale property of Z are related to almost sure explosion of P • x 0 .
Let us shortly comment on the role played by the initial value.
Lemma 2.9. Suppose that a is locally Hölder continuous and ξ, a(x)ξ > 0 for all x ∈ R d and ξ ∈ R d \{0}. Then, the following hold: Proof: This follows from the fact that x → P x (θ < ∞) is harmonic and the maximum principle, see [2, Lemmata 1.2, 1.4].
Corollary 2.10. Suppose that a satisfies the assumptions from Lemma 2.9 and that c, ac is locally Hölder continuous. Then the following hold: For conservative martingale problems this observation has been reported in [1, Corollary 1].
Many analytic conditions for almost sure explosion and non-explosion of martingale problems are known, see, e.g., [22,28,36]. Due to Corollary 2.7, these lead to analytic conditions for P For the one-dimensional case an analytic characterization of Our time-change argument requires that c = 0, which is not needed in [6,24]. In return, our approach is robust w.r.t. the dimension.
In the following section we present three applications of Theorem 2.2 and Corollary 2.7. First, we derive a Khasminskii-type integral test for absolute continuity and singularity, second we derive a Feller-type integral test for explosion of a multidimensional time-changed Brownian motion, and third we outline an application of Corollary 2.7 to mathematical finance in which c = 0 is usually naturally satisfied.

3.1.
A Khasminskii-Test for Absolute Continuity/Singularity. In this section we assume that (S1) -(S5) from Section 2 hold. We now formulate analytic conditions for a dimension-dependent constant C d > 0. This observation and Theorem 2.2 imply the following: By the standard linear growth condition for non-explosion, we have The following corollary shows that in case g is locally Hölder continuous and a least of quadratic growth the convergence criterion in Corollary 3.2 is optimal.
Corollary 3.3. Suppose that g is locally Hölder continuous and Proof: We define a Id, b 0 and c g − 1 2 e 1 , where e 1 is the first unit vector. Let Q x 0 be the unique solution to the MP (a, c, x 0 ), see Proposition A.1 in Appendix A. Note that c, ac = g −1 is a strictly positive continuous function. Corollary 2.7 yields that P x 0 (θ = ∞) = 1 if and only if W x 0 ⊥ Q x 0 . It follows from [1, Corollary 4] 1 that This completes the proof.
Remark 3.4. The growth condition (3.1) and R d g −1 (x) x 2−d dx = ∞ do not exclude themselves: In case (3.1) holds, we have The following proposition explains that in general the growth condition (3.1) is sharp.

(3.2)
Set B R (x) {y ∈ R d : x − y < R} and note that the balls (B Rn (x n )) n∈N are disjoint, because where we use (3.2) and in particular that 3R n+1 > 4R n . Define It is clear that g is Borel and locally bounded away from zero and infinity. If x ∈ B Rn (x n ) we have x n − R n ≤ x and In other words, (i) holds.
Thus, the Borel-Cantelli lemma yields that W 0 -a.a. paths of X hit only finitely many elements of (B Rn (x n )) n∈N . Recalling that Brownian motion is transient for d ≥ 3, i.e. that W 0 -a.a. paths of X leave bounded domains forever in finite time, we conclude that W 0 -a.s.
Note that where ω d is the volume of the unit ball in R d . We conclude that W 0 -a.s.
It is left to verify (iii). Using (3.2), the fact that f (x) = x 2−d is harmonic on R d and the mean-value property of harmonic functions, we obtain This implies (iii) and the proof is complete.
In case g is radially symmetric, the growth condition on g is not needed: Corollary 3.6. Suppose that g(x) = s( x ) for a Borel function s : R + → (0, ∞) which is locally bounded away from zero and infinity. The following hold: Proof: Due to [12, Theorem 2] and [38, Corollary 3], for every Borel function z : R + → R + the following are equivalent: The claims now follow directly from Theorem 2.2.

3.3.
On the Absence of Arbitrage in Diffusion Markets. Suppose that a is continuous and strictly positive definite. Then, there exists a unique solution Q x 0 to the MP (a, 0, x 0 ) due to Proposition A.1 in Appendix A. In addition, we assume that Q x 0 is conservative and we define a process S = (S 1 , . . . , S d ) by . . , d. We think of S as discounted price process in a financial market with d risky assets.
The assumption that a is strictly positive definite corresponds to the assumption that S has non-vanishing volatility, which is a typical assumption in mathematical finance. Because Q x 0 solves a martingale problem with zero drift, the process S is a nonnegative local Q x 0 -martingale. Hence, we call Q x 0 a local martingale measure. We are interested whether S is a UI Q x 0 -martingale, in which case we call Q x 0 a martingale measure. This question is of importance in mathematical finance as it determines the existence or absence of certain arbitrage opportunities, see [11,34] for more details. We note that S i equals Z as defined in (2.2) for −c set to be the i-th unit vector e i . Thus, with b ae i we are in the setting of Section 2. In particular, (S1) -(S5) hold by Proposition A.1 in Appendix A and the assumptions on a. Consequently, Corollary 2.7 implies the following: is the unique solution to the MP (a −1 ii a, a −1 ii ae i , x 0 ). Applying this corollary for all i = 1, . . . , d, we obtained explosion criteria for Q x 0 to be a martingale measure. Based on results from [22,28,36] one can also formulate analytic conditions. We leave the formulation of such conditions to the reader.
Finally, let us stress that the results for finite and infinite time horizons are very different. For example, in case d = 1, the probability measure Q 1 x 0 solves the MP (1, 1, x 0 ), which is obviously conservative, and S = S 1 is a no UI Q x 0 -martingale, while it is a Q x 0 -martingale if and only if

Proof of Theorem 2.2
In this section we prove Theorem 2.2, i.e. we prove the following: Theorem. Let f : R d → (0, ∞) be Borel and locally bounded away from zero and infinity. There exist R d ∆ -valued right-continuous measurable processes Y and U such that for every x 0 ∈ R d and every solution P x 0 to the MP (a, b, x 0 ) the following hold: (i) Y has P x 0 -a.s. continuous paths.
Existence and uniqueness hold simultaneously for the MPs (a, b, x 0 ) and (f −1 a, f −1 b, x 0 ). Let x 0 ∈ R d and let P x 0 be a solution to the MP (a, b, x 0 ). To simplify the notation, we denote P ≡ P x 0 . We start the proof by defining Y . For t ∈ R + we set The functions T, L : R + → [0, ∞] are increasing. Because f is locally bounded, we have T θn∧n < ∞ for all n ∈ N. Using this and the strict positivity of f, we see that T is finite, absolutely continuous and strictly increasing on [0, θ). Moreover, because lim tրθ T t = T θ by the monotone convergence theorem, T is everywhere continuous. We also note that L is finite, continuous and strictly increasing on [0, T θ ) and everywhere right-continuous, and L Ts = s for s < θ and T Lt = t for t < T θ , see [29, pp. 7 -9]. In particular, we have lim It is easy to see that Y is right-continuous and measurable. Because {t < T θ } = {L t < θ}, for every t < T θ we have Y t ∈ R d and consequently, T θ ≤ θ(Y ). Noticing that θ(Y ) ≤ T θ by definition, we obtain that The following lemma shows that Y has almost surely continuous paths, which is part (i) in Theorem 2.2.
We set and on {σ m n < ∞} we further set M K ·∧γ+σ m n − K σ m n , I ·∧γ+σ m n σ m n 2 X s , b(X s ) + tr(a(X s )) ds.
Using that for every t ∈ R + on {σ m 2 y, b(y) + tr(a(y)) t αt on {σ m n < ∞}, we obtain that For every t ∈ R + we have t ∧ γ + σ m n < θ on {σ m n < ∞}. Consequently, Applying the definition of the martingale problem with f (x) = x 2 yields that for every k ∈ N the process K ·∧θ k ∧k is a P -martingale. Note that for every t ∈ R + sup k∈N |K (t∧γ+σ m n )∧θ k ∧k − K σ m n ∧θ k ∧k |I {σ m n <∞} ≤ 2(n + 1) 2 + αt.

(4.4)
It is well-known that σ m n and τ m n are (F o t ) t≥0 -stopping times, see [13, Proposition 2.1.5]. We note that t ∧ γ + σ m n , which is set to be ∞ in case σ m n = ∞, is an (F o t ) t≥0 -stopping time, too. To see this, note that for all s ∈ R + , which follows because for any (F o t ) t≥0 -stopping times ρ and τ the following hold: The Dambis, Dubins-Schwarz theorem (see e.g., [16,Theorem 16.4]) yields that on a standard extension of the filtered probability space (Ω, F , F σ m n , P (·|F o σ m n )(ω)), which we ignore in our notation for simplicity, there exists a one-dimensional Brownian motion B such that P (·|F o y, a(y)y ∨ 1 t βt, t ∈ R + , In summary, (4.2) and (4.5) imply that We note that the law of τ under P (·|F o σ m n )(ω) only depends on the Wiener measure, which means that C is a constant independent of n, m and ω. Note also that C < 1. Now, we obtain for all n ∈ Z + By induction, we conclude Letting n → ∞ and using the dominated convergence theorem yields that We conclude that P -a.s. Σ c 2 ⊆ Θ. The proof is complete.
The other proof of Lemma 4.1 requires no uniqueness assumption on P and no continuity assumptions on b and/or a, which are often required for x → P x to be continuous, see [28,36].
For n ∈ N set γ n T θn∧n and note that L γn = θ n ∧n. It follows from [29, Proposition V. 1.4] that for all t ∈ R + and n ∈ N In other words, we have for all n ∈ N  Note that L t is an F-stopping time. Define the time-changed filtration G = (G t ) t≥0 (F Lt ) t≥0 . Because (L t ) t≥0 is right-continuous, also G is right-continuous, and because θ n ∧n is an F-stopping time, [15,Lemma 10.5] implies that γ n = T θn∧n is a G-stopping time and that t → L t∧γn is an increasing sequence of F-stopping times.
We set Recall that, by the definition of the MP (a, b, x 0 ), the process Kf (X s )ds is a P -martingale.
Recall further that L t∧γn ≤ θ n ∧ n. Using (4.7) and the optional stopping theorem, for s < t, n ∈ N and f ∈ C 2 (R d ) we obtain that P -a.s.
This yields that Kf (Y r )dr f(Y s ) is a P -martingale for the filtration G. In particular, by [30, Lemma II.67.10], it is a P -martingale for the P -augmentation of G, which we denote by G P . We redefine the process Y on a P -null set such that it gets everywhere continuous paths. With abuse of notation we denote the redefined process still by Y . Note that θ n (Y ) is a G P -stopping time. Recalling that P -a.s. γ n ր T θ = θ(Y ) and t ∧ θ n (Y ) < θ(Y ), the dominated convergence theorem yields for all s < t, n ∈ N and f ∈ C 2 (R d ) that P -a.s. .
Using the tower rule, we conclude that We now introduce the process U and verify (ii) in Theorem 2.2. For t ∈ R + we define A t inf(s ∈ R + : S s > t), Again, it is easy to see that U is right-continuous and measurable. Using (4.1) and (4.6), we obtain P -a.s. for all t ∈ R + S t • Y = In particular, P -a.s. S θ • Y = θ. Now, we obtain P -a.s. for t ∈ R + A t • Y = inf(s ∈ R + : L s > t) = T t , which implies X At • Y = X L T t = X t for all t < S θ • Y = θ. In summary, we conclude that P -a.s. U • Y = X, i.e. that (ii) in Theorem 2.2 holds. Finally, let us explain that if the MP (f −1 a, f −1 b, x 0 ) has at most one solution, then P is the unique solution to the MP (a, b, x 0 ). For n ∈ N let 0 ≤ t 1 < t 2 < · · · < t n < ∞ and G 1 , . . . , G n ∈ B(R d ∆ ). Suppose that Q is a second solution to the MP (a, b, x 0 ). Then, the push-forwards P • Y −1 and Q • Y −1 both solve the MP (f −1 a, f −1 b, x 0 ) and we deduce from the uniqueness assumption and (ii) in Theorem 2.2 that P (X t 1 ∈ G 1 , . . . , X tn ∈ G n ) = P • Y −1 (U t 1 ∈ G 1 , . . . , U tn ∈ G n ) = Q • Y −1 (U t 1 ∈ G 1 , . . . , U tn ∈ G n ) = Q(X t 1 ∈ G 1 , . . . , X tn ∈ G n ).
By a monotone class argument, P = Q. The proof is complete.

Appendix A. A Few Existence and Uniqueness Results
In this appendix we collect some existence and uniqueness results for martingale problems. We assume that b : R d → R d and a : R d → S d are locally bounded and we formulate the following conditions: (A1) b and a are continuous. (A2) det(a) is locally bounded away from zero. (A3) a is continuous and ξ, a(x)ξ > 0 for all x ∈ R d and ξ ∈ R d \{0}.
(A4) b is locally Lipschitz continuous and a has a locally Lipschitz continuous root. We use this opportunity and illustrate that Theorem 2.2 can also be used to obtain existence and uniqueness criteria for martingale problems.
Proposition A.1. Let x 0 ∈ R d . If (A1) or (A2) holds, then there exists a solution P x 0 to the MP (a, b, x 0 ). If (A3) or (A4) holds, then there exists a unique solution P x 0 to the MP (a, b, x 0 ).
Proof: Let f : R d → (0, ∞) be a continuous function such that fa and fb are bounded. Such a function can be constructed as follows: Set x ≤k a(x) ∨ 1, and let z : R + → (0, ∞) be a continuous function z ≤ g. Then, f(x) z( x ) has the claimed properties. In case one of (A1) -(A3) holds, the MP (fa, fb, x 0 ) has a (conservative) solution and in case (A3) holds the solution is even unique. With these observations at hand, Theorem 2.2 implies that existence holds for the MP (a, b, x 0 ) under either of (A1) -(A4) and that uniqueness holds under (A3). That uniqueness also holds under (A4) is well-known, see [ [28,Theorems 1.13.1]. Local integrability conditions, which are weaker than (A2), for the existence of a solution with not necessarily continuous paths (the left-limit at θ needs not to exist) are given in [20,Theorem 3.1].