A class of F -doubly stochastic Markov chains ∗

We deﬁne a new class of processes, very useful in applications, F -doubly stochastic Markov chains which contains among others Markov chains. This class is fully characterized by some martingale properties, and one of them is new even in the case of Markov chains. Moreover a predictable representation theorem holds and doubly stochastic property is preserved under natural change of measure.


Introduction
Our goal is to find a class of processes with good properties which can be used for modeling different kind of phenomena. So, in Section 2 we introduce a class of processes, which we call F-doubly stochastic Markov chains. The reason for the name is that there are two sources of uncertainty in their definition, so in analogy to Cox processes, called doubly stochastic Poisson processes, we chose the name "F-doubly stochastic Markov chains". This class contains Markov chains, compound Poisson processes with jumps in Z, Cox processes, the process of rating migration constructed by Lando [16] and also the process of rating migration obtained by the canonical construction in Bielecki and Rutkowski [3]. We stress that the class of F-doubly stochastic Markov chains contains processes that are not Markov. In the following we use the shorthand "F-DS Markov chain" for the "F-doubly stochastic Markov chain". Note that an F-doubly stochastic Markov chain is a different object than a doubly stochastic Markov chain which is a Markov chain with a doubly stochastic transition matrix. On the end of this section we give examples of F-doubly stochastic Markov chains. Section 3 is devoted to investigation of basic properties of F-doubly stochastic Markov chains. In the first part we prove that an F-DS Markov chain C is a conditional Markov chain and that any F-martingale is a F ∨ F C -martingale. This means that the immersion property for (F, F ∨ F C ), so called hypothesis H, holds. Moreover, the family of transition matrices satisfies the Chapman-Kolmogorov equations. In the second part and until the end of the paper we restricted ourselves to a class of F-DS Markov chains with values in a finite set = {1, . . . , K}. We introduce the notion of intensity of an F-DS Markov chain and formulate conditions which ensure its existence. In section 4 we prove that an F-DS Markov chain C with intensity is completely characterized by the martingale property of the compensated process describing the position of C as well as by the martingale property of the compensated processes counting the number of jumps of C from one state to another (Theorem 4.1). The equivalence between points iii) and iv) in Theorem 4.1 in a context of Markov chains has not yet been known according to the best of our knowledge. In a view of the above characterizations, the F-DS Markov chains can be described as the class of processes that behave like time inhomogeneous Markov chains conditioned on ∞ . Moreover these equivalences and the fact that the class of F-DS Markov chains contains the most of F conditional F ∨ F C Markov chains used for modeling in finance, indicate that the class of F-DS Markov chains is a natural, and very useful in applications, subspace of F conditional F ∨ F C Markov chains. Next, an F-DS Markov chain with a given intensity is constructed. Section 5 is devoted to investigation of properties of distribution of C and the distribution of sojourn time in fixed state j under assumption that C does not stay in j forever. We find some conditional distributions which among others allows to find a conditional probability of transition from one state to another given that transition occurs at known time. In Section 6 a kind of predictable representation theorem is given. Such theorems are very important for applications, for example in finance and backward stochastic differential equations (see Pardoux and Peng [18] and El Karoui, Peng and Quenez [7]). By the way we prove that F-DS Markov chain with intensity and arbitrary F adapted process do not have jumps at the same time. Our results allows to describe and investigate a credit risk for a single firm. In such case the predictable representation theorem (Theorem 6.5) generalize the Kusuoka theorem [14]. In the last section we study how replacing the probability measure by an equivalent one affects the properties of an F-DS Markov chain.
Summing up, the class of F-DS Markov chains is a class with very good and desirable properties in modeling. It can be applied to model rating migration in financial markets. More precisely, it can be used for modeling a credit rating migration process and which contains processes usually taken for this purpose. This allows us to include rating migration in the process of valuation of defaultable claims and generalize the case where only two states are considered: default and non-default (for such generalizations see Jakubowski and Niewęgłowski [12]). These processes can also be applied in other fields where system evolves in a way depending on random environment, e.g., in insurance.

Definition and examples
In this section we introduce and investigate a new class of processes, which will be called F-doubly stochastic Markov chains. This class contains among others Markov chains and Cox processes. We assume that all processes are defined on a complete probability space (Ω, , P). We also fix a filtration F satisfying usual conditions, which plays the role of a reference filtration.

Definition 2.1.
A càdlàg process C is called an F-doubly stochastic Markov chain with state space ⊂ Z = {. . . , −1, 0, 1, 2, . . .} if there exists a family of stochastic matrices P(s, t) = (p i, j (s, t)) i, j∈ for 0 ≤ s ≤ t such that 1) the matrix P(s, t) is t -measurable, and P(s, ·) is F progressively measurable, 2) for any t ≥ s ≥ 0 and every i, j ∈ we have (2.1) The process P will be called the conditional transition probability process of C.
The equality (2.1) implies that P(t, t) = I a.s. for all t ≥ 0. Definition 2.1 extends the notion of Markov chain with continuous time (when ∞ is trivial). A process satisfying 1) and 2) is called a doubly stochastic Markov chain by analogy with Cox processes (doubly stochastic Poisson processes). In both cases there are two sources of uncertainty. As mentioned in the Introduction, we use the shorthand "F-DS Markov chain" for the "F-doubly stochastic Markov chain". Now, we give a few examples of processes which are F-DS Markov chains.

Example 1.
(Compound Poisson process) Let X be a compound Poisson process with jumps in Z, i.e., where N is a Poisson process with intensity λ, Y i is a sequence of independent identically distributed random variables with values in Z and distribution ν, moreover (Y i ) i≥1 and N are independent. By straightforward calculations we see that: a) X is an F-DS Markov chain with F = F N and b) X is an F-DS Markov chain with respect to F being the trivial filtration, and with deterministic transition matrix given by the formula From these examples we have seen that the conditional transition probability matrix depends on the choice of the reference filtration F, and P(s, t) can be either continuous with respect to s, t or discontinuous.

Example 2.
(Cox process) The process C with càdlàg trajectories such that for some F-adapted process λ such that λ ≥ 0, t 0 λ s ds < ∞ for all t ≥ 0 we call a Cox process. This definition implies that , so the increments and the past (i.e. C s ) are conditionally independent given ∞ . Therefore for j ≥ i, Thus satisfy conditions 1) and 2) of Definition 2.1. A Cox process C is therefore an F-DS Markov chain with = N. Usually in a definition of Cox process there is one more assumption on intensity, namely ∞ 0 λ s ds = ∞ a.s. Under this assumption C has properties similar to Poisson process and is called conditional Poisson process (or doubly stochastic Poisson process). So our definition of Cox process is a slight generalization of a classical one.  3 Basic properties

General case
In this subsection we consider the case of an arbitrary countable state space . We study basic properties of transition matrices and martingale invariance property of F with respect to F ∨ F C . Moreover we prove that the class of F-DS Markov chains is a subclass of F conditional F ∨ F C Markov chains.
For the rest of the paper we assume that C 0 = i 0 for some i 0 ∈ . We start the investigation of F-DS Markov chains from the very useful lemma describing conditional finite-dimensional distributions of C, which is a counterpart of the well known result for Markov chains.
Proof. The proof is by induction on n. For n = 1 the above formula obviously holds. Assume that it holds for n, arbitrary 0 ≤ u 0 ≤ . . . ≤ u n and (i 0 , . . . , i n ) ∈ n+1 . We will prove it for n + 1 and by the induction assumption applied to u 1 ≤ . . . ≤ u n+1 and (i 1 , . . . , i n+1 ) ∈ n+1 we know that the left hand side of (3.1) is equal to Using ∞ -measurability of family of transition probabilities (P(s, t)) 0≤s≤t<∞ , and the definition of F-DS Markov chain, we obtain and this completes the proof. As a consequence of our assumption that C 0 = i 0 and Lemma 3.1 we obtain Proposition 3.3. If C is an F-DS Markov chain, then for arbitrary 0 ≤ u 1 ≤ . . . ≤ u n ≤ t and (i 1 , . . . , i n ) ∈ n we have

HYPOTHESIS H:
For every bounded ∞ -measurable random variable Y and for each t ≥ 0 we have It is well known that hypothesis H for the filtrations F and F C is equivalent to the martingale invariance property of the filtration F with respect to F ∨ F C (see Brémaud  Now, we will show that each F-DS Markov chain is a conditional Markov chain (see [3, page 340] for a precise definition). For an example of a process which is an F conditional F ∨ F C Markov chain and is not an F-DS Markov chain we refer to Section 3 of Becherer and Schweizer [2].

Proposition 3.5. Assume that C is an F-DS Markov chain. Then C is an
Proof. We have to check that for s ≤ t, By the definition of an F-DS Markov chain,  for i ∈ . The process H i t tells us whether at time t the process C is in state i or not. Let H t := (H α t ) α∈ , where denotes transposition. We can express condition (2.1) in the definition of an F-DS Markov chain, for t ≤ u, in the form or equivalently The next theorem states that the family of matrices P(s, t) = [p i, j (s, t)] i, j∈ satisfies the Chapman-Kolmogorov equations.

Theorem 3.6. Let C be an F-DS Markov chain with transition matrices P(s, t). Then for any u
so on the set C s = i we have Proof. It is enough to prove that (3.6) holds on each set C s = i , i ∈ . So we have to prove that

H s P(s, u) = H s P(s, t)P(t, u).
By the chain rule for conditional expectation, equality (3.5) and the fact that and this completes the proof.

The case of a finite state space
From this subsection, until the end of the paper we restrict ourselves to a finite set , i.e. = {1, . . . , K}, with K < ∞. It is enough for the most of applications, e.g., in finance to model markets with rating migrations we use processes with values in a finite set. In this case H t := (H 1 t . . . , H K t ) . We recall the standing assumption that C 0 = i 0 for some i 0 ∈ .
The crucial concept in this subsection and in study of properties of F-DS Markov chains is the concept of intensity, analogous to that for continuous time Markov chains. Definition 3.7. We say that an F-DS Markov chain C has an intensity if there exists an F-adapted matrix-valued process Λ = (Λ(s)) s≥0 = (λ i, j (s)) s≥0 such that: 1) Λ is locally integrable, i.e. for any 2) Λ satisfies the conditions: the Kolmogorov forward equation: for all v ≤ t, A process Λ satisfying the above conditions is called an intensity of the F-DS Markov chain C.
It is not obvious that if we have a solution to the Kolmogorov backward equation then it also solves the Kolmogorov forward equation. This fact follows from the theory of differential equations, namely we have Proof. The existence and uniqueness of solutions of the ODE's (3.11) and (3.12) follows by standard arguments. To deduce that X (t) = Y −1 (t) we apply integration by parts to the product X (t)Y (t) of finite variation continuous processes and get This ends the proof since X (t) = Y −1 (t) implies that Z(t, t) = I for every t ≥ 0.

Corollary 3.9. If an F-DS-Markov chain C has intensity, then the conditional transition probability process P(s, t) is jointly continuous at (s, t) for s ≤ t.
Proof. This follows immediately from Theorem 3.8, since and both factors are continuous in s and t, respectively.
Theorem 3.8 gives us existence and uniqueness of solutions to (3.9) and (3.10). Next proposition provides the form of these solutions.  .7) and (3.8). Then the solution to (3.9) is given by the formula and the solution to (3.10) is given by Proof. It is a special case of Theorem 5 in Gill and Johansen [8], see also Rolski et Proposition 3.11. Let P = (P(s, t)), 0 ≤ s ≤ t, be a family of stochastic matrices such that the matrix P(s, t) is t -measurable, and P(s, ·) is F-progressively measurable. Let Λ = (Λ(s)) s≥0 be an F-adapted matrix-valued locally integrable process such that the Kolmogorov backward equation (3.9) and Kolmogorov forward equation (3.

10) hold. Then i) For each s ∈ [0, t] there exists an inverse matrix of P(s, t) denoted by Q(s, t). ii) There exists a version of Q(·, t) such that the process Q(·, t) is a unique solution to the integral (backward) equation dQ(s, t) = Q(s, t)Λ(s)ds, Q(t, t) = I. (3.13)
This unique solution is given by the following series: iii) There exists a version of Q(s, ·) such that the process Q(s, ·) is a unique solution to the integral This unique solution is given by the following series: where X , Y are solutions to the random ODE's (3.11), (3.12) and moreover Y = X −1 . Therefore the matrix P(s, t) is invertible and its inverse Q(s, t) is given by Q(s, t) = X (t)Y (s). ii) We differentiate Q(s, t) with respect to the first argument and obtain (3.13). Uniqueness of solutions to (3.13) follows by standard arguments based on Gronwall's lemma. Formula (3.14) is derived analogously to a similar formula for P(s, t) in § 8.4.1, page 348 of Rolski et al. [20].
iii) The proof of iii) is analogous to that of ii).
In the next theorem we prove that under some conditions imposed on the conditional transition probability process P, an F-DS Markov chain C has intensity. Using this intensities we can construct martingale intensities for different counting processes building in natural way from F-DS Markov chains (see Theorem 4.1). Therefore, Theorem 3.12 is in a spirit of approaches of Dellacherie (Meyer's Laplacian see Delacherie [6], Guo and Zeng [9] ) and of Aven [1]. Theorem 3.12 generalizes for F-DS Markov chains results from [1].
Theorem 3.12 (Existence of Intensity). Let C be an F-DS-Markov chain with conditional transition probability process P. Assume that 1) P as a matrix-valued mapping is measurable, i.e.

2) There exists a version of P which is continuous in s and in t.
3) For every t ≥ 0 the following limit exists almost surely and is locally integrable. Then Λ is the intensity of C.
Proof. By assumption 3 the process Λ is well defined and by 1) it is (R + ×Ω, (R + )⊗ ) measurable. By assumption 3, Λ(t) is t+ -measurable, but F satisfies the usual conditions, so Λ(t) is tmeasurable. It is easy to see that (3.8) holds. It remains to prove that equations (3.9) and (3.10) are satisfied. Fix t. From the assumptions and the Chapman-Kolmogorov equations it follows that for Therefore ∂ + ∂ v P(v, t) exists for a.e. v and is (R + × R + × Ω, (R + × R + ) ⊗ ) measurable. Using assumptions 2 and 3 we finally have Since elements of P(u, t) are bounded by 1, and Λ is integrable over [v, t] (by assumption 3), we see that ∂ + ∂ u P(u, t) is Lebesgue integrable on [v, t], so (see Walker [21]) Hence, by (3.18), we have and this is exactly the Kolmogorov backward equation (3.9).
Similar arguments apply to the case of right derivatives of P(v, t) with respect to the second variable.
which gives (3.10), Now, we find the intensity for the processes described in Examples 3 and 4.

Example 5.
If C t = min N t , K , where N is a Cox process with càdlàg intensity processλ, then C has the intensity process of the form

Martingale properties
We prove that under natural assumptions, belonging of a process X to the class of F-DS Markov chains is fully characterized by the martingale property of some processes strictly connected with X . These nice martingale characterizations allow to check by using martingales whenever process is an ii) The processes   "i) ⇒ ii)" Assume that C is an F-DS Markov chain with intensity process (Λ(t)) t≥0 . Fix t ≥ 0 and set N s := P(s, t) H s for 0 ≤ s ≤ t. (4.8) The process C satisfies (3.5), which is equivalent to N being a G martingale for 0 ≤ s ≤ t. Using integration by parts and the Kolmogorov backward equation ( Therefore, by the G martingale property of N , we see that M is a G local martingale. "ii) ⇒ i)" Assume that the process M associated with C and Λ is a G martingale. Fix t ≥ 0. To prove that C is an F-DS Markov chain it is enough to show that for some process (P(s, t)) 0≤s≤t the process N defined by (4.8) is a G martingale on [0, t]. Let P(s, t) := X (s)Y (t) with X , Y being solutions to the random ODE's (3.11) and (3.12). We know that P(·, t) satisfies the integral equation (3.9) (see Theorem 3.8). We also know that P(s, t) is t -measurable (Remark 3.10) and continuous in t, hence F progressively measurable. Using the same arguments as before, we find that (4.9) holds. So, using the martingale property of M we see that N is a local martingale. The definition of N implies that N is bounded (since H and P are bounded, see Last and Brandt [17, §7.4]). Therefore N has an integrable supremum, so it is a G martingale, which implies that C is an F-DS Markov chain with transition matrix P. From Theorem 3.8 it follows that Λ is the intensity matrix process of C.
"ii) ⇔ iii)" and "iii) ⇔ i v)" These equivalences follows from Lemmas 4.3 and 4.4 below, respectively, with A = G given by (4.1). The proof is complete.

Remark 4.2. The equivalence between i) and ii) in the above proposition corresponds to a well known martingale characterization of a Markov chain with a finite state space. In a view of this characterization of F-DS Markov chains with intensities, they can be described as the class of processes that conditioned on ∞ behave as time inhomogeneous Markov chains with intensities. Hence, we can also see that the name F-doubly stochastic Markov chain well describe this class of processes. The equivalence between iii) and iv) in a context of Markov chains has not yet been known according to the best of our knowledge.
The equivalence between points ii), iii) and iv) in Theorem 4.1 is a simple consequence of slightly more general results, which we formulate in two separate lemmas. It is worth to note that equivalences in lemmas below follow from general stochastic integration theory and in the proofs we do not use the doubly stochastic property.
Here and in what follows we use the notation j =i = j∈ \{i} . Indeed, from (4.4) it follows that Next, by (4.3), Proof. Integration by parts formula gives indeed

d L t = Q(0, t) d H t + d(Q(0, t) )H t = Q(0, t) d H t − Q(0, t) Λ(t) H t d t = Q(0, t) d M t .
We know that Q is A predictable and locally bounded. So, if M is an A local martingale, then L is also an A local martingale.
On the other hands, taking P(0, t) as an unique solution to the integral equation Therefore, if L is an A local martingale, then M is also an A local martingale.

Corollary 4.5. If C is an F-DS Markov chain, then M i are G local martingales with
Proof. This follows from the fact that the M i are adapted to G, and G is a subfiltration of G. [16], then M i are G local martingales, so C is an F-DS Markov chain.

Remark 4.6. The process C obtained by the canonical construction in [3] is an F-DS Markov chain. This is a consequence of Theorem 4.1, because Λ in the canonical construction is bounded and calculations analogous to those in [3, Lemma 11.3.2 page 347] show that M i are G martingales. In a similar way one can check that if C is a process of rating migration given by Lando
The martingale M is orthogonal to all square integrable F martingales.  7) is strongly orthogonal to all square integrable F martingales.
Proof. Denote by N an arbitrary R d -valued, square integrable F martingale. Since C is an F-DS Markov chain we have, by Proposition 3.5, that H hypothesis holds and therefore N is also F ∨ F C martingale. Since M is an F ∨ F C martingale, we need to show that the process N j M i is an F ∨ F C martingale for every i ∈ and j ∈ {1, . . . , d}. Fix arbitrary t ≥ 0 and any s ≤ t, then and hence the result follows.
Now we construct an F-DS Markov chain with intensity given by an arbitrary F adapted matrixvalued locally bounded stochastic process which satisfies condition (3.8).
Proof. We assume that on a probability space (Ω, , P) with a filtration F we have a family of Cox processes N i, j for i, j ∈ with intensities (λ i, j (t)) such that the N i, j are conditionally independent given ∞ (otherwise we enlarge the probability space). We construct on (Ω, , P) an F-DS Markov chain C with intensity (Λ(t)) t≥0 and given initial state i 0 . It is a pathwise construction inspired by Lando [16]. First, we define a sequence (τ n ) n≥1 of jump times of C and a sequence (C n ) n≥0 which describes the states of rating after change. We define these sequences by induction. We put If τ 1 = ∞, then C never jumps soC 1 := i 0 . If τ 1 < ∞, then we putC 1 := j, where j is the element of \C 0 for which the above minimum is attained. By conditional independence of N i, j given ∞ , the processes N i, j have no common jumps, soC 1 is uniquely determined. We now assume that τ 1 , . . . , τ k ,C 1 , . . . ,C k are defined, τ k < ∞ and we construct τ k+1 as the first jump time of the Cox processes after τ k , i.e. τ k+1 := min If τ k+1 = ∞ thenC k+1 =C k , and if τ k+1 < ∞ we putC k+1 := j, where j is the element of \C k for which the above minimum is attained. Arguing as before, we see that τ k+1 andC k+1 are well defined.
Having the sequences (τ n ) n≥0 and (C n ) n≥0 we define a process C by the formula This process C is càdlàg and adapted to the filtration A = ( t ) t≥0 , where t : and hence it is also adapted to the larger filtration A = ( t ) t≥0 , t : Recall that t = ∞ ∨ C t , so G ⊆ A. Therefore each M i, j is also a G local martingale, since M i, j is G adapted. Hence, using Theorem 4.1, we see that C is an F-DS Markov chain. Remark 4.9. Suppose that in the above construction τ k < ∞ a.s. and C τ k = i. Then

Distribution of C and sojourn times
In this section we investigate properties of distribution of C and the distribution of sojourn time in a fixed state j under assumption that C does not stay in j forever. These properties give, among others, some interpretation to and to the ratios The first result says that the sojourn time of a given state has an exponential distribution in an appropriate time scale.
If τ k < ∞ a.s., then the random variable is independent of τ k−1 and E k is exponentially distributed with parameter equal to 1. Moreover, (E i ) 1≤i≤k is a sequence of independent random variables.
Proof. For arbitrary j ∈ define the process that counts number of jumps from j Y j is a G local martingale by Theorem 4.1. Moreover, a sequence of stopping times defined by In what follows we denote The definition of σ n implies that Y n is a bounded G martingale for every n. Using the Itô lemma on the interval ]τ k−1 , τ k ] we get For every n the boundedness of (Y n ) and (e iuY n ) imply that the process is a uniformly integrable G martingale. Therefore, by the Doob optional sampling theorem we have Hence using (5.4) and facts that ∆N j τ k ∧σ n = 1 {τk≤σn} and ∆N j s∧σ n = 0 for τ k−1 < s < τ k we infer that on the set C τ k−1 = j : and therefore Applying Lebesgue's dominated convergence theorem and the fact that σ n ↑ +∞ yields Which implies the first statement of theorem. The second statement follows immediately from the above considerations.
As a consequence of the Proposition 5.1 we have the following result which states that the exit times are in some sense exponentially distributed. This is a generalization of the well known property of Markov chains.
The next proposition describes the conditional distribution of the vector (C τ k , τ k −τ k−1 ) given τ k−1 .

Proposition 5.3. Let C be an F-DS
Markov chain with an intensity Λ and τ k be given by (5.2). If τ k < ∞ a.s., then we have on the set C τ k−1 = i .
Proof. First we note that RHS of (5.7) is well defined. Fix arbitrary j ∈ . Let N j be the process that counts number of jumps of C to the state j, i.e. and Then Y j is a G local martingale by Theorem 4.1. Therefore there exist a sequence of G stopping times σ n ↑ +∞ such that (Y j t∧σ n ) t≥0 is an uniformly integrable martingale. By Doob's optional sampling theorem we have on the set we can pass to the limit on the LHS of (5.8) and obtain The RHS of (5.8) we can write as a sum where By a monotone convergence theorem and fact that σ n ↑ +∞ we obtain On the other hand lim n→∞ I 2 (n) = 0, a.s.
Summing up, we have proved the following equality holds on the set C τ k−1 = i To finish the proof it is enough to transform the RHS of (5.9). Corollary 5.2 and Fubbini theorem yield In the last equality we use (5.6). The proof is now complete.

Remark 5.4. Using Corollary 5.2 in the same way as in the last part of the proof of Proposition 5.3 we obtain on the set C
which, by (5.7), is equal to P(C τ k = j, τ k − τ k−1 ≤ t τ k−1 ). This together with j∈ \{i} suggest that the ratio can be treated as a conditional probability of transition from i to j given that transition occurs at the time τ k . The next corollary confirms this suggestion.

Corollary 5.5. On the set C τ k−1 = i , under assumptions of Proposition 5.3, we have
Proof. In a view of Proposition 5.3 and Corollary 5.2 we have on the set C τ k−1 = i : which implies (5.10).

Predictable representation theorem
In this section we prove a predictable representation theorem in the form useful for applications. At first we study a hypothesis which often appears in literature as hypothesis that a given ρ avoids F stopping times. This hypothesis says that for every F stopping time σ it holds P(ρ = σ) = 0.
The third equality follows from measurability of . Moreover P(E k+1 = x) = 0 since random variable E k+1 has an exponential distribution (see Proposition 5.1), so the fourth equality holds, and the assertion follows.
This proposition immediately implies  Proof. Proposition 3.4 implies that N is an F N ∨ F C martingale and, by Theorem 4.1, the process M is F N ∨ F C martingale. For every square integrable N T ∨ C T measurable random variable X and T > 0 we have to find an F N ∨ F C predictable stochastic process such that (6.1) holds with (N , M ) instead of N . Fix arbitrary T > 0. In the proof we will use a monotone class theorem. Let where R, S are F N ∨ F C predictable stochastic processes .
We claim that is a monotone vector space. Obviously, it is a vector space over R and 1 ∈ . From closedness of the space of integrals follows that the bounded limit of monotone sequence of elements from belongs to . Let It is easy to see that is a multiplicative class. To finish the proof it is enough to prove that ⊂ , because by a monotone class theorem, contains all bounded functions that are measurable with respect to the σ-algebra generated by i.e. L ∞ ( N T ∨ C T ) ⊂ . Hence, by standard arguments we obtain Therefore, we need to derive predictable representation for an arbitrary random variable in . Fix X ∈ . Without loss of generality we can assume that t 1 ≤ . . . ≤ t n . Let Z be a martingale defined by Z t := E(X N t ∨ C t ), so it has the form Fix m ∈ {1, . . . , n}. For K dimensional vectors e i k with 1 on i k -th component and zero otherwise, and t ∈ (t m−1 , t m ] we have where (R m t ) d N t for some F-predictable stochastic process R m . Hence, using (4.10) and Proposition 6.1, we have we obtain the desired predictable representation.

Change of probability and doubly stochastic property
Now, we investigate how changing the probability measure to an equivalent one affects the properties of an F-DS Markov chain. We start from a lemma where η 1 is an ∞ -measurable strictly positive random variable and η 2 is an ∞ ∨ C T * -measurable strictly positive random variable integrable under P. Let (η 2 (t)) t∈[0,T * ] be defined by the formula η 2 (t) := E P (η 2 | ∞ ∨ C t ), η 2 (0) = 1. (7.2) Then (N (t)) t∈[0,T * ] is a G martingale (resp. local martingale) under Q if and only if (N (t)η 2 (t)) t∈[0,T * ] is a G martingale (resp. local martingale) under P.
Proof. ⇒ By the abstract Bayes rule and the fact that η 1 is ∞ measurable and hence also u measurable for all u ≥ 0, we obtain, for s < t, ⇐ The proof is similar.
Therefore a factorization η(t) = η 1 (t)η 2 (t) as in Lemma 7.1 holds, since η 1 is ∞ measurable. As an immediate consequence we find that C is an F-DS Markov chain under Q with intensity [λ Q ] i, j = ((1 + κ i, j )λ i, j ) and moreover the process defined by W * t := W t − t 0 γ u du is a Brownian motion under Q.