No arbitrage in continuous financial markets

We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Itô process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We also show that for Markov switching models the minimal martingale measure preserves the independence of the noise and we study how the minimal martingale measure can be modified to change the structure of the switching mechanism. Our main mathematical tools are new criteria for the martingale and strict local martingale property of certain stochastic exponentials.


Introduction
The absence of arbitrage is of fundamental interest in many areas of financial mathematics. Our goal is to provide a systematic discussion for a financial market with one risky asset modeled via its discounted price process P = (P t ) t∈[0,T ] , which we assume to be either the stochastic exponential of an Itô process, i.e. to have dynamics For semimartingale markets the classical concepts of no arbitrage are the notions of no free lunch with vanishing risk (NFLVR) as defined by Delbaen and Schachermayer [15,16] and no feasible free lunch with vanishing risk (NFFLVR) as defined by Sin [52]. The difference between (NFLVR) and (NFFLVR) is captured by the concept of a financial bubble in the sense of Cox and Hobson [9]. For our market it is well-known that (NFLVR) is equivalent to the existence of an equivalent local martingale measure (ELMM), see [16], and that (NFFLVR) is equivalent to the existence of an equivalent martingale measure (EMM), see [6,52,55]. The no arbitrage condition used in the stochastic portfolio theory of Fernholz [22] is no relative arbitrage (NRA). In complete markets Fernholz and Karatzas [21] showed that (NRA) is equivalent to the existence of a strict martingale density (SMD). A weaker concept is no unbounded profit with bounded risk (NUPBR), which is known to be equivalent to the existence of a strict local martingale density (SLMD), see [7]. (NUPBR) is considered to be the minimal notion needed for portfolio optimization, see [35].
The first findings of this article are integral tests for the existence and non-existence of SMDs, ELMMs and EMMs. For (1.1) the tests are formulated in terms of Markovian upper and lower bounds on the volatility coefficient σ and for (1.2) the tests depend on x → σ (x, j) with j in the state space of the Markov chain ξ . The main novelty of our results is that they apply in the presence of multiple sources of risk. Beside the Markov switching framework, this is for instance the case in diffusion models with a change point, which represents a change of the economical situation caused for instance by a sudden adjustment in the interest rates or a default of a major financial institution. In general, the question whether (NFLVR) and/or (NFFLVR) hold for a model with a change point is difficult, see [24] for some results in this direction. Our integral tests provide explicit criteria, which are easy to verify. For many applications of the Markov switching model (1.2) it is important to know how the change to an ELMM affects the dynamics of the Markov chain ξ . As a second contribution, we study this question form a general perspective for independent sources of risk modeled via martingale problems. In particular, we show that the minimal local martingale measure (MLMM), see [25], preserves the independence and the laws of the sources of risk. To our knowledge, this property has not been reported in the literature. A third contribution of this article are integral tests for the martingale property of certain stochastic exponentials driven by Itô processes or switching diffusions. These characterizations are our key tools to study the absence of arbitrage.
We comment on related literature. For continuous semimartingale models the absence of arbitrage has been studied by Criens [11], Delbaen and Shirakawa [17], Lyasoff [40] and Mijatović and Urusov [43]. Criens, Delbaen and Shirakawa and Mijatović and Urusov proved integral tests for the existence of SMDs, ELMMs and EMMs in diffusion frameworks. Our results can be viewed as generalizations to an Itô process or Markov switching framework. For a model comparable to (1.1), Lyasoff proved that the existence of an ELMM is determined by the equivalence of a probability measure to the Wiener measure. The structure of this characterization is very different from our results. In Sect. 3.3 below we comment in more detail on the results in [11,17,40,43]. The martingale property of stochastic exponentials is under frequent investigation. At this point we mention the articles of Blanchet and Ruf [3], Cheridito et al. [5], Criens [11] and Kallsen and Muhle-Karbe [34]. Criens used arguments based on Lyapunov functions and contradictions to verify the martingale property of certain stochastic exponential in a multi-dimensional diffusion setting. We transfer these techniques to a general Itô process setting. Cheridito et al. and Kallsen and Muhle-Karbe related the martingale property of a stochastic exponential to an explosion probability via a method based on the concept of local uniqueness as defined in [30]. This technique traces back to work of Jacod and Mémin [29] and Kabanov et al. [31,32]. We use a similar argument for the Markov switching setting. The main difficulties are the proofs of explosion criteria and local uniqueness. Both approaches have a close relation to the work of Blanchet and Ruf, where a tightness criterion for the martingale property of non-negative local martingales has been proven. The connection between Lyapunov functions, explosion and tightness is for instance explained in [12].
Let us also comment on consecutive problems and extensions of our results: In case the discounted price process P is a positive Itô process of the type our results on the martingale property of stochastic exponentials can be used to obtain characterizations for no arbitrage with a similar structure as for the model (1.2). Moreover, in case P is the stochastic exponential of a diffusion with Markovian switching, i.e.
our martingale criteria yield conditions for no arbitrage with a similar structure as for (1.1). It is also interesting to ask about multi-dimensional models. In this case, results in the spirit of [11] can be proven by similar arguments as used in this article. However, the conditions are rather complicated to formulate and space consuming. Therefore, we restrict ourselves to the one-dimensional case.
The article is structured as follows. In Sect. 2 we give conditions for the martingale and strict local martingale property of certain stochastic exponentials. In Sect. 3.1 we study the model (1.1) and in Sect. 3.2 we study the model (1.2). In Sect. 4 we show that the MLMM preserves independence and laws for sources of risk and we explain how the MLMM can be modified to affect the law of an additional source of risk. The proofs are collected in the remaining sections.

Martingale property of stochastic exponentials
Fix a finite time horizon 0 < T < ∞ and let ( , F , F, P) be a complete filtered probability space with right-continuous and complete filtration F = (F t ) t∈ [0,T ] . Moreover, fix a state space I (l, r ) with −∞ ≤ l < r ≤ +∞.
In the following two sections we provide conditions for the martingale and strict local martingale property of certain stochastic exponentials.

The general case
Assume that S = (S t ) t∈[0,T ] is an I -valued Itô process with deterministic initial value S 0 ∈ I and dynamics where W = (W t ) t∈[0,T ] is a one-dimensional Brownian motion and b = (b t ) t∈[0,T ] and σ = (σ t ) t∈[0,T ] are real-valued progressively measurable processes. It is implicit that b and σ are such that the integrals are well-defined, i.e. a.s. We assume that λ \ ⊗ P-a.e. σ = 0, which latter will correspond to the assumption that we consider an asset price process with a non-vanishing volatility.
Let c = (c t ) t∈[0,T ] be a real-valued progressively measurable process such that a.s.
is a true or a strict local martingale. Here, E denotes the stochastic exponential. The structure of Z is very important in mathematical finance, because Z is the prototype of a strict local martingale density, see Lemma 3.2 below. Let a, a : I → (0, ∞), u, u : I → R and ζ : [0, T ] → R + be Borel functions such that .
where x 0 ∈ I is fixed. Let l n l, r n r be sequences such that l < l n+1 < l n < r n < r n+1 < r .
The first main result of this section is the following: Theorem 2.1 Assume the following: is a localizing sequence for Z , i.e. Z ·∧τ n is a martingale for every n ∈ N. We use the convention that inf(∅) ∞.
Then, Z is a martingale.
The proof of this theorem is given in Sect. 5.
Next, we provide a counterpart to Theorem 2.1. Let H be the set of all Borel functions h : R + → R + which are starting at zero, are strictly increasing and satisfy = ∞ for all ε > 0, and let K be the set of all Borel functions κ : R + → R + , which are starting at zero, are strictly increasing and concave and satisfy In case ( f , g) is one of the pairs (u, a), (u, a), . . . we say that ( f , g) satisfies the Yamada-Watanabe (YW) conditions, if for every n ∈ N there exist h n ∈ H and κ n ∈ K such that and for all x, y ∈ [l n , r n ] The second main result of this section is the following: Theorem 2.3 Assume one of the following conditions:

3)
and and Then, Z is a strict local martingale.
The proof of this theorem is given in Sect. 5. In Sect. 2.3 below we comment on the assumptions of Theorems 2.1 and 2.3 and related literature.

Markov switching case
In this section we consider a special case of the setting from Sect. 2.1 and assume that S is a switching diffusion. Before we introduce the setting in detail, we clarify terminology: A process is called a Feller-Markov chain if it is a Markov chain which is a Feller process in the sense that the corresponding transition semigroup is a self-map on the space of continuous functions vanishing at infinity. For conditions implying that a Markov chain is Feller-Markov we refer to [2]. It is also important to stress that whenever we have fixed a filtration and a Markov chain, we presume that the Markov chain is Markovian for the given filtration. All non-explained terminology for Markov chains, such as irreducible, recurrent, etc., can be found in [44]. We assume that S = (S t ) t∈[0,T ] is an I -valued Itô process with deterministic initial value S 0 ∈ I and dynamics where L S denotes the local time of S. The lemma is proven.
We are interested in the martingale property of the non-negative local martingale This definition coincides with (3.3) for the choices c = c(S, ξ) and N = 0. Before we state the main result of this section, we fix some notation. Because L 2 loc (I ) ⊂ L 1 loc (I ), (2.5) and (2.6) imply that Thus, we can set for (x, j) ∈ I × J and a fixed x 0 ∈ I . We say that σ satisfies the Engelbert (ES2) For every compact set K ⊂ I there are Borel functions g : K → R and h : R → [0, ∞] and a constant c > 0 such that the following properties are satisfied: We say that the Markov chain ξ is recurrent if it is a recurrent Markov chain when extended to the infinite time interval R + .
The following theorem gives an almost complete answer to the question when Z is a true or strict local martingale. A proof is given in Sect. 6. Theorem 2.5 (i) Suppose that c is bounded on compact subsets of I × J , that σ satisfies the ES conditions for all j ∈ J and that Then, Z is a martingale. (ii) Assume that ξ is recurrent and that there exists a j ∈ J such that σ satisfies the ES conditions for j and Then, Z is a strict local martingale.

Remark 2.6
The proof of Theorem 2.5 (ii) is based on a contradiction argument. In case (2.8) holds and Z is a martingale there exists an I -valued switching diffusion with an explosive regime j. The recurrence of ξ simplifies the proof that this switching diffusion reaches the regime j, which leads to a contradiction. In case the initial regime j 0 is already explosive, more precisely if σ satisfies the ES conditions for j 0 and lim x r v(x, j 0 ) < ∞ or lim x l v(x, j 0 ) < ∞, the proof needs no change even without the assumption that ξ is recurrent.
Noting that ξ is recurrent in case N < ∞, we obtain the following: In financial applications, N can be interpreted as the number of states of the business cycle and therefore N < ∞ is a reasonable assumption.

Comments on related literature
The martingale property of non-negative local martingales is under frequent investigation.
To the best of our knowledge, for a general Itô process or Markov switching setting Theorems 2.1, 2.3 and 2.5 are the first results which provide integral tests for the martingale property of certain stochastic exponentials.
For the diffusion case a complete characterization of the martingale property of the non-negative local martingale has been proven in [43]. We stress that in [43] the diffusion S is allowed to explode, which is a feature not included in our framework. Provided S is non-explosive, the main theorem of [43] shows that Z is a martingale if and only if where u b+cσ σ 2 and v is defined as in (2.2). The same condition is implied by either Theorems 2.1 and 2.3, or Corollary 2.7. For the strict local martingale property we require that σ satisfies the ES conditions, which are not imposed in [43].
The key idea underlying Theorems 2.1, 2.3 and 2.5 is a local change of measure combined with either a Lyapunov-type argument (in case of Theorem 2.1), a comparison with onedimensional diffusions (in case of Theorem 2.3) or a local uniqueness property (in case of Theorem 2.5).
The idea of using a local change of measure is not new. It has for instance been used in [5,11,13,50,53]. The Lyapunov and comparison arguments were inspired by [11], where a multi-dimensional diffusions has been studied. To use the ideas in our general setting, we prove a new Lyapunov condition for Itô processes and we transport the comparison arguments from a multi-dimensional diffusion setting to a one-dimensional Itô process framework, see Sect. 5 below. The idea of relating local uniqueness to the martingale property of a stochastic exponential traces back to [29,31,32]. More recently, the method was used in [5,11,34,53]. Although the terminology suggests the converse, local uniqueness is a strong version of uniqueness in law. In the proof of Theorem 2.5 we deduce local uniqueness from pathwise uniqueness by a Yamada-Watanabe-type argument.

On the absence and existence of arbitrage
Let 0 < T < ∞ be a finite time horizon and let ( , F , F, P) be a complete filtered probability space with right-continuous and complete filtration F = (F t ) t∈[0,T ] . We consider a financial market consisting of one risky asset with discounted price process P = (P t ) t∈[0,T ] , which is assumed to be a positive continuous semimartingale with deterministic initial value.
Recall the following classical terminology: A probability measure Q is called an equivalent In the following we study existence and non-existence of SMDs, ELMMs and EMMs in case P is either the stochastic exponential of an Itô process or a positive switching diffusion. In case P is a positive Itô process or the stochastic exponential of a real-valued switching diffusion similar results can be deduced from the martingale criteria in Sect. 2.

Stochastic exponential model
Suppose that P is the stochastic exponential of the real-valued Itô process S = (S t ) t∈[0,T ] with deterministic initial value S 0 ∈ R and dynamics are real-valued progressively measurable processes such that the stochastic integrals in (3.1) are well-defined. We assume that λ \ ⊗ P-a.e. σ = 0, which corresponds to the assumption that P has a non-vanishing volatility.

Absence of arbitrage
In the following we study when a SMD, ELMM or EMM exists. As a minimal condition we assume that (NUPBR) holds. This is equivalent to the existence of a market price of risk θ = (θ t ) t∈[0,T ] , i.e. a real-valued progressively measurable process such that a.s.
We define the continuous local martingale Integration by parts and (3.2) yield that which shows that Z P is a local martingale or, equivalently, that Z is a SLMD. We observe the following: (O1) If Z P is a martingale, then Z is a SMD by definition. In summary, to prove the existence of a SMD, ELMM and EMM we have to identify conditions for the martingale property of Z P and Z . The following is the main result of this section: Theorem 3.1 Suppose the following: is a localizing sequence for Z . (L2) There are Borel functions a : R → (0, ∞) and ζ : [0, T ] → R + such that Then, Z is a martingale, Q defined by dQ dP Z T is an ELMM and is a Q-Brownian motion such that

If in addition
then Q is an EMM and Z is a SMD.
Proof We apply Theorem 2.1 with I R, l n −n, r n n and c −θ . Note that (L1) equals (M1). Furthermore, set u( which shows that (M3) holds due to [36,Problem 5.5.27]. We conclude that Z is a martingale and that Q is an ELMM by (O2).
Next, we assume that (3.7) holds. We apply Theorem 2.1 with I R, l n −n, r n n and c σ − θ to show that the local martingale Hence, (M3) holds for Z . We conclude that Z is a martingale and the proof is complete.
In our setting there might exist several ELMMs and it is an important question which ELMM should be chosen for applications. The ELMM from Theorem 3.1 is the minimal local martingale measure (MLMM) as defined in [25]. 1 For financial interpretations of the MLMM we refer to [25] and for a general overview on possible applications we refer to [23]. In Theorem 4.3 below we discover a new property of the MLMM: The MLMM preserves independence and laws of sources of risk.
In the following paragraph we relate the assumptions (L1) and (L2) to so-called weakly equivalent local martingale measures (WELMM) as introduced in [37]. We explain the connection from a general point of view under the assumptions that F = F T and that (NUPBR) holds. With slight abuse of notation, let Z = (Z t ) t∈[0,T ] be a SLMD with localizing sequence (τ n ) n∈N . For every n ∈ N we can define a probability measure Q n by the Radon-Nikodym derivative dQ n dP Z T ∧τ n . It is easy to see that Q n is an ELMM for the stopped process P ·∧τ n . In other words, for every n ∈ N the notion (NFLVR) holds for all admissible strategies which invest riskfree after τ n . Roughly speaking, this observation suggests that (NFLVR) holds in case we can take the limit n → ∞. As explained in Section 2.4.2 of [37], Alaoglu's theorem yields that (Q n ) n∈N has an accumulation point Q for the weak * topology on the dual of L ∞ ( , F , P), which is a finitely additive probability such that Q(A) = 0 for all A ∈ F with P(A) = 0, see the "Appendix" of [14]. We use the sans-serif typeface to highlight that Q is not necessarily a probability measure, because it may fail to be countably additive. Note that Q = Q n on F τ n for every n ∈ N. Using this fact, it follows that for all A ∈ F with Q(A) = 0 we also have P(A) = 0, which shows that Q and P have the same null-sets. Indeed, if for all n ∈ N. This implies P(A ∩ {τ n > T }) = 0 and, because P-a.s. τ n ∞ as n → ∞, we conclude that P(A) = 0. Following [37], we call Q a WELMM. The main difference between WELMMs and ELMMs, and therefore between (NUPBR) and (NFLVR), is that a WELMM is not necessarily a measure.
The idea of condition (L1) is to identify a WELMM, which, as explained above, is a natural candidate for an ELMM. Assuming that (τ n ) n∈N is given by (3.5) means controlling the MPR via the size of the asset. This assumption is reasonable from a modeling perspective, because, as explained by Lyasoff [40, p. 488], "excessively large expected instantaneous net returns from risky securities entail excessively large demands for money (to invest in such securities), which, in turn, means higher and higher interest rates, which, in turn, means lower and lower market price of risk". In the diffusion settings of Mijatović and Urusov [42], (L1) is implied by the local integrability condition [42, Eq. 3.2] on the MPR, see [43,Lemma 6.3].
Condition (L2) takes care on the countable additivity of the candidate WELMM, which corresponds to problems arising when n → ∞. Indeed, Q is countably additive if and only if which is also the condition we check in the proof of Theorem 2.1. If Q is countably additive, then (3.8) follows from the monotone convergence theorem and the fact that P-a.s. τ n ∞ as n → ∞. Conversely, assume that (3.8) Thus, Q is continuous at zero, which implies that it is countably additive.

Existence of a financial bubble
In Theorem 3.1 we gave conditions for the existence of an ELMM. In this section, we derive a counterpart to (3.7), which implies the existence of a financial bubble in the sense of [9].
As we explain next, the question when no SMD exists is strongly connected to the question when a non-negative local martingale is a strict local martingale. We recall the following: In case Z is a SMD, (3.9) holds and is a martingale by definition. If this is not the case, we have a contradiction and no SMD exists.
The following is the main result of this section: Then, no SMD exists.
Proof We use Theorem 2.3 with I R and u 1 to show that Z P as defined in (3.10) is a strict local martingale. Because θ is a MPR, Thus, the conditions from part (ii) of Theorem 2.3 hold and we conclude that Z P is a strict local martingale. Consequently, as explained above, no SMD exists.
The conditions (3.7) and (3.11) provide a test for the MLMM to be an EMM or not. In a diffusion setting the conditions boil down to a single sufficient and necessary condition, which is also given in [11, Proposition 5.2].

Example: diffusion models with a change point
Fontana et al. [24] study (NUPBR) and (NFLVR) for a model with a change point. The authors are interested in the influence of filtrations, which represent different levels of information. Under a weak form of the H -hypothesis the model can be included into our framework. More precisely, in this case S is of the form where τ is a stopping time. The coefficient σ (i) is assumed to be positive, continuous and Lipschitz continuous in the second variable uniformly in the first, see [24, Condition I]. Theorem 3.1 provides local conditions for (NFLVR). In particular, for the special cases described in [24, Section 3.3], Theorem 3.1 yields that (NFLVR) always holds, because where μ (i) is locally bounded. This extends the observation from [24] that (NUPBR) holds in these cases. Furthermore, if in addition to (3.12) for i = 1, 2 then even (NFFLVR) holds. The notion (NFFLVR) has not been studied in [24].

Diffusion model with Markov switching
In this section, we assume that P is a positive continuous semimartingale with deterministic initial value P 0 ∈ (0, ∞) and dynamics We can interpret N as the number of all possible states of the business cycle. The assumption of irreducibilily means that we exclude all states of the business cycle which are not attainable from the initial state. We assume ξ to be a Feller process for technical reasons. In case N < ∞ any Markov chain with values in J is a Feller process, because all real-valued functions on J are continuous and vanishing at infinity. Due to Lemma A.1 in the "Appendix", the sources of risk ξ and W are independent. The lemma even shows that it is not possible to model ξ and W as Markov processes for a superordinate filtration without their independence. This observation gives a novel interpretation for the independence assumption, which is typically interpreted as the price process being influenced by the business cycle and an additional independent source of risk represented by the driving Brownian motion.

Absence and existence of arbitrage
We impose the following two assumptions: The coefficient b is bounded on compact subsets of (0, ∞) × J , σ 2 is bounded away from zero on compact subsets of (0, ∞) × J and σ satisfies the ES conditions for all j ∈ J , see Sect. 2.2 for this terminology.
We define which is a Borel map bounded on compact subsets of (0, ∞) × J . The process θ t θ(P t , ξ t ) is a MPR. We define the continuous local martingale Z as in (3.3). Note that the observations (O1) -(O3) in Sect. 3.1 also hold in this setting. We call the E(L)MM Q with Radon-Nikodym derivative dQ dP = Z T the minimal (local) martingale measure (M(L)MM). The following theorem provides conditions for the existence of the M(L)MM and for Z to be a SMD.
Then, Z is a martingale and the probability measure Q defined by the Radon-Nikodym derivative dQ dP Z T is an ELMM. Moreover, B as defined in (3.6) is a Q-Brownian motion such that

14)
then Q is an EMM.
Proof The claim follows similar to the proof of Theorem 3.1 when Theorem 2.5 is used instead of Theorem 2.1.
Theorem 2.5 suggests that in case ξ is recurrent, the conditions in Theorem 3.4 are sufficient and necessary. The following theorem makes this precise.
Theorem 3.5 Suppose that ξ is recurrent.
then Z is a strict local martingale and the MLMM does not exist.
then Z is no SMD. In particular, the MMM does not exist.
Proof The claim follows similar to the proof of Theorem 3.3 when Theorem 2.5 is used instead of Theorem 2.3.
Recalling that in case N < ∞ the Markov chain ξ is recurrent, we obtain the following: We will see in the next section that in case one of the frozen markets allows arbitrage, it is not possible to find a risk-neutral market in which the business cycle has Markovian dynamics.

Non-existence of structure preserving ELMMs and EMMs
Let L sp the set of all ELMMs Q such that ξ is an irreducible recurrent Feller-Markov chain on ( , F , F, Q) and let M sp be the set of all EMMs in L sp . The main result of this section is the following: In Sect. 4 we show that an equivalent change to the MLMM does not affect the Markov chain ξ . Thus, Theorem 3.7 generalizes Theorem 3.5.

Example: Markov switching CEV model
We consider a version of the CEV model (see [10]) with Markov switching. Take β : J → (0, ∞) and assume that for all j ∈ J . Then, the discounted asset price process P exists due to Theorem 6.1 below.
In case N < ∞, Corollary 3.6 shows the following:

Comments on related literature
For continuous semimartingale markets the existence and non-existence of SMDs, ELMMs and EMMs has been studied in [11,17,40,43]. We comment on these works in more detail.
In [17,43] a one-dimensional diffusion framework has been considered. We discuss the results from [43] and refer to [43,Remark 3.2] for comments on the relation between [17] and [43]. In [43] it is assumed that the price process P = see also [36,Definition 5.5.20]. In the following we assume that P cannot explode to zero. In [43] the notions (NFLVR) and (NFFLVR) are also studied in case P can explode to zero and (NFLVR), (NFFLVR) and (NRA) are further studied for the infinite time horizon. For the non-explosive case the results from [43] are as follows: [43,Theorem 3.11]. Applying Corollary 3.14 with N = 1 shows versions of (a) -(c) under slightly more restrictive regularity assumptions on b and σ . The novelty of Corollary 3.14 or more generally Theorems 3.4 and 3.5 is their scope of application.
A multi-dimensional diffusion setting has been studied in [11]. We explain the onedimensional version: Assume that the price process P = (P t ) t∈[0,T ] is the stochastic exponential of where b, σ : R → R are locally bounded Borel functions such that σ 2 is locally bounded away from zero. In this setting, [11,Propositions 5.1] shows that (NFLVR) always holds and [11,Proposition 5.2] Under slightly different regularity assumptions on b and σ , the same observation follows from Theorems 3.1 and 3.3. The novelty of Theorems 3.1 and 3.3 is that no diffusion structure is needed. In particular, the coefficients b and σ are allowed to depend on the path of S or several sources of risk. In [11] the main interest lies in the multi-dimensional setting. We stress that it is possible to extend our results to a multi-dimensional framework. The structure of the conditions will be similar as in [11].
In [40] the price process P = (P t ) t∈[0,T ] is assumed to be the stochastic exponential of where X = (X t ) t∈[0,T ] is a continuous process, α and θ are suitable processes such that the integrals are well-defined and λ \ ⊗ P-a.e. α = 0. The process X is called information process. This setting is closely related to those from Sect.

Modifying minimal local martingale measures
In Sect. 3.2.1 we proved conditions for the existence of the minimal (local) martingale measure in a Markov switching framework. We ask the following consecutive questions: 1. Does the MLMM change the dynamics of the Markov chain? 2. Is it possible to modify the MLMM such that the dynamics of the Markov chain are changed in a tractable manner?
In this section we answer these questions from a general perspective under an independence assumption, which holds in our Markov switching framework.

Martingale problems
To characterize additional sources of risk in our financial market, we introduce a martingale problem.
Let J be a Polish space, define D(R + , J ) to be the space of all càdlàg functions R + → J and D to be the σ -field generated by the coordinate process X = (X t ) t≥0 , i.e. X t (ω) = ω(t) for ω ∈ D(R + , J ) and t ∈ R + . We equip D(R + , J ) with the Skorokhod topology, which renders it into a Polish space. It is well-known that D is the Borel σ -field on D(R + , J ). We refer to [20,30] for more details. Let Due to [20, Proposition 2.1.5], ρ n is a D o -stopping time and, due to [20,Problem 4.27], ρ n ∞ as n → ∞. We will use the sequence (ρ n ) n∈N as a localizing sequence for test martingales of our martingale problem. We fix this sequence, because for some arguments we need a common localizing sequence consisting of D o -stopping times.
The input data for our martingale problem is the following: where PM denotes the space of all D-progressively measurable processes. (iii) An initial value j 0 ∈ J . (iv) A time horizon 0 < T ≤ ∞.
We use the convention that in case T = ∞ the interval [0, T ] is identified with R + .
, supporting a càdlàg, adapted, J -valued process ξ = (ξ t ) t∈[0,T ] . We say that ξ is a solution process to the martingale problem (A, L, j 0 , T ), if for all f ∈ A and n ∈ N the process Martingale problems were introduced by Stroock and Varadhan [54] in a diffusion setting. Martingale problems for semimartingales were studied in [27] and Markovian martingale problems with a Polish state space were studied in [20]. Our definition is unifying in the sense that it deals with non-Markovian processes and a Polish state space. Most of the conditions for existence and uniqueness given in [20,27,54] also apply to our setting.

How to modify the MLMM
Fix a finite time horizon 0 < T < ∞ and let ( , F , F, P) be a complete filtered probability space with right-continuous and complete filtration F = (F t ) t∈[0,T ] , which supports a solution process ξ = (ξ t ) t∈[0,T ] to the martingale problem (A, L, j 0 , T ). Let W = (W t ) t∈[0,T ] be a one-dimensional Brownian motion such that σ (W t , t ∈ [0, T ]) and σ (ξ t , t ∈ [0, T ]) are independent. We think of W and ξ as two independent sources of risk influencing the market. The independence assumption is satisfied when ξ is a Feller-Markov chain, see Lemma A.1 in the "Appendix".
In the following theorem we find a new property of the MLMM. To wit, we show that the MLMM preserves the independence of the sources of risk and their laws. Because the M(L)MM is often used for pricing, this observation is important for analytical and numerical computations. We prove the following theorem in Sect. 7. We ask further whether it is possible to modify the MLMM such that the law of ξ can be affected in a tractable manner. An answer to this question is provided by the next theorem. A proof can be found in Sect. 8.

Theorem 4.4 Let f ∈ A be strictly positive and suppose that the process
is a martingale. Set Suppose that for every g ∈ A * and n ∈ N there exists a constant C = C(g, n) > 0 such that a.s.
Define the probability measure Q by the Radon-Nikodym derivative dQ dP Z T . Then, σ (ξ t , t ∈ [0, T ]) and σ (W t , t ∈ [0, T ]) are Q-independent, W is a Q-Brownian motion and ξ is a solution process for the martingale problem (A * , L * , j 0 , T ) on ( , F , F, Q). We explain an application of Theorem 4.4: Suppose that the MLMM exists. Then, using the change of measure described in Theorem 4.4, the MLMM can be changed further such that the law of ξ gets affected as described in the theorem, while the local martingale property of the price process is preserved. We stress that in this manner the MLMM induces a family of ELMMs, which is often infinite. In a Markov switching framework with N < ∞ the following proposition explains how the Q-matrix of the driving Feller-Markov chain can be changed.

Proof See [45, Proposition 5.1].
A useful criterion for the martingale property of (4.3) is given by Theorem 4.9 below. We consider it as an extension of results from [5,27,54]. In the following X = (X t ) t≥0 denotes the coordinate process on D(R + , J ).  Note that

Definition 4.7 A set
Because C 0 (J ) equipped with the uniform metric is a separable metric space, G is a separable metric space when equipped with the taxicap uniform metric. Hence, we find a countable set Due to [20, Proposition 4.3.1], A is a determining set for the martingale problem (A, L, ∞).
A proof for the following theorem can be found in Sect. 8.
where K maps A * into the space of Borel functions J → R. Finally, assume that the martingale problem (A * , L * , ∞) is well-posed and that (ρ n (ξ )) n∈N is a localizing sequence for the local martingale Roughly speaking, this theorem shows that in Markovian settings we can modify the law of ξ whenever the martingale problem (A * , L * , ∞) is well-posed.

Proof of Theorems 2.1 and 2.3
The following section is divided into three parts. In the first part we prove Lyapunov-type conditions for non-explosion of Itô processes, in the second part we prove non-existence conditions for Itô processes and in the third part we deduce Theorems 2.1 and 2.3.

Criteria for non-explosion
In this section we pose ourselves into a version of the setting from Sect. 2.1. Let I = (l, r ) be as in Sect. 2.1 and ( , F ) be a measurable space which supports three real-valued processes . For every n ∈ N we fix a probability measure Q n and a right-continuous Q n -complete filtration F n = (F n t ) t∈[0,T ] on ( , F ) such that S, b and σ are F n -progressively measurable. We set τ n as in Theorem 2.1, i.e. where l n l, r n r are sequences such that l < l n+1 < l n < r n < r n+1 < r . Moreover, suppose that Q n -a.s.
is a Brownian motion on ( , F , F n , Q n ). It is implicit that the integrals are well-defined. We also assume that λ \ ⊗ Q n -a.e. σ = 0 for all n ∈ N (5.1) and we fix a Borel function ζ : [0, T ] → R + such that ζ ∈ L 1 ([0, T ]).

A Lyapunov criterion
In this section we give a Lyapunov-type condition for lim sup n→∞ Q n (τ n = ∞) = 1.
For f ∈ C 1 (I , R) with locally absolutely continuous derivative, it is well-known that there exists a λ \ -null set N f ⊂ I such that f has a second derivative f on I \N f . In this case, we set

Suppose there exists a λ \ -null set N ⊂ I such that
Then, (5.2) holds.
Proof Let L S be the local time of the continuous Q n -semimartingale S ·∧τ n . The occupation times formula yields that Q n -a.s.
which implies that Q n -a.s λ \ ({t ∈ [0, τ n ∧ T ]: S t ∈ N }) = 0. We will use this fact in the following without further reference. Set Using a generalized version of Itô's formula (see [49, Lemma IV.45.9]), we obtain that the process is a local Q n -martingale. We deduce from (5.4) and the fact that non-negative local martingales are supermartingales, that Q n -a.s.
W.l.o.g. we assume that S 0 ∈ (l 1 , r 1 ). Note that for all n ∈ N we have Q n -a.s. S τ n ∈ {l n , r n } on {τ n ≤ T }. We conclude that for all n ∈ N

By (5.3) there exists a sequence (n
which implies (5.2). The proof is complete.
Recall from Sect. 2 that in case ( f , g) is one of the pairs (u, a), (u, a) we set dudy, x ∈ I , (5.5) for a fixed x 0 ∈ I . The main result of this section is the following: Moreover, for all n ∈ N assume that for Then, (5.2) holds.

Criteria for non-existence
In this section we give a converse to Theorem 5.2. As in Sect. 2, let I = (l, r ) with −∞ ≤ l < r ≤ +∞ and let a : I → (0, ∞) and u, u : I → R be Borel functions such that 1 a + |u| + |u| ∈ L 1 loc (I ). If where it is implicit that the integrals are well-defined. Then, there exists no pair (Q, B) ∈ I such that for Proof (i). We use a comparison and contradiction argument as in the proof of [11,Theorem 4.1]. For contradiction, assume that (Q, B) ∈ I is such that (5.8) holds. W.l.o.g. we assume that F is Q-complete. In the following we work on ( , F , F, Q). Because a is positive and continuous and a.s is a.s. finite, continuous and strictly increasing, which implies that the same holds for the function provided the integrals are well-defined. Moreover, the process We deduce from Lemma 5.4 that a.s.
We will use this observation in the following without further reference. Applying Lemma 5.4 with yields that a.s. (5.10) Using again Lemma 5.4, we obtain that a.s. for all t ∈ [0, T ] Due to Lemma 5.4 and (5.10), we obtain that a.s. for all t ∈ [0, T ] Consequently, B is a continuous local F φ -martingale with a.s.
i.e. an F φ -Brownian motion due to Lévy's characterization. We summarize that Using a standard extension of ( , F , F φ , Q) we can extend (B t ) t∈[0,T ] to a Brownian motion (B t ) t≥0 , see, e.g., the proof of [47, Theorem V.1.7]. We will use the following terminology: When we say that (V t ) t≥0 is a continuous [l, r ]valued process we mean that all its paths are continuous in the [l, r ]-topology and absorbed in {l, r }, i.e. that V t = V τ (V ) for all t ≥ τ (V ) inf(t ∈ R + : V t / ∈ I ). This convention is in line with [36,Definition 5.5.20].
Definition 5.5 Let μ : I → R and v : I → R be Borel functions. We say that an SDE 11) where (B * t ) t≥0 is a one-dimensional Brownian motion, satisfies strong existence and uniqueness up to explosion, if on any complete probability space , which supports a Brownian motion (B * t ) t≥0 and an I -valued F o 0 -measurable random variable ψ, there exists a up to indistinguishability unique adapted continuous [l, r ]-valued process (V t ) t≥0 such that a.s.
It is implicit that the integrals are well-defined. The process (V t ) t≥0 is called the solution process to (5.11) with driver (B * t ) t≥0 .
Due to [19,Remark 4.50 (2), Theorem 4.53], the SDE satisfies strong existence and uniqueness up to explosion. Consequently, there exists a solution process (Y t ) t≥0 to (5.12) with driver (B t ) t≥0 . The following lemma is proven after the proof of Theorem 5.3 is complete.
(ii). For contradiction, assume that (Q, B) ∈ I is such that (5.9) holds. By the same arguments as in part (i), there exists a process (Y t ) t≥0 such that T ] reaches l with positive probability and again the pathwise ordering gives a contradiction.
Proof of Lemma 5. 6 There are functions h n ∈ H and κ n ∈ K such that for all x, y ∈ [l n , r n ] We set Note that for all t ∈ (0, T ] we have Thus, [47,Lemma IX.3.3] implies that the local time of Y ·∧ρ n − S φ ·∧ρn in the origin is a.s. zero. We deduce from Tanaka's formula that a.s.
Taking expectation, using the martingale property of the Brownian part of Y ·∧ρ n − S φ ·∧ρn and Jensen's inequality yields that for all t ∈ [0, T ] Finally, Bihari's lemma (see [11,Lemma E.2]) yields that for all t ∈ [0, T ] Consequently, due to the continuous paths of Y and S φ , the claim follows.

Proof of Theorem 2.1
Because non-negative local martingales are supermartingales, Z is a martingale if and only if E P [Z T ] = 1. By (M1), we can define Q n by the Radon-Nikodym derivative dQ n dP = Z T ∧τ n . We note that the assumption λ \ ⊗ P-a.e. σ = 0 implies (5.1). Due to Girsanov's theorem, there exists a Q n -Brownian motion B n = (B n t ) t∈[0,T ] such that The monotone convergence theorem yields that In view of (M2) and (M3), Theorem 5.2 yields that lim sup n→∞ Q n (τ n = ∞) = 1.
Thus, E P [Z T ] = 1 and the proof is complete.

Proof of Theorem 2.3
For contradiction, assume that (Z t ) t∈[0,T ] is a martingale. Define a probability measure Q by the Radon-Nikodym derivative dQ dP Z T . By Girsanov's theorem, there exists a Q-Brownian motion B = (B t ) t∈[0,T ] such that Consequently, in case (SL1) holds we obtain a contradiction to part (i) of Theorem 5.3 and in case (SL2) holds we obtain a contradiction to part (ii) of Theorem 5.3. The proof is complete.

Proof of Theorem 2.5
The section is split into two parts: First, we prove existence, non-existence and local uniqueness for switching diffusions and second, we deduce Theorem 2.5.

Existence and non-existence criteria
As in Sect. 2.2, let I = (l, r ) with −∞ ≤ l < r ≤ +∞ and J = {1, . . . , N } with 1 ≤ N ≤ ∞. Moreover, let u : I × J → R and σ : I × J → R\{0} be Borel functions such that We fix x 0 ∈ I and set v(x, j) for (x, j) ∈ I × J . Let ( , F , F, P) be a filtered complete probability space with a rightcontinuous and complete filtration F = (F t ) t≥0 , which supports a Brownian motion W = (W t ) t≥0 , a J -valued irreducible continuous-time Feller-Markov chain ξ = (ξ t ) t≥0 and an I -valued F 0 -measurable random variable φ. The main result of this section is the following: Then, there exists an adapted I -valued continuous process (Y t ) t≥0 such that

3)
where it is implicit that the integrals are well-defined. (ii) Assume there exists a j ∈ J such that σ satisfies the ES conditions for j and Let 0 < T ≤ ∞ be a time horizon. If ξ is recurrent, then there exists no adapted I -valued continuous process Y = (Y t ) t∈[0,T ] such that (6.3) holds.
Proof The case N = 1 concerns classical diffusions for which all claims are known, see [4,19,36] for details. We prove the claim under the assumption N ≥ 2.
(i). We define the jump times of ξ inductively by Because ξ is irreducible, we have a.s. γ n < ∞ (see [33,Theorem 10.19]) and a.s. γ n −γ n−1 > 0 for all n ∈ N. We follow the idea from the proof of [26, Theorem IV.9.1] and construct the process Y explicitly from solutions to the SDEs where W = (W t ) t≥0 is a Brownian motion. For the construction we require a strong existence and uniqueness property, which we explain next. Fix j ∈ J . It follows from [19,Remark 4.50 (2), Theorem 4.53] and Feller's test for explosion (see [36,Theorem 5.5.29]) that the SDE (6.4) has a weak solution and that it satisfies pathwise uniqueness for all deterministic initial values. We conclude from [33,Theorem 18.14] that there exists a Borel function F j : I × C(R + , R) → C(R + , I ) such that for any one-dimensional Brownian motion W = (W t ) t≥0 and any I -valued random variable ψ, which is independent of σ (W t , t ∈ R + ), the process X j = F j (ψ, W ) is a solution process to (6.4) with X j 0 = ψ, which is adapted to the completion of the natural filtration of W and ψ, see [36, Definition 5.2.1]. The function F j is independent of the law of ψ and universally adapted (see [33, p. 346] for a definition).
Set W n W ·+γ n − W γ n . Due to [47, Proposition V.1.5] and Lévy's characterization, W n is a Brownian motion for the filtration F n (F t+γ n ) t≥0 . In particular, W n is independent of F γ n . By induction, define Moreover, set The process Y is I -valued and continuous and, as we explain next, it is also F-adapted. Define H t Y n t−γ n 1 {γ n <t} . We claim that (H t ) t≥0 is F-progressively measurable. Because t → Y n t−γ n 1 {γ n <t} is left-continuous and s → Y n t−s 1 {s<t} is right-continuous, an approximation argument shows that is suffices to explain that (h t ) t≥0 (Y n t−ζ 1 {ζ <t} ) t≥0 is F-adapted for any F-stopping time ζ which takes values in the countable set 2 −m N for some m ∈ N and satisfies ζ ≥ γ n . Let G ∈ B(R) and set N m,t 2 −m N ∩ [0, t). We have Here, we use that {Y n t−k ∈ G} ∈ F t−k+γ n ⊆ F t−k+ζ and that F t−k+ζ ∩ {ζ = k} ∈ F t . Thus, (H t ) t≥0 is F-progressively measurable and consequently (Y t ) t≥0 is F-adapted.
We note that γ n − γ n−1 = inf(t ∈ R + : ξ t+γ n−1 = ξ γ n−1 ), which is an F n−1 -stopping time. Thus, Y n−1 γ n −γ n−1 is F γ n -measurable and therefore independent of σ (W n t , t ∈ R + ). This yields that the process X n, j F j (Y n−1 Thus, due to classical rules for time-changed stochastic integrals (see [47, By induction, a.s. for t ∈ [γ n , γ n+1 ] Therefore, the process Y satisfies the SDE and the proof of (i) is complete.
(ii). For contradiction, assume that Y satisfies (6.3). Let j ∈ J be such that Because ξ is recurrent, we have a.s. δ < ∞, see [44,Theorem 1.5.7]. Due to the strong Markov property of ξ and [33,Lemma 10.18], for all G ∈ B(R + ) it holds that P(ζ − δ ∈ G) = − G q j j e q j j x dx, (6.5) where q j j < 0 is the j-th diagonal element of the Q-matrix of ξ . Recall our convention that we call a process V = (V t ) t≥0 to be continuous and [l, r ]valued in case all paths are continuous in the [l, r ]-topology and absorbed in {l, r }, i.e. that V t = V τ (V ) for all t ≥ τ (V ) inf(t ∈ R + : V t / ∈ I ). It follows from [19,Remark 4.50 (2), Theorem 4.53] that the SDE (6.4) satisfies strong existence and uniqueness up to explosion in the sense of Definition 5.5.
Consequently, there exists a continuous [l, r ]-valued process X = (X t ) t≥0 such that where W δ W ·+δ∧T − W δ∧T is a Brownian motion for the filtration F δ (F t+δ∧T ) t≥0 . We prove the following lemma after the proof of (ii) is complete.
Because on {τ (X ) < ∞} we have X τ (X ) / ∈ I , Lemma 6.2 implies that The proof of the following lemma is given after the proof of (ii) is complete.
We conclude from Lemma 6.3 and Galmarino's test (see [30,Lemma III.2.43]) that for all Because strong existence and uniqueness up to explosion holds for the SDE (6.4), for a.a.

Lemma 6.4
For all G ∈ B(R + ) we have a.s.
Using (6.7), the monotone convergence theorem and then (6.9), we obtain that implies that for λ \ -a.a. s ∈ (0, T ) we have P(δ ≤ T − s) = 0. However, because ξ is irreducible, we have P(ξ t = j) > 0 for all t > 0. This is a contradiction and the proof of (ii) is complete.

Proof of Lemma 6.2 Define
which shows that ι is an F δ -stopping time. Moreover, we have for all s, t ∈ R + Thus, the random time s ∧ ι + δ ∧ T is an F-stopping time. We deduce from classical rules for time-changed stochastic integrals that a.s. for all t ∈ R + Because the SDE (6.4) satisfies strong existence and uniqueness up to explosion, Lemma 6.3 implies that a.s. Y t∧ι+δ∧T = X t∧ι for all t ∈ R + . On {ζ ≤ T } ⊆ {δ ≤ T } we have ι = ζ − δ and the claim follows.
We set The process Z has continuous paths and similar arguments as used in the proof of Theorem 6.1 (i) show that it is F-adapted. Let θ Z n inf(t ∈ R + : Z t / ∈ (l n , r n )).
Next, we discuss what happens on the set Classical rules for time-changed stochastic integrals yield that on {τ < t ∧ θ Z n } a.s.
We conclude that Z is a solution process of the SDE (5.11) with driver W and initial value ψ. By the strong existence and uniqueness hypothesis, we conclude that a.s. Z = V . The definition of Z implies the claim.
Proof of Lemma 6.4 Denote the Wiener measure with initial value x ∈ R by W x and by μ j the law of a Feller-Markov chain with the same Q-matrix as ξ and initial value j ∈ J . Let C be the σ -field on C(R + , R) generated by the coordinate process. We deduce from Lemma A.
is Borel for every F ∈ D ⊗ C and that the process (ξ, W ) is a strong Markov process in the following sense: For all F ∈ D ⊗ C and every a.s. finite stopping time θ we have a.s.
For all A ∈ D and F ∈ C the strong Markov properties of ξ, W and (ξ, W ) imply that a.s.
The proof is complete.

Local uniqueness
For the space of continuous functions from R + into I or R, we denote by C the σ -field generated by the coordinate process. Moreover, we denote by C o (C o t ) t≥0 the filtration generated by the corresponding coordinate process and by C (C t ) t≥0 its right-continuous version. The image space will be clear from the context. Let where U ⊆ I and V ⊆ J are open: Let u : I × J → R and σ : I × J → R\{0} be Borel functions such that (6.1) holds, σ satisfies (6.2) and the ES conditions for all j ∈ J (see Sect. 2.2 for this terminology). In other words, we ask that the conditions from part (i) of Theorem 6.1 hold.
For i = 1, 2, let ( i , F i , F i , P i ) be a filtered probability space with right-continuous complete filtration F i = (F i It is implicit that the stochastic integrals are well-defined. We stress that ξ 1 and ξ 2 have the same law, because they have the same Q-matrix, see Example 4.2.
The main observation of this section is the following: Proof We follow the Yamada-Watanabe-type idea used in [28]. Define * and for i = 1, 2 Denote the Wiener measure by W and denote the unique law of ξ i by μ. Due to Lemma A.1 in the "Appendix", we have When the space of continuous functions is equipped with the local uniform topology it is a Polish space and the corresponding Borel σ -field is generated by the coordinate process. Thus, there exist regular conditional probabilities is a Q n -Brownian motion such that We deduce from Lemma A.1, Example 4.2 and Theorem 4.3 that under Q n the process ξ remains a Feller-Markov chain with unchanged Q-matrix. W.l.o.g. we extend W , ξ and F to the infinite time interval R + . Applying Theorem 6.1 with u b + cσ yields that on ( , F , F, P) there exists an adapted continuous I -valued process X = (X t ) t≥0 such that We set It follows from Lemma 6.5 and Theorem 6.6 that Consequently, using Galmarino's test, we obtain that lim n→∞ Q n (τ n = ∞) = lim n→∞ P(ρ n = ∞) = 1.
Now, it follows as in the proof of Theorem 2.1 that Z is a martingale.
(ii). This result follows similar as Theorem 2.3, where Theorem 6.1 has to be used instead of Theorem 5.3. We omit the details.

Proof of Theorem 4.3
Step 1 Let g ∈ A and set Due to the definition of the martingale problem (A, L, T ), the process M g is a local martingale with localizing sequence (ρ n (ξ )) n∈N . Thus, the quadratic variation process [M g , W ] is welldefined. Our first step is to show that a.s. [M g , W ] = 0. We explain that W M g is a local martingale for the completed right-continuous version of the natural filtration of ξ and W .
The independence assumption yields that Step 2), the optional stopping theorem implies that for all stopping times ψ ≤ T Consequently, by [47,Proposition II.1.4], M g ·∧ρ m (ξ ) is a Q-martingale. Because Q ∼ Q, this implies that on ( , F , F, Q) the process ξ is a solution process for the martingale problem (A, L, j 0 , T ). The uniqueness assumption for the martingale problem (A, L, j 0 , T ) implies that for all where G 1 , . . . , G n ∈ B(J ) and 0 ≤ t 1 < · · · < t n ≤ T . We fix such that Q( ) > 0 and define Using the definition of Q, (7.2), the fact that K f is a Q-martingale and the optional stopping theorem, we obtain Because ζ was arbitrary, we conclude that K f is a Q-martingale. Furthermore, Q(B 0 = 0) = 1 follows from the fact that B is a Q-Brownian motion. Finally, due to [20,Proposition 4.3.3], the process B is a Q-Brownian motion. We conclude that for all F 1 , . . . , F k ∈ B(R) and 0 ≤ s 1 < · · · < s k ≤ T . By the definition of Q, we have proven that which implies that the σ -fields σ (ξ t , t ∈ [0, T ]) and σ (B t , t ∈ [0, T ]) are Q-independent. The proof is complete.
The P-independence of σ (ξ t , t ∈ [0, T ]) and σ (W t , t ∈ [0, T ]) and the uniqueness of the Wiener measure yield that Q( 1 ∩ 2 ) = E P Z T 1 1 ∩ 2 9 Proof of Theorem 4.9 Let (X t ) t≥0 be the coordinate process on D(R + , J ) and denote Define by μ P • ξ −1 a Borel probability measure on D(R + , J ). We have to show that It follows from [27,Lemma 2.9] that M f is a local μ-martingale with localizing sequence (ρ n ) n∈N . For all n ∈ N, define a Borel probability measure μ n on D(R + , J ) via the Radon-Nikodym derivative The following lemma is proven after the proof of Theorem 4.9 is complete. This completes the proof.
Proof of Lemma 9. 1 We adapt the proof of [30, Theorem III.2.40]. To simplify our notation, we set T ∧ ρ n ρ. We denote by μ j the unique law of a solution process to the martingale problem (A * , L * , j, ∞).
Step 1 We show that j → μ j (G) is Borel for all G ∈ D, following the strategy outlined in [54,Exercise 6.7.4]. Recall that we assume that A * contains a countable determining set A. Let P be the space of Borel probability measures on D(R + , J ) equipped with the topology of convergence in distribution. Note that a D o -adapted process is a D-martingale if and only if it is a D o -martingale. The implication ⇒ follows from the downward theorem (see [48, Theorem II.51.1]) and the implication ⇐ follows from the tower rule. For g ∈ A set Define I to be the set of all ν ∈ P such that ν • X −1 0 = δ j for some j ∈ J . Moreover, let M be the set of all ν ∈ P such that E ν (K g t∧ρ m − K g s∧ρ m )1 G = 0, for all g ∈ A, all rational s < t, m ∈ N and G in a countable determining class of D o s . By the uniqueness assumption, {μ j , j ∈ J } = I ∩ M. Because the set {δ j , j ∈ J } is Borel due to [8,Theorem 8.3.7] and ν → ν • X −1 0 is continuous, I is Borel. The set M is Borel due to [1,Theorem 15.13]. We conclude that {μ j , j ∈ J } is Borel. Let : {μ j , j ∈ J } → J be defined by (μ j ) = j for all j ∈ J . We note that is a continuous injection. Thus, the inverse map −1 is Borel due to Kuratovski's theorem ( [8,Proposition 8.3.5]). This means that j → μ j (G) is Borel for all G ∈ D.
Step 2 Because μ n ∼ μ, we have μ n (X 0 = j 0 ) = 1. As in the proof of Theorem 4.4, we see that for all g ∈ A * the process K g ·∧ρ is a μ n -martingale.
It follows from [30, Lemma III.2.47] that ν is a probability measure, i.e. that ν is defined unambiguously. Our goal is to show that ν solves the martingale problem (A * , L * , j 0 , ∞).
Providing an intuition, ν is the law of in case X 1 is sampled according to μ n and X 2 is sampled according to μ j with j = X 1 ρ(X 1 ) . In other words, we extend μ n to a solution of the global martingale problem. For G ∈ D o 0 we can choose G = G × D(R + , J ). Consequently, ν(X 0 = j 0 ) = μ n (X 0 = j 0 ) = 1.
Acknowledgements Open Access funding provided by Projekt DEAL. The author thanks the referee for excellent reports which helped to improve the article substantially. Moreover, he is grateful to the associate editor for many valuable suggestions.
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Itô's formula yields that K is a martingale. The process M is a martingale by Dynkin's formula. Thus, integration by parts yields that (A.1) Because ξ has only finitely many jumps in a finite time interval, the process M is of finite variation over finite time intervals. Thus, since K has continuous paths, it follows that a.s. [M, K ] = 0. In view of (A.1), we conclude that M K is a martingale. Now, recalling Example 4.2, we can argue as in Step 3 of the proof of Theorem 4.3 to obtain the claimed independence.