On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein-Uhlenbeck processes

We establish sufficient conditions for the existence, and derive explicit formulas for the $\kappa$'th moments, $\kappa \geq 1$, of Markov modulated generalized Ornstein-Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process. Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.

Assuming that the driving Markov chain J is defined on a finite state space S and is ergodic with stationary distribution π, it has been shown in [7,Thm. 3.3(a) & Rem.3.4.2]that the MMGOU process admits a non-trivial stationary distribution if and only if the integral (0,t] e ξ * s− dL * s converges P * π -almost surely as t → ∞ to some finite-valued random variable V ∞ .Hereby, ((ξ * , L * ), J * ) denotes the time-reversal of the Markov additive process ((ξ, L), J) .In this case, the stationary distribution of the MMGOU process under P π is uniquely determined as the distribution of Necessary and sufficient conditions for P * π -a.s.convergence of integrals of the form (1.3) in terms of the characteristics of the appearing processes have been given in [6].Besides, by [7,Thm. 3.3(b)], the MMGOU process can admit trivial stationary distributions whenever V degenerates to a continuous-time Markov chain that is piecewise constant with discrete stationary distribution.This behaviour will typically be excluded in this paper.
In the special case that (ξ, η) (equivalently (U, L)) is a bivariate Lévy process, the MMGOU process reduces to the generalized Ornstein-Uhlenbeck (GOU) process, originally introduced in [13] and subsequently studied and applied by various authors, such as [3,5,20,21,24,26,31], to name just a few.Second-order properties of GOU processes like moments and tail distributions are specifically studied in [4], as well as in [3,21,24] for special cases.Another special case of the MMGOU process that has gained attention in the last few years is the Markov modulated Ornstein-Uhlenbeck (MMOU) process.This can be obtained by specifying that U t = −γ Jt t is a piecewise linear MAP, while L t = σ Jt B t + α Jt t is a Markov modulated Brownian motion with drift, where (γ j , α j , σ j ) are constants with σ j > 0, γ j ≥ 0, and γ j > 0 for at least one j ∈ S. The MMOU process has been introduced and studied first in [18], where the authors also derive the mean and (autoco-)variance function of the MMOU process using methods from stochastic analysis, while a recursion for higher moments is derived using Laplace transforms.Other articles like e.g.[25] and [40] focused on stationarity properties of MMOU processes.More precisely, in [25], assuming α j ≡ 0, conditions for stationarity of MMOU processes and some expressions for the stationary distribution as scale mixtures are derived.In [40], again assuming α j ≡ 0, expressions for the Fourier transform of the density of an MMOU process with two-state Markov switching (i.e.|S| = 2) are provided.As the stationary distribution of the MMGOU process is the distribution of an exponential functional, at this point we also mention [34], where moments of exponential functionals with deterministic integrator and with a general process with independent increments in the exponential are studied.
In this paper we derive new existence results and formulas for the running mean and the autocovariance function of the MMGOU process, thus generalizing the results for the MMOU process given in [18].Under suitable conditions we prove that the autocovariance function is exponentially decreasing.This coincides with well-known analogue results for the (G)OU process as well as for the related discrete-time autoregressive AR(1) time series.Further, we derive existence results and formulas for integer moments of the stationary distribution of the MMGOU process.
Here the first two moments will be stated explicitly, while we provide a recursion formula for higher order moments.All formulas are given in terms of the characteristics of the driving MAP.In order to prove these results, after recalling various preliminaries on MAPs in Section 2, in Section 3.1 we collect existence results for (exponential) moments of MAPs.Additionally, in this section we prove a general existence result for moments of stochastic integrals with respect to MAPs (Theorem 3.5).This theorem relies on the one hand on a new result on the existence of moments of Lévy driven stochastic integrals up to a stopping time, and on the other hand on a

The additive component
From the definition of the MAP (X, J) one can show, cf.[1], that there exists a sequence of independent R d -valued Lévy processes {X (j) , j ∈ S} with respect to F, each with characteristic triplet {(γ X (j) , Σ 2 X (j) , ν X (j) ), j ∈ S} with respect to the standard truncation function 1 {|x|≤1} , and such that, whenever J t = j on some time interval (t 1 , t 2 ), the additive component (X t ) t 1 <t<t 2 behaves in law as X (j) .In this paper, we will often consider MAPs (X, J) with a bivariate additive component X = (ζ, χ).Thus the corresponding Lévy processes (ζ (j) , χ (j) ) are two-dimensional and their characteristics (γ and a Lévy measure ν X (j) on R 2 \ {(0, 0)}.
Moreover, whenever the driving chain J jumps for the n-th time at a time T n , say from state i to state j, it induces an additional jump Z ij X,n for X, whose distribution F ij X depends only on (i, j) and neither on the jump time nor on the jump number, and it is independent of any other occurring random elements.As the Lévy processes {X (j) , j ∈ S} have càdlàg paths, this implies that (X, J) always admits a càdlàg modification and we will therefore assume any MAP (X, J) to be càdlàg from now on.Moreover the above yields the following path decomposition of the additive component where X 1 = (X 1,t ) t≥0 describes the Lévy process type behaviour of X, while X 2 = (X 2,t ) t≥0 encodes the additional jumps.Conversely, every process (X, J) such that (J t ) t≥0 is a continuoustime Markov chain with state space S and X has a representation as in (2.2), is a MAP.
We notice that the process X 1 in (2.2) is a semimartingale whose characteristics depend on J, cf.[16].This holds also for X 2 , it being an adapted process of finite variation.Therefore, X is a semimartingale, too, and we can define stochastic integrals with respect to X.
Furthermore, according to [14, Prop. 2 and Eq. ( 41)], we have where v j (t) denotes the time J spends in state j up to time t, and N i→j t is the number of jumps

The Markovian component
We assume throughout, that the Markovian component J is irreducible, hence ergodic.We denote the intensity matrix of J by Q = (q ij ) i,j∈S , and recall that its entries satisfy q ij > 0 for all i, j ∈ S, i = j, and q ii = − j∈S\{i} q ij .As J is assumed to be ergodic, it admits a unique stationary distribution that we denote by π = (π j ) j∈S and we write (2.4) For a fixed state j ∈ S we define the return times to j iteratively by In a similar way, we define the exit times from the state j by Clearly, return times and exit times are stopping times with respect to F. The sequences (τ re n+1 (j)− τ re n (j)) n≥1 and (τ ex n+1 (j) − τ ex n (j)) n≥1 are i.i.d.under any P i .Moreover, under P j , we may set τ re 0 (j) = 0, which yields an i.i.d.sequence (τ re n+1 (j) − τ re n (j)) n≥0 .
Throughout, let (N t ) t≥0 be the counting process that describes the number of jumps of J up to time t.Further, let {T n , n ≥ 1} be the sequence of jump times of J and set T 0 = 0, such that in particular N t = ∞ n=1 1 {Tn≤t} .We also define the counting processes The ergodicity of J and the finiteness of the state space S imply that J is positive recurrent, that is E j [τ re 1 (j)] < ∞ for all j ∈ S, cf.[29,Chapter 3.5].In our setting the return times have finite moments of all orders, i.e.E j [τ re 1 (j) k ] < ∞ for all k ∈ N. As we were unable to find a suitable reference for this fact, we state it here as lemma and provide a short proof.
Lemma 2.1.Let J be an ergodic Markov chain on S with |S| < ∞.
As under P 1 the variable τ ex 1 (1) ∼ Exp(|q 11 |) has finite moments of all orders and the same holds true for τ (see e.g.[10,Cor. 3.1.18]),this immediately implies the statement by an application of the binomial theorem.
We introduce the localization process Λ = (Λ t ) t≥0 corresponding to the Markovian component J of the MAP (X, J) given by Λ t := e Jt = (1 {Jt=j} ) j∈S , t ≥ 0, where e j denotes the j-th unit vector, such that (2.6) For any real-valued process Y we write Ŷt := Y t Λ t and note that this implies 1 ⊤ Ŷt = Y t , with 1 denoting the column vector of 1's.As Λ is a regular jump Markov process, by (see [15, Appendix B, Lemma 1.1]) there exists an |S|-dimensional square integrable F-martingale M with respect to P, such that where we make the following important notational convention for multivariate stochastic integrals: where We notice that, from (2.7), M is a martingale which is bounded on compact time intervals, since, for every t > 0, we have In particular, by [28, Eq. ( 14)], for every predictable H such that sup t≥0 |H t | ∈ L 1 (P), we have E[sup 0≤s≤t | (0,s] H u dM u |] < ∞ for all t ≥ 0. Therefore, we have shown the following. Lemma 2.3.For every predictable process H with sup t≥0 |H t | ∈ L 1 (P) the integral process ( (0,t] H s dM s ) t≥0 is a centered martingale.
Clearly, M is a martingale also with respect to P j , for any j ∈ S, and hence with respect to P π .

The matrix exponent
The matrix exponent Ψ X of a MAP (X, J) with univariate additive component X is the matrix in C |S|×|S| defined as for all w ∈ C such that the right hand side exists, cf.[1, Prop.XI.2.2] or [14].Hereby and in the following, diag(a j ) denotes a diagonal matrix with entries a j , j = 1, . . ., n, "•" means elementwise multiplication (of matrices) and ψ j (w) = log E[e wX (j) 1 ] is the Laplace exponent corresponding to the Lévy process X (j) .The matrix exponent determines the characteristic function of the additive component X since, cf.[14, Sec.A.1], E i e wXt 1 {Jt=j} = e ⊤ j e tΨ X (w) e i . (2.10)

Related processes
As explained in [7] We also recall the dual MAP (X * , J * ) of (X, J), sometimes also called its time-reversal, which is defined as the process satisfying where π * = π is the unique stationary distribution of J * , P * j (•) = P(•|J * 0 = j) and P * π defined accordingly; see [14,Lem. 21] for details.Moreover, note that by [14,Sec. A.2], the matrix exponent Ψ X * of the dual process (X * , J * ) fulfills the relation for all w where the matrix exponents are defined.We denote the return and exit times of the dual process by τ re n (j) * and τ ex n (j) * , n ∈ N, j ∈ S.
3 Moments of (integrals with respect to) MAPs In this paper, we aim to circumvent the eigenvalue theory and use the Lévy system of a MAP instead, to derive moment formulas of MAPs as well as stochastic integrals with respect to MAPs that directly depend on the semimartingale characteristics of the processes.Such formulas for the mean and variance of the additive component of (X, J) will be presented in Theorems 3.8 and 3.10 below, after establishing conditions for finiteness of moments in the upcoming subsection.

Existence of moments
In [14,Thm. 34] conditions for the existence of the mean of the additive component of a MAP are collected.The following lemma generalizes parts of these results to general κ'th moments, κ ≥ 1.Its proof is postponed to Section 5.
Lemma 3.1.Let (X, J) be an F-MAP and fix κ ≥ 1, t > 0. Then the following are equivalent: The next lemma provides conditions for the existence of exponential moments of the additive component of (X, J), that will later be useful.Again its proof is given in Section 5.
Further insight into the exponential moments of a MAP can also be obtained from the matrix exponent and its eigenvalue with maximal real part.This is illustrated by the following proposition whose proof is to be found in Section 5. ) For our study of moments of the MMGOU processes in Section 4 below, it will be crucial to study moments of stochastic integrals with respect to MAPs up to a certain stopping time.To this aim we continue by showing finiteness of moments of the stochastic integral (0,τ ] H s− dX s for an R × S-valued F-MAP (X, J), a càdlàg F-adapted process H, and a stopping time τ .We make the following assumption.
Assumption 3.4.The stopping time τ has finite moments of every order, that is, E[τ n ] < ∞ for every n ∈ N, and the càdlàg F-adapted process H satisfies for ε = 0 if τ is bounded, and for some ε > 0 otherwise.
If H and τ satisfy Assumption 3.4 but τ is not bounded, by Hölder's inequality with exponents p = (κ + ε)/κ and exponent q = p/(p − 1) = (κ + ε)/ε, we immediately derive the estimate The next theorem is the main result of this section: To prove Theorem 3.5 we use the decomposition X = X 1 + X 2 of the MAP X given in (2.2) and consider the integrals with respect to X 1 and X 2 separately.As X 1 is a concatenation of Lévy processes, our studies of this part rely on a generalization of [4,Lem. 6.1] where moments of stochastic integrals with respect to Lévy processes up to a fixed time have been considered.The proof of [4, Lem.6.1] could not be generalized to integration up to a general stopping time and in Section 5 below we therefore provide an alternative proof.
In order to deal with the integral with respect to the additional jumps of X, following [30], observe that we have the representation with the integer-valued random measures µ ij Z , i, j ∈ S, i = j, defined by We stress that the definition of µ ij Z in [30] (Π i Z in the notation used in [30]) is flawed: In [30, Eq. ( 6)] the Dirac measure δ Z i n (dx) is replaced by P Z i n (dx).This however is inconsistent with (3.3).In the proof of Theorem 3.5 we will apply [27,Thm. 1] to the integral with respect to X 2 .Therefore, we need the explicit form of the F-dual predictable projection ν ij Z of µ ij Z , which we are now going to compute.Recall that F ij X denotes the cdf of the law of Z ij X,1 , and that denotes the number of jumps of J from state i to state j up to t ≥ 0. By [15, Appendix B, p. 361] (see also [39, p. 290]) the process N i→j − (0,•] 1 {J s− =i} q ij ds is a martingale, hence the F-dual predictable projection φ ij of N i→j is given by From this we derive the following Lemma, see also [23,Sec. 2] and [8] for related results and special cases.Its proof is given in Section 5.
Lemma 3.7.For any i, j ∈ S, i = j, the F-dual predictable projection ν ij Z of µ ij Z is given by We are now ready to prove Theorem 3.5.For this recall that by Jensen's inequality for any non-negative numbers a 1 , . . ., a n and κ ≥ 1, we have Proof of Theorem 3.5.By (2.2) and Minkowski's inequality we get where for the first summand an application of (3.5) yields (3.7) As, by the assumptions of the theorem, we have for all j ∈ S by Lemma 3.1, and the process H1 {J=i} satisfies Assumption 3.4, this is finite by Lemma 3.6.We now consider the second summand in (3.6) and note that by Lemma 3.1, from our assumptions, we have 3) and (3.5) it follows where we define µ ij Z := µ ij Z − ν ij Z with ν ij Z as in Lemma 3.7.Hereby, due to the specific form of ν ij Z , we get where the last estimate follows by (3.2) with η = κ, and with ε > 0 if τ is not bounded.This shows that the second expectation in (3.8) is finite for all i, k ∈ S, i = k.Concerning the first expectation in (3.8), using [27, Thm.1], we get for some constants C 1 κ , C 2 κ ∈ (0, ∞).Hence, using (3.2) with η = 1 or η = κ/2, and with ε > 0 if τ is not bounded, we get as in (3.9) Therefore, the right-hand side of (3.8) and hence of (3.6) is finite, and the proof of the theorem is complete.

Explicit expressions for the moments
We can now provide an explicit formula for the mean of the additive component of a MAP.The proof of this theorem is given in Section 5.
Theorem 3.8.Let (X, J) be an and in particular with the expectation matrix Moreover, X # := X − ǫ[X] (0,•] Λ s ds is an F-martingale under P. The following representation of the expectation of a MAP could also have been derived from [1, Cor.XI.2.5].We mention it here, as the proof in that source is not fully given.It is an immediate consequence of (3.12) upon noticing that e Q ⊤ s • π = π for all s ≥ 0, and due to [35,Eq. (25.7)].
Corollary 3.9.Let (X, J) be an F-MAP on R × S and suppose that We continue by providing an explicit expression of the variance of the additive component of a MAP in the next theorem.
Theorem 3.10.Let (X, J) be an then for all j ∈ S, t > 0 with ǫ [X, X] = diag Var(X The proof of Theorem 3.10 as given in Section 5 relies on an application of integration by parts and the upcoming Lemma 3.11 which will also be useful for our computations in Section 4. Its proof can also be found in Section 5. Lemma 3.11.Let (X, J) be an F-MAP on R × S and H an F-adapted càdlàg process.Assume that Then for all j ∈ S it holds and with the expectation matrix ǫ[X] defined in (3.13).
Remark 3.12.Via Itô's formula and similar computations as in the proof of Theorem 3.10 one can also derive closed form expressions for higher moments of the additive component of a MAP, which are then expressed in terms of powers of the expectation matrix, the correlation matrix diag(Var(X (j) )), and multiple integrals with integrand e Q ⊤ t .In order to avoid overly lengthy computations we restrain from giving any details, but instead also refer to the recent work [2] where recursive expressions of moments of MAPs similar to the above are presented in Prop.3.

Application to Markov modulated GOU processes
In this final section we consider the MMGOU process (V t ) t≥0 solving the SDE (1.2) for some bivariate MAP ((U, L), J).In Section 4.1 we compute the running mean and the autocovariance function of the MMGOU process under suitable conditions that imply their existence.Afterwards, Section 4.2 focuses on stationary MMGOU processes: We derive suitable conditions for the existence of the moments of the stationary distribution, as well as explicit moment formulas.Similar results as in this section in the special case of the MMOU process have been obtained in [18] and [25].However, in the latter, no expressions in terms of the driving processes are provided.For the special case of the GOU process, moments of the stationary distribution have been considered in [24] (existence) and [4] (formulas).Related results can also be found in [37], where moments of the stationary distribution of reflected Markov modulated OU processes are computed, and in [34] where moments of exponential functionals of additive processes are studied.
Throughout, we assume ∆U > −1.This allows us to use the explicit representation (1.1) of the MMGOU process with the MAP ((ξ, η), J) that is one-to-one related with ((U, L), J) by as shown in [7,Prop. 2.11].Note that in absence of additional jumps, that is if Z ij U,n ≡ Z ij L,n ≡ 0, the conditional independence of ξ and η (or analogously of U and L) given J implies L t = η t for all t ≥ 0. Furthermore, the relation between U and ξ in (4.1) is equivalent to where (E(U ) t ) t≥0 is the (Doléans-Dade) stochastic exponential of U , i.e. the unique solution of the SDE dZ t = Z t− dU t , t ≥ 0 with Z 0 = E(U ) 0 = 1.This intimate relationship between U and ξ implies that their moments are also closely related.For |S| = 1, i.e. for U and ξ being Lévy processes, this has been elaborated in [4,Prop. 3.1].The upcoming two results consider the general case |S| ≥ 1.
Lemma 4.1.Let (ξ, J) be a MAP and define the MAP (U, J) via (4.2).Then, for any κ ≥ 1, we have Proof.The fact that (ξ, J) is a MAP if and only if (U, J) is a MAP with ∆U > −1 has been shown in [7].Further, by Lemma 3.1 we know that The Lévy processes U (j) , j ∈ S, have finite κ's moment if and only if |x|>1 |x| κ ν U (j) (dx) < ∞, j ∈ S, cf.[35,Thm. 25.17].Via the path decomposition (2.2) we observe that (4.1) implies and hence E[|U again by [35,Thm. 25.17].Summing up, we observe that 1 ] < ∞ for all j ∈ S and E[e −κZ ij ξ,1 ] < ∞ for all (i, j) ∈ S 2 such that q ij > 0. Finally, by an analogue computation as in (5.17), we observe that this is in turn equivalent to E π e −κξ 1 < ∞, which finishes the proof.Proposition 4.2.Let (ξ, J) be a MAP and define the MAP (U, J) via (4.2).Then for any k < ∞ for some t ≥ 0 we have the identity In particular it holds Proof.On the one hand, as the matrix exponent Ψ ξ fulfills (2.10), relation (4.2) immediately implies On the other hand, from the SDE for the Doléan-Dade stochastic exponential, we have ∆E(U ) = E(U ) − ∆U .Thus we compute via the binomial theorem Moreover, since dE(U ) c t = E(U ) t− dU c t , we have Thus Itô's formula (cf.[32, Thm.II.32]) yields where we define Notice that (Y (k) , J) is an F-MAP for any k as can easily be seen by the path description (2.2).Moreover, due to the given form of Y (k) , we conclude by Lemmas 3.1 and 3.2 that our assumptions imply Thus, via partial integration as in (2.8), applying (3.16) and arguing via Lemma 2.3, we obtain As E(U ) 0 = 1, this ODE is solved uniquely by Multiplying with 1 ⊤ proves (4.3) due to (4.5).Finally we get ǫ[Y (1) is additive.This proves (4.4).

Mean and autocovariance structure of the MMGOU process
We start with a technical lemma which provides sufficient conditions for the existence of moments of the MMGOU process.
Assume that for some κ ≥ 1 and all j ∈ S Proof.Using (1.1) and the inequality (3.5) we have where we used the fact that V 0 is chosen independently of ((ξ t , η t ), J t ) t≥0 .While the second summand is finite by assumption and Lemma 3.2, for the first summand we note that by As the dual processes ξ * and L * inherit the moments of ξ and L, we may conclude via (4.1) and the results in Section 3 that under our conditions Hence finiteness of the first summand follows from Theorem 3.5 and this implies the statement.Finally, if the assumptions hold for κ = 2, then is also finite by the above.
Theorem 4.4.Consider the MMGOU process (V t ) t≥0 driven by the bivariate MAP ((ξ, η), J) and solving the SDE (1.2) for the bivariate MAP ((U, L), J) in (4.1).Assume that for all j ∈ S Then for all t ≥ 0 and all j ∈ S Further, using a Fubini argument and (3.16), this implies since Inserting (2.6) yields the ODE which is solved uniquely by Multiplying this expression by 1 ⊤ yields (4.6).
The next theorem proves that the autocovariance function of the MMGOU decreases exponentially if the leading eigenvalue of the matrix exponent Ψ ξ (−1) is negative.
Proof.We start to consider and derive via partial integration over (s, t] Λ u− dV u , V s (4.9) Hereby, via (1.2) we have Λ u− dL u , V s , and while due to (2.7) Inserting the above in (4.9) and applying (3.16) we obtain by a straightforward computation Moreover, again using (2.7) and Lemma 2.3, we derive Equations (4.10) and (4.11) together can now be reformulated in matrix and differential form as with unique solution as given in (4.8).The autocovariance function is thus decreasing exponentially if and only if all eigenvalues of K have non-positive real part.Denoting I = diag(1), we see that As, in our setting, all eigenvalues of the intensity matrix Q have non-positive real part, cf.[29], it is sufficient to consider the eigenvalues of Q ⊤ + ǫ[U ], i.e. the eigenvalues of Ψ ξ (−1) by (4.4).This finishes the proof.

Stationary MMGOU processes
As already mentioned in the introduction, under our standing assumption that J is defined on a finite state space S and is ergodic with stationary distribution π, the MMGOU process (1.1) admits a non-trivial stationary distribution if and only if the integral (0,t] e ξ * s− dL * s converges P * π -almost surely as t → ∞ to some finite-valued random variable V ∞ .In this case the stationary distribution of the MMGOU process under P π is uniquely determined as the distribution of V ∞ given in (1.3).In this section we analyse the moments of V ∞ .We start by providing conditions for their existence before presenting explicit formulas for some integer moments.Theorem 4.6.Consider the MMGOU process (V t ) t≥0 driven by the bivariate MAP ((ξ, η), J) and solving the SDE (1.2) for the bivariate MAP ((U, L), J) in (4.1).Assume that lim t→∞ ξ t = ∞ P π -a.s. and that there exists κ ≥ 1 such that for all j ∈ S Then (V t ) t≥0 has a stationary solution with distribution Proof.As shown in [7,Thm. 3.3], there exists a finite random variable Fix j ∈ S and let N re, * t denote the number of returns of J * to state j up to time t > 0. Then under P * j for any t ≥ 0 we can rewrite the exponential integral as  Then under P * j the sequence We can thus follow the lines of the proof of [24, Prop.4.1] and derive via Hölder's inequality that where for κ = ⌊κ⌋ the second factor can be omitted.To show the absolute convergence of the two sums, note that for any ℓ ≥ 1 it holds .
Passing to the limit as n → ∞ this converges absolutely due to assumption (4.12).Since we already proved the existence of the limit in (4.13) this implies the statement.
Under the above derived conditions for the existence of the κ'th moment of the stationary distribution, the following theorem provides explicit formulae for the first and second moment.
Theorem 4.8.Consider the MMGOU process (V t ) t≥0 driven by the bivariate MAP ((ξ, η), J) and solving the SDE (1.2) for the bivariate MAP ((U, L), J) in (4.1).Assume the conditions of Corollary 4.7 are fulfilled for κ = 1, or κ = 2, respectively.Then the first two moments of the stationary distribution V ∞ of the MMGOU process are given by Proof.To compute the mean, we follow the lines of the proof of Theorem 4.4, where we note that all needed conditions are met due to Lemma 3.
This immediately implies as stated, where we note that by (4.4) , which is invertible as by assumption λ ξ max (−1) < 0. For the second moment, following the first lines of the proof of Theorem 3.10 we derive Hereby notice that Lemma 2.3 was applicable since E j [sup 0<s≤t |V s | 2 ] < ∞ follows from our assumptions due to Lemma 4.3.By stationarity it holds ) then yields by a straightforward computation using (3.16) Inserting the formula (4.15) for the first moment this proves This easily yields the given formula for (4.4), and since λ ξ max (−2) < 0, this matrix is invertible.
Remark 4.9.With a similar procedure as used in the proof of Theorem 4.8 and using ideas as in the proof of Proposition 4.2 one can derive a recursion formula for higher moments of the stationary distribution, given they exist.For k > 2 this yields the recursion assuming that all needed moment conditions are fulfilled.Note that by Proposition 4.2 the inverted matrix in the recursion equals Ψ ξ (−k) and hence the inverse exists, whenever λ ξ max (−k) < 0. Further, we observe that for continuous processes U and L the recursion stops with the k − 2'nd moment.This agrees with the recursion for the moments of the MMOU process derived in [18,Sec. 3.4].
5 Additional proofs for the results in Section 3

Proofs for the results in Section 3.1
Proof of Lemma 3.1.Throughout the proof we set t = 1.The proof for general t > 0 works alike.Obviously (i) implies (ii), while the equivalence of (ii) and (iii) is immediate since To prove the remaining conclusions we follow ideas from the proof of [14,Thm. 34]: Assume first (iii).Fix any j ∈ S and consider the event O, that J has no transition in [0, 1].Then for all j ∈ S. Now consider the event I, that the first transition of J happens before t = 1 and the second after t = 1.Then Hence, especially, for every (i, j) ∈ S 2 , and we can conclude with (3.5) that for any such s ∈ [0, 1] This implies (iv).Finally, we show that (iv) implies (i).So, assume (iv) and note that by [35,Thm 25.18] our assumptions imply that E sup 0<s≤1 |X (j) s | κ < ∞ for all j ∈ S. By (2.3) we obtain Via Minkowski's inequality and (3.5) this yields where, for the second summand, we used the independence of N 1 and Z iℓ X,n , that Z iℓ X,n ∼ Z iℓ X,1 , and that N 1 has finite moments of all orders.
Proof of Lemma 3.2.W.l.o.g.we consider sup s∈[0,1] .The proof for general t < ∞ is similar.As earlier on, let N i→j t denote the number of transitions of J from the state i to the state j up to time t > 0. The Lévy processes X (j) , j ∈ S, are independent and also independent of the variables Z ij X,ℓ , and of the counting processes N i→j for all i, j ∈ S. Thus by monotonicity of the exponential function and using (2.3), we get (5.17) The first factor does not involve the initial distribution of J, and by [35,Thms. 25.18 and 25.3] for all j ∈ S E sup It thus remains to prove that the second factor in (5.17) is finite if and only if E e κ|Z ij X,1 | < ∞ for all i, j ∈ S. Since (N i→j 1 ) i,j∈S is independent of the i.i.d.sequences {Z ij X,ℓ , ℓ ∈ N}, conditioning on (N i→j 1 ) i,j∈S = (n ij ) i,j∈S =: n, yields which is finite if and only if E e κ|Z ij X,1 | < ∞ for all (i, j) ∈ S 2 such that q ij > 0.
Proof of Proposition 3.3.Recall first that the transition probabilities of the embedded discrete time Markov chain of J are given by p ij = q ij |q ii | for any i, j ∈ S. Further, the corresponding Lévy processes X (j) and the additional jumps Z ij are all independent, and independent of the holding times of J in any state.Moreover, for an exponentially distributed random variable T with rate q, independent of a Lévy process (Y t ) t≥0 , the Laplace transform of Y T , whenever it exists, is given by, and Clearly, by our assumptions, R (and hence L) is well-defined for the chosen value of κ.Using the notations as introduced in Section 2.2 we note that T N τ re 1 (1) = τ re 1 (1) and hence, using the defined matrices, we have We thus observe that E 1 [e κX τ re 1 (1) ] < ∞ if and only if the geometric series in (5.18) converges.By [9,Prop. 9.4.13 and Cor. 9.4.10]this holds in particular if L row = L ∞,∞ < 1.This in turn is equivalent to (3.1) by the definition of L, thus finishing the proof of (i).For (ii) assume that E 1 [e κX τ re 1 (1) ] < ∞ such that the series in (5.18) Recall that for any square matrix A ∈ R d×d the adjugate matrix A † = (A † ij ) i,j=1...,d is defined via where 1) is the matrix that results from deleting the ith row and jth column of A. Since for an invertible matrix A it holds A −1 = (det A) −1 A † , and since R 11 = 0, (5.18) yields (5. 19) Further, a Laplace expansion of det(I − R) leads to (5.20) Thus, combining (5.19) and (5.20) we obtain To show the claim, it remains to verify that if λ X max (κ) < 0, then the ratio on the right hand side of (5.21) is always positive.To this end we rewrite both determinants in terms of det Ψ X (κ) which gives via (2.9) As Ψ X (κ) is assumed to be negative definite, by Silvester's criterion (see e. Proof of Lemma 3.6.Let (γ, σ 2 , ν), W σ , and µ denote the characteristic triplet, the Gaussian part, and the jump measure of the Lévy process X, respectively.Furthermore, we denote by µ(dt, dx) := µ(dt, dx) − dtν(dx) the compensated jump measure of X.We notice that in the given setting (0,•] R x(µ(ds, dx)−ν(dx)ds) is a martingale and, for κ ≥ 2, it is a square integrable martingale.
For the Brownian integral in (5.24), by the Burkholder-Davis-Gundy inequality, cf.[33, Cor.IV.(4. 2)], we get for some constant C κ ∈ (0, ∞) where the last estimate follows from (3.2) with η = κ/2 and ε > 0, if τ is not bounded.It remains to consider the integral with respect to the compensated jump measure in (5.24).By [27, Thm.1] there exist constants κ are finite by the assumptions on X.By (3.2) with η = 1 or η = κ/2, and with ε > 0 if τ is not bounded, we see that the right-hand side in the previous estimate is finite.The proof is complete.
Hereby P(F) denotes the σ-algebra of F-predictable sets.
Assume that W (ω, t, x) = B t (ω)f (x), where (B t ) t≥0 , B t ≥ 0, is a bounded predictable process and f ≥ 0 a bounded Borel function, then As B Tn is F Tn− -measurable, B being predictable, and where in the last identity we used the properties of the dual predictable projection φ ij given in (3.4).Hence, we obtain (5.25) for this special choice of W . Since the class of predictable functions W of the chosen form generates P(F), the monotone class theorem (cf.[32, Thm.I.1.8])now yields (5.25) for every non-negative predictable bounded function W . Finally, consider an arbitrary predictable function W ≥ 0. Then W n := W ∧ n is bounded and by the previous step (5.25) holds for W n .By monotone convergence this implies (5.25) for W and hence the statement.

Proofs for the results in Section 3.2
Proof of Theorem 3.8.To prove (3.11) we use the integration by parts formula (2.8) for X, where we first show that the R |S| -valued local martingale (0,•] X s− dM s in (2.8) is an R |S|valued centered martingale with respect to P j , for every j.Indeed, we have for every fixed T > 0, because of Lemma 3.1.Thus, the claim follows by Lemma 2.3 and we obtain that E j [ (0,t] X s− dM s ] = 0 with 0 denoting the column vector of zeros.So, from (2.8), we obtain Using that X 0 = 0 and ∆Λ j = ±1, and recalling the notation introduced in (2.5), this yields Inserting this and (5.27) in (5.26), and applying a Fubini argument we obtain the following inhomogeneous ODE for E j [ Xt ] as function of t This is solved uniquely by By our standard assumption X 0 = 0 a.s. the first term vanishes and we obtain (3.11).Moreover, multiplying (3.11) with 1 ⊤ from the left yields (3.12), since 1 ⊤ e Q ⊤ r = 1 ⊤ .To finish the proof of the lemma, note that since E π [|X t |] < ∞, we have E j [|X # t |] < ∞ for all j ∈ S. We may thus compute Proof of Lemma 3.11.Recall that the process X # := X − ǫ[X] (0,•] Λ s ds defined in Theorem 3.8 is a martingale.Hence, X # := 1 ⊤ X # is a martingale, too, and the stochastic integral (0,t] H s− dX # s is a local martingale.Moreover, it is a true centered martingale, since by the triangle inequality, by the definition of X # , and by Theorem 3.5, we have Thus, Recall from Section 2 that ([X, X] t , J) t≥0 is again an F-MAP and therefore by Theorem 3.8 Thus via (3.16) we obtain i,j∈S , as stated.Finally, the stated formula for the variance is easily derived from (3.14) via of J from i to j over the time interval [0, t].Hereby, Y d = Z reads Y and Z have the same distribution.

e
ξ u− dη u under P π is equal in law to − (0,s] e ξ * u− dL * u under P * π .