Geometric Ergodicity of the multivariate COGARCH(1,1) Process
Robert Stelzer Johanna Vestweber

Transport Effects on Multiple-Component Reactions in Optical Biosensors
This work was done with the support of the National Science Foundation under award number NSF-DMS 1312529. The first author was also partially supported by the National Research Council through an NRC postdoctoral fellowship.
    Ryan M. Evans        David A. Edwards 

    Received: date / Accepted: date
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For the multivariate COGARCH(1,1) volatility process we show sufficient conditions for the existence of a unique stationary distribution, for the geometric ergodicity and for the finiteness of moments of the stationary distribution by a Foster-Lyapunov drift condition approach. The test functions used are naturally related to the geometry of the cone of positive semi-definite matrices and the drift condition is shown to be satisfied if the drift term of the defining stochastic differential equation is sufficiently “negative”.  We show easily applicable sufficient conditions for the needed irreducibility and aperiodicity of the volatility process living in the cone of positive semidefinite matrices, if the driving Lévy process is a compound Poisson process. 



AMS Subject Classification 2010: Primary:  60J25    Secondary:   60G10, 60G51





Keywords: 
Feller process, Foster-Lyapunov drift condition, Harris recurrence, irreducibility, Lévy process, MUCOGARCH, multivariate stochastic volatility model



§ INTRODUCTION

General autoregressive conditionally heteroscedastic (GARCH) time series models, as introduced in <cit.>, are of high interest for financial economics. They capture many typical features of observed financial data, the so-called stylized facts (see <cit.>). A continuous time extension, which captures the same stylized facts as the discrete time GARCH model, but can also be used for irregularly-spaced and high-frequency data, is the COGARCH process, see e.g. <cit.>. The use in financial modelling is studied e.g. in <cit.> and the statistical estimation in <cit.>, for example. Furthermore, an asymmetric variant is proposed in <cit.> and an extension allowing for more flexibility in the autocovariance function in <cit.>.  

To model and understand the behavior of several interrelated time series as well as to price derivatives on several underlyings or to assess the risk of a portfolio multivariate models for financial markets are needed. The fluctuations of the volatilities and correlations over time call for employing stochastic volatility models which in a multivariate set-up means that one has to specify a latent process for the instantaneous covariance matrix. Thus, one needs to consider appropriate stochastic processes in the cone of positive semi-definite matrices.  Many popular multivariate stochastic volatility models in continuous time, which in many financial applications is preferable to modelling in discrete time, are of an affine type, thus falling into the framework of <cit.>.  Popular examples include the Wishart (see e.g. <cit.>) and the Ornstein-Uhlenbeck type stochastic volatility model (see <cit.>, for example). 
 Thus the price processes have two driving sources of randomness and the tail-behavior of their volatility process is typically equivalent to the one of the driving noise (see <cit.>). A very nice feature of GARCH models is that they have only one source of randomness and their structure ensures heavily-tailed stationary behavior even for very light tailed driving noises (<cit.>).
In discrete time one of the most general multivariate GARCH versions (see <cit.> for an overview) is the BEKK model, defined in <cit.>, and the multivariate COGARCH(1,1) (shortly MUCOGARCH(1,1)) process introduced and studied in <cit.>, is the continuous time analogue, which we are investigating further in this paper. 



The existence and uniqueness of a stationary solution as well as the convergence to the stationary solution is of high interest and importance. Geometric ergodicity ensures fast convergence to the stationary regime in simulations and paves the way for statistical inference. By the same argument as in <cit.> geometric ergodicity and the existence of some p-moments of the stationary distribution provide exponential β-mixing for Markov processes. This in turn can be used to show a central limit theorem for the process, see for instance <cit.>, and so allows to prove for example asymptotic normality of estimators (see e.g. <cit.> in the context of univariate COGARCH(1,1) processes). In a similar way <cit.> employ the results of the present paper to analyse moment based estimators of the parameters of MUCOGARCH(1,1) processes.

In many applications involving time series (multivariate) ARMA-GARCH models (see e.g. <cit.>) turn out to be adequate and geometric ergodicity is again key to understand the asymptotic behaviour of statistical estimators. In continuous time a promising analogue currently investigated in <cit.> seems to be a (multivariate)  CARMA process (see e.g. <cit.>) driven by a (multivariate) COGARCH process. The present paper also lays  foundations for the analysis of such models.

For the univariate COGARCH process geometric ergodicity was shown by <cit.> and <cit.> discussed it for the BEKK GARCH process. In <cit.> for the MUCOGARCH process sufficient  conditions for the  existence of a stationary distribution are shown by tightness arguments, but the paper failed to establish uniqueness or convergence to the stationary distribution. In this paper we deduce under the assumption of irreducibility sufficient conditions for the uniqueness of the stationary distribution, the convergence to it with an exponential rate and some finite p-moment of the stationary distribution of the MUCOGARCH volatility process Y. To show this we use the theory of Markov process, see e.g. <cit.>. A further result of this theory is, that our volatility process is positive Harris recurrent. If the driving Lévy process is a compound Poisson process, we show easily applicable conditions ensuring irreducibility of the volatility process in the cone of positive semidefinite matrices. 

Like in the discrete time BEKK case the non-linear structure of the SDE prohibits us from using well-established results for random recurrence equations like in the one-dimensional case and due to the rank one jumps establishing irreducibility is a very tricky issue. To obtain the latter <cit.> in discrete time   used techniques from algebraic geometry (see also <cit.>) whereas we use a direct probabilistic approach playing the question back to the  existence of a density for a Wishart distribution. However, we restrict ourselves to processes of order (1,1) while in the discrete time BEKK case general orders were considered. The reason is that on the one hand order (1,1) GARCH processes seem sufficient in most applications and on the other hand multivariate COGARCH(p,q) processes can be defined in principle (<cit.>), but no reasonable conditions on the possible parameters are known. Already in the univariate case these conditions are quite involved (cf. <cit.>, <cit.>). On the other hand we look at the finiteness of an arbitrary p-th moment (of the volatility process) and use drift conditions related to it, whereas <cit.> only looked at the first moment for the BEKK case. In contrast to <cit.> we avoid any vectorizations, work directly in the cone of positive semi-definite matrices and use test functions naturally in line with the geometry of the cone.

After a brief summary of some preliminaries, notations and Lévy processes, the remainder of the paper is organized as follows: In Section 3 we recall the definition of the MUCOGARCH(1,1) process and some of its properties of relevance later on. In Section 4 we present our first main result: sufficient conditions ensuring the geometric ergodicity of the volatility process Y. Furthermore, we compare the conditions for geometric ergodicity to previously known conditions for (first order) stationarity.  Moreover, we discuss the applications of the obtained results and illustrate them by exemplary simulations. In Section <ref> we establish sufficient conditions for the  irreducibility and aperiodicity of Y needed to apply the previous results on geometric ergodicity. Section <ref> first gives a brief repetition of the Markov theory we use and the proofs of our results are developed. 

 





§ PRELIMINARIES
 
Throughout we assume that all random variables and processes are defined on a given filtered probability space 
(Ω,ℱ, , (ℱ_t)_t∈𝒯) with 𝒯= in the discrete time case and 𝒯=^+ in the 
continuous one. Moreover, in the continuous time setting we assume the usual conditions (complete, right continuous filtration) to be satisfied. 

For Markov processes in discrete 
and continuous time we refer to <cit.> and respectively <cit.>.
A summary of the most relevant notions and results from Markov processes is given in Section <ref> for the convenience of the reader.  




 §.§ Notation


The set of real m × n matrices is denoted by M_m,n() or only by M_n() if m=n. For the invertible n × n matrices
we write GL_n(). The linear subspace of symmetric matrices we denote by _n, by _n^+ the closed cone of   positive semi-definite 
matrices and the open cone of positive definite matrices by _n^++. Further we denote by I_n the n × n identity matrix. 

We introduce the natural ordering on _n and denote it by ≼, that is for A,B  ∈_n it holds A≼ B  ⇔  B-A ∈_n^+.
The tensor (Kronecker) product of two matrices A,B is written as A⊗ B. $⃗ denotes the well-known vectorization operator that maps 
then×nmatrices to^n^2by stacking the columns of the matrices below another.  The spectrum of a matrix is denoted byσ(·)and the spectral radius byρ(·). For a matrix with only real eigenvaluesλ_max(·)andλ_min(·)denote the largest and the smallest eigenvalue.(x)is the real part of a complex number. Finally,A^⊤is the transpose of a matrixA∈M_m,n().

By . _2we denote both the Euclidean norm for vectors and the 
corresponding operator norm for matrices and by._Fthe 
Frobenius norm for matrices. 



Furthermore, we employ an intuitive notation with respect to the (stochastic) integration with matrix-valued integrators referring to any of 
the standard texts (e.g. <cit.>) for a comprehensive treatment of the theory of stochastic integration. For anM_m,n()-valued Lévy processL, andM_d,m()resp.M_n,p()- valued  processesX,Yintegrable with respect toL, the term∫_0^t X_s   dL_s Y_sis to be understood as thed×p(random) matrix with(i,j)-th entry∑_k=1^m ∑_l=1^n ∫_0^t X_s^ik   dL_s^kl Y_s^lj.
If(X_t)_t∈^+is a semi-martingale in^mand(Y_t)_t∈^+one in^nthen the quadratic variation([X,Y]_t)_t∈^+is 
defined as the finite variation process inM_m,n()with components[X,Y]_ij,t=[X_i,Y_j]_tfort∈^+andi=1,…,m,j=1,…,n. 



 §.§ Lévy processes

Later on we use Lévy processes (see e.g. <cit.>) in^dand in the symmetric matrices_d. 

We consider a Lévy processL=(L_t)_t∈^+(whereL_0=0a.s.)
in^ddetermined by its characteristic function in the 
Lévy-Khintchine formE[e^i⟨u,L_t⟩]=exp{tψ_L(u)}fort∈^+with

    ψ_L(u)=i⟨γ_L,u⟩-1/2⟨ u,τ_Lu⟩+∫_ ^d
    (e^i⟨ u,x⟩-1-i⟨ u,x⟩ I_[0,1]({x)) ν_L(dx),   u∈ ^d,

whereγ_L∈^d,τ_L∈_d^+and the Lévy measureν_Lis a measure on^dsatisfyingν_L({0})=0and∫_^d(x^2∧1) 
ν_L(dx)<∞.Ifν_Lis a finite measure,Lis a compound Poisson process.
Moreover,⟨·,·⟩denotes the usual Euclidean scalar product on^d.

We always assumeLto be càdlàg and denote its jump measure byμ_L, i.e.μ_Lis the Poisson random measure on^+×^d∖{0}given by μ_L(B)=♯{s≥0: (s,L_s-L_s-)∈B}
for any
measurable setB⊂^+×^d∖{0}.Likewise,μ̃_L(ds,dx)=μ_L(ds,dx)-dsν_L(dx) denotes the compensated jump measure.


Regarding matrix-valued Lévy processes, we will only encounter matrix subordinators (see <cit.>), i.e. Lévy processes with paths in_d^+. Since matrix subordinators are  of finite variation and(X^*Y)(withX,Y∈_danddenoting the usual trace functional) defines a
scalar product on_dlinked to the Euclidean scalar product on^d^2via(X^*Y)=(⃗X)^*(⃗Y)=⟨(⃗Y), (⃗X)⟩, the characteristic function of a matrix subordinator can be represented as

    E(e^i(L_t^*Z))   =exp(tψ_L(Z)),  Z∈_d,  ψ_L(Z)=i (γ_L    Z)+∫__d^+(e^i(XZ)-1)ν_L(dX)

with driftγ_L∈_d^+and Lévy measureν_L.


The discontinuous part of the quadratic variation of any Lévy processLin^d[L,L]^d_t:=∫_0^t∫_ ^dxx^*μ_L(ds,dx)=∑_0≤ s≤ t(Δ L_s)(Δ L_s)^*,

is  a matrix subordinator with drift zero and Lévy measure
ν_[L,L]^d(B)=∫_^dI_B(xx^*)ν_L(dx)
for all Borel setsB⊆_d.


§ MULTIVARIATE COGARCH(1,1) PROCESS

In this section we present the definition of the MUCOGARCH(1,1) process and some relevant  properties mainly based on <cit.>.


 Let L be an ^d-valued Lévy process, A,B ∈ M_d() and C ∈_d^++. 
The MUCOGARCH(1,1) processG=(G_t)_t ≥ 0 is defined as the solution of 

    dG_t   = V_t-^1/2dL_t
    V_t   = Y_t + C 
     dY_t   = (BY_t- + Y_t-B^⊤) dt + A V_t-^1/2 d [L,L]^d_tV_t-^1/2 A^⊤,

with initial values G_0∈^d and Y_0∈_d^+.  

The process Y=(Y_t)_t≥ 0 is called MUCOGARCH(1,1) volatility process. 

Since we only consider MUCOGARCH(1,1) processes, we 
often simply write MUCOGARCH. 

Equations (<ref>) and (<ref>) directly give us an SDE for the covariance matrix processV:

    dV_t = (B(V_t- -C)+(V_t--C)B^⊤)dt + AV_t-^1/2 d[L,L]^d_t 
    V_t-^1/2 A^⊤.


Providedσ(B) ⊂(-∞,0) + i, we see thatV, as long as no jumps occur,  
returns to the levelCat an exponential rate determined byB. Since all 
jumps are positive semidefinite,Cis not a mean level but a lower bound. 


To have the MUCOGARCH process well-defined, we have to know that a unique 
solution of the SDE system exists and the solution ofY(andV) does 
not leave the set_d^+. In the following we always understand that our processes live on_d. Since_d^++is an open subset of_d, we now are in the most natural setting for SDEs and we get:


Let A,B∈ M_d(), C∈_d^++  and L be a d-dimensional  Lévy process. 
Then the SDE (<ref>) with  initial value Y_0∈_d^+ has a unique positive semi-definite solution (Y_t)_t∈^+.
The solution (Y_t)_t∈^+ is locally bounded and of finite variation. 

Moreover, it satisfies 
    Y_t=e^BtY_0e^B^⊤ t+∫_0^te^B(t-s)A(C+Y_s-)^1/2d[L,L]_s^d (C+Y_s-)^1/2A^⊤ e^B^⊤(t-s)

for all t ∈^+ and thus Y_t≽ e^BtY_0e^B^⊤ t 
for all t∈^+.
 
In particular, whenever (<ref>) is started with an initial valueY_0∈_d^+(or (<ref>) withV_0≽C) the solution stays in_d^+(_d^++C) at all times. This can be straightforwardly seen from (<ref>) and the fact that for anyM∈M_d()maps of the formX↦MXM^⊤map_d^+into itself (see <cit.> for a more detailed discussion).

At first sight, it appears superfluous to introduce the processYinstead of directly working with the processV. However, in the following it is more convenient to work withY, as then the matrixCdoes not appear in the drift and the state space of the Markov process analysed is the cone of positive semi-definite matrices itself and not a translation of this cone.


  (i) The MUCOGARCH process (G,Y) as well as its volatility process Y alone are  
temporally homogeneous  strong Markov processes on ^d×_d^+ and 
_d^+, respectively, and they have the weak -Feller property.

  (ii)Y is non-explosive and has the weak -Feller property. Thus it is a Borel right process.
<cit.> is (i).
	
	 -Feller property of Y: Let f∈(_d^+). We have to show that ∀ 
	t≥0^t f(x) → 0,   for  x→∞,
	where we understand x →∞ in the sense of x_2→∞.
	Since ^t f(x)=(f(Y_t(x))), where x denotes the starting point Y_0=x, it is enough to show that Y_t goes to infinity for x →∞:
	
    Y_t_2 ≥    e^BtY_0 e^B^⊤ t_2 	≥ e^-Bt^-2_2 x_2 →∞ for x_2 →∞.

	
As argued in Section	<ref> any  -Feller process is a Borel right process and the non-explosivity property is shown in the proof of Theorem 6.3.7 in 
	<cit.>. 

For ^d-valued solutions to Lévy-driven stochastic differential equations 
dX_t = σ(X_t-)dL_t<cit.> gives necessary and sufficient conditions for the rich Feller property, which includes the -Feller property, if σ is continuous and of (sub-)linear growth. But since our direct proof is quite short, we prefer it instead of trying to adapt the result of <cit.> to our state space. 



































§ GEOMETRIC ERGODICITY OF THE MUCOGARCH VOLATILITY PROCESS Y


In Theorem <cit.> sufficient conditions for the existence of a stationary distribution for the volatility processYorV(with certain moments finite) are shown, but neither the uniqueness of the stationary distribution nor that it is a limiting distribution are obtained. Our main theorem now gives sufficient conditions for geometric ergodicity and thereby for the existence of a unique stationary distribution to which the transition probabilities converge exponentially fast in total variation (and in stronger norms).

On the proper closed convex conethe trace:  →^+is well-known to define a norm which is also a linear functional. So do actually all the maps→^+,  X↦(ηX)withη∈_d^++, as_d^+is also generating, self-dual and has interior_d^++(cf. <cit.>, for instance). The latter can also be easily seen using the following Lemma which is a consequence of <cit.>.

Let X∈_d, Y∈_d^+. Then

    λ_min(X)(Y)≤(XY)≤λ_max(X)(Y).


To be in line with the geometry of_d^+we thus use the above norms to define appropriate test functions and look at the trace norm for the finiteness of moments (which, of course, is independent of the actually employed norm).

Forp=1andp ≥2it is shown in <cit.>, that the finiteness of(L_1_2^2p)and(Y_0_2^p)implies the finiteness of(Y_t_2^p)for allt. We improve this to allp >0. 

	Let Y be a MUCOGARCH volatility proces and  p >0. If ((Y_0)^p) < ∞, ∫_y_2≤ 1 y _2 ^2∧2pν_L(dy) < ∞  and (L_1_2^2p) < ∞, then ((η Y_t)^p)< ∞ for all t ≥ 0,  η∈_d^++ and t ↦((η Y_t)^p) is locally bounded. 

Observe that(L_1_2^2p) < ∞is equivalent to∫_y_2≥1  y _2 ^2p ν_L(dy) < ∞and that∫_y_2≤1  y _2 ^2∧2p ν_L(dy) < ∞is always true forp≥1and otherwise means that the2p-variation ofLhas to be finite.


Let Y be a MUCOGARCH volatility process  which is μ-irre­ducible with the support of μ having 
non-empty interior and aperiodic.

If one of the following conditions is satisfied

  (i) setting p=1 there exists an η∈_d^++ such that ∫_y_2≥ 1y_2^2ν_L(dy)<∞ and 
	
    η B+B^⊤η+ A^⊤η A∫_^d  yy^⊤ν_L(dy)_2∈-_d^++,

  (ii) setting p=1 there exists an η∈_d^++ such that ∫_y_2≥ 1y_2^2ν_L(dy)<∞ and 
	
    η B+B^⊤η+ λ_max(A^⊤η A)∫_^d yy^⊤ν_L(dy)∈-_d^++,

  (iii) there exist a p∈ (0,1]  and an η∈_d^++ such that 	∫_^d y _2 ^2pν_L(dy) < ∞ and 
	
    ∫_^d( ( 1+K_η,Ay^2_2)^p - 1)ν_L(dy)
    	+K_η,B p  < 0,

	where K_η,B:=max_x∈_d^+,(x)=1 ((η B+B^⊤η)x)/(η x) and K_η,A:=max_x∈_d^+,(x)=1 (A^⊤η Ax)/(η x),
 
  (iv) there exist a p∈ [1,∞) and an η∈_d^++ such that ∫_y_2≥ 1y_2^2pν_L(dy)<∞ and 
	
    ∫_^d(2^p-1( 1+K_η,Ay^2_2)^p - 1)ν_L(dy)
    	+K_η,B p  < 0,
	
where K_η,B, K_η,A are as in (iii),
 
  (v) there exists a p∈ [1,∞) and an η∈_d^++ such that ∫_y_2≥ 1y_2^2pν_L(dy)<∞ and 
 
    max{2^p-2,1}K_η,A∫_^dy^2_2(  1 +y^2_2K_η,A)^p-1ν_L(dy)+K_η,B    < 0

 where K_η,B, K_η,A are as in (iii),
 
then a  unique stationary distribution for the MUCOGARCH(1,1)
volatility process Y exists, Y is positive Harris recurrent, geometrically ergodic (even (η·)^p+1 uniformly ergodic) and the 
stationary distribution has a finite p-th moment.

  (i) For p=1 the cases (i) and (ii) give us quite strong conditions comparable to the ones known for affine processes in  (cf. <cit.>). For p≠ 1 it seems that the non-linearity of our SDE implies that we need to use inequalities that are somewhat crude.
	
  (ii) Condition (<ref>) demands that the driving Lévy process is compound Poisson for p>1. To overcome this restriction is the main motivation for considering case (v).
	
  (iii) For p=1 the Conditions  (<ref>), (<ref>), (<ref>) agree. 
	
	In dimension one they also agree with Conditions (<ref>) and (<ref>). Moreover, then the above sufficient conditions agree with the necessary and sufficient conditions of <cit.> for a univariate COGARCH(1,1) process to have a stationary distribution with finite first moments, as follows from <cit.>.
	
	
  (iv) 	 Observing that 
    K_η,A∫_^dy^2_2ν_L(dy)+K_η,B≥max_x∈_d^+,(x)=1 ((η B+B^⊤η+A^⊤η A∫_^d  yy^⊤ν_L(dy)_2)x)/(η x),

	the self-duality of  shows that for p=1 the equivalent Conditions (<ref>), (<ref>), (<ref>) imply  that (<ref>) holds. 
	
	Conversely the upcoming Examples <ref>, <ref> show that (<ref>) and  (<ref>) are less restrictive than the equivalent Conditions (<ref>), (<ref>), (<ref>) and that (<ref>) does not imply  (<ref>) and vice versa. 

  (v) 	Arguing as in <cit.>, one sees that if (<ref>) is satisfied for some p>0 it is also satisfied for all  smaller ones. Using similar elementary arguments, the same can be shown for (<ref>) and for (<ref>) this property is obvious.
	
	Note, however, that ∫_^d y _2 ^2pν_L(dy) < ∞ for some 0<p≤ 1 does not imply that ∫_^d y _2 ^2p̅ν_L(dy) is finite for 0<p̅≤ p.
	
  (vi) From the exercise on p. 98 of <cit.> (taking e.g. A=[  1  0; 10  1 ] there) we see immediately that if (<ref>),  (<ref>), (<ref>), (<ref>), or (<ref>) is satisfied for one η∈_d^++ it can be violated for η̅∈_d^++, η̅≠η. Hence, the possibility to choose η∈_d^++ freely is important.
	
  (vii) In the Case (iii) ∫_^d K_η,A^py^2p_2ν_L(dy)
	+K_η,B p  < 0, implies (<ref>), as (x+y)^p-x^p≤ y^p for x,y≥ 0 and p∈(0,1].
		
The above Conditions (<ref>), (<ref>), (<ref>) appear enigmatic at a first sight. However, essentially they are related to the drift being negative in an appropriate way.

  (i) If one of the Conditions (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) is satisfied, then η B+B^⊤η∈-_d^++.
		
  (ii) If one of the Conditions (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) is satisfied, then (σ(B))<0.
		
  (iii) If (σ(B))<0, then there exists an η∈_d^++ such that η B+B^⊤η∈-_d^++.
	
(i)	This is obvious for (<ref>), (<ref>).
	In the other cases we get that K_η,B<0 must hold. Therefore max_x∈_d^+,(x)=1 ((η B+B^⊤η)x)<0. By the self-duality of  this shows η B+B^⊤η∈-_d^++.

(ii) and (iii) now follow  from <cit.>.

So in the end what we demand is that the drift is “negative” enough to compensate for a positive effect from the jumps.  The Conditions (<ref>), (<ref>) can be related to eigenvalues of linear maps on_d.

  (i) Condition (<ref>) holds if and only if the linear map _d→_d, X↦ XB+B^⊤ X +A^⊤ X A ∫_^d yy^⊤ν_L(dy)_2 has only eigenvalues with strictly negative real part.
		
  (ii) Condition (<ref>) holds if the linear map _d→_d, X↦ XB+B^⊤ X +(A^⊤ X A)∫_^d  yy^⊤ν_L(dy) has only eigenvalues with strictly negative real part.
	
	(i)	Follows from <cit.>, as it is easy to see that the map is quasi monotone increasing and as a linear map and its adjoint have the same eigenvalues. 
	
	(ii) Follows analogously after noting that 0≤λ_max( A^⊤ X A)≤(A^⊤ X A).
[Relation to first order stationarity]
	We say that a process is first order stationary, if it has finite first moments at all times, if the first moment converges to a finite limit independent of the initial value as time goes to infinity and if the first moment is constant over time when the process is started at time zero with an initial value whose  first moment equals the limiting value.
	
	According to <cit.> sufficient conditions for asymptotic first-order stationarity of the MUCOGARCH volatility Y are:
	
  (a) there exists a constant σ_L ∈^+ such that ∫_^d y y^⊤ν_L(dy) = σ_L I_d,
		
  (b)σ(B) ⊂ (-∞, 0) + i,
		
  (c)σ( B ⊗ I + I ⊗ B + σ_L(A ⊗ A)  ) ⊂ (-∞, 0) + i.
	An inspection of the arguments given there shows that (c) only needs to hold for the linear operator B ⊗ I + I ⊗ B + σ_L(A ⊗ A)  restricted to the set (⃗_d). Under (a) ∫_^d yy^⊤ν_L(dy)_2=σ_L and devectorizing thus shows that B ⊗ I + I ⊗ B + σ_L(A ⊗ A) is just the linear operator in Lemma <ref> (i). 
	
	Hence, under (a) our Condition (<ref>) implies (b) and (c). So our conditions for geometric ergodicity with a finite first moment  are certainly not worse than the previously known conditions for just first order stationarity. 


The constantsK_η,B, K_η,Aare related to changing norms and may be somewhat tedious to obtain. The following lemma follows immediately from Lemma <ref> and implies that we can replace them by eigenvalues in the Conditions  (<ref>), (<ref>), (<ref>).

It holds that:

  * K_η,B≤λ_max(η B+B^⊤η)/λ_min(η).
	
  * K_η,A≤λ_max(A^⊤η A)/λ_min(η).
	
  * K_I_d,B=λ_max(B+B^⊤).
	
  * K_I_d,A=λ_max(A^⊤ A).

If B is symmetric, we have that A⊗ A_2=A_2^2=λ_max(A^TA) and that λ_max(B+B^⊤)=2λ_max(B). So we see that in this case our Condition (<ref>) implies Condition (4.4) of <cit.>  for the existence of a stationary distribution for p=k=1.

 For a symmetric B and p=k>1 the conditions are very similar only that an additional factor of 2^p-1 appears inside the integral in the conditions ensuring geometric ergodicity. So the previously known conditions for the existence of a stationary distribution with a finite p-th moment for p>1 may be somewhat less restrictive than our conditions for geometric ergodicity with a finite p-th moment for p>1.  But in contrast to <cit.> we do not  need to restrict ourselves to B being diagonalizable and integer moments p. 

	Let us now consider that the driving Lévy process is a compound Poisson process with rate γ>0 and jump distribution P_L. So P_L is a probability measure on ^d and ν_L=γ P_L.
	
	As in many applications where discrete time multivariate GARCH processes are employed the noise is taken to be an iid standard normal distribution, a particular choice would be to take P_L as the standard normal law.
	
	However, we shall only assume that P_L has finite second moments, mean zero and covariance matrix Σ_L. Observe that the upcoming Section <ref> shows that the needed irreducibility and aperiodicity properties hold as soon as P_L has an absolutely continuous component with a strictly positive density around zero (again definitely satisfied when choosing the standard normal distribution).
	
	Then we have that
	
    ∫_^d  yy^⊤ν_L(dy)_2   =γΣ_L_2=γλ_max(Σ_L),
    ∫_^dy^2_2ν_L(dy)   =γ(Σ_L).

	
	If we assume that B=β I_d, A=α I_d for some α,β∈ and η=I_d, then (<ref>) and (<ref>) are equivalent to
	
    2β+α^2γλ_max(Σ_L)<0

	whereas 
	(<ref>), (<ref>), (<ref>) are becoming
	
    2β+α^2γ(Σ_L)<0

	for p=1. Note that in this particular set-up it is straightforward to see that the choice of η has no effect on  (<ref>), (<ref>), (<ref>), (<ref>). The latter is also the case for (<ref>) if Σ_L is additionally assumed to be a multiple of the identity. 
	
	As, for example, λ_max(I_d)=1, but (I_d)=d, this also illustrates that Conditions 	(<ref>), (<ref>), (<ref>) are considerably more restrictive than Conditions (<ref>) and (<ref>) unless we are in the univariate case. 
	
	For p≠ 1  it is not sufficient to only specify the mean and variance of the jump distribution to check Conditions 	(<ref>), (<ref>), (<ref>). For a concrete specification of P_L it is, however, straightforward to check them by (numerical) integration.
		
				We consider the same basic set-up as in Example <ref> in dimension d=2.
				
				If we take A=I_2, B=[ -2  0;  0 -4 ], γ=1, Σ_L=[ 3 0; 0 6 ] and η=I_2, then Condition (<ref>) is violated whereas (<ref>) is satisfied.
				
				If we take A=[ √(3)    0;    0 √(6) ] , B=[ -2  0;  0 -4 ], γ=1, Σ_L=I_2 and η=I_2, then Condition (<ref>) is violated whereas (<ref>) is satisfied.
				
				In both cases Conditions 	(<ref>), (<ref>), (<ref>) are violated for p=1.
			
			An inspection of our proofs shows, that we can also consider the stochastic differential equation
			
    dY_t = (BY_t- + Y_t-B^⊤) dt + A  (C+Y_t-)^1/2 dL_t(C+Y_t-)_t-^1/2 A^⊤

			with  A,B ∈ M_d() and C ∈_d^++, initial value Y_0∈_d^+ and L being a d× d matrix subordinator with drift γ_L=0 and Lévy measure ν_L.
			
			All results we obtained in this section remain valid when replacing ∫_^d by ∫_, yy^T by y, y_2^2 by y_2 and (L_1_2^2p) < ∞ by (L_1_2^p) < ∞.
			
		
			In principle, it is to be expected that the results can be generalized with substantial efforts to general order MUCOGARCH(p,q) processes (cf. <cit.> for discrete time BEKK models). However, as explained in the introduction, they have only been defined briefly and no applicable conditions on the admissible parameters are known nor is the natural state space of the associated Markov process (which should not be some power of the positive definite cone).  As can be seen from the discrete time analogue, the main issue is to find a suitable Lyapunov test function and suitable weights need to be introduced. For orders different from (1,1) the weights known to work in the discrete time setting are certainly not really satisfying except when the Lyapunov test function is related to the first moment (the case analysed in <cit.>).  On the other hand in applications very often order (1,1) processes provide rich enough dynamics already. Therefore, we refrain from pursuing this any further.
		
		
		The geometric ergodicity results of this section are of particular relevance for at least  the following:
		
  *  Model choice:

			In many applications  stationary models are called for and one should use models which have a unique stationary distribution. Our results are the first giving sufficient criteria for a MUCOGARCH model to have a unique stationary distribution (for the volatility process).
			
  *  Statistical inference:
<cit.> investigate  in detail a moment based estimation method for the parameters of MUCOGARCH(1,1) processes. This involves establishing identifiability criteria, which is a highly non-trivial issue, and calculating moments explicitly, which requires assumptions on the moments of the driving Lévy process. However, the asymptotics of the estimators derived there hinge  centrally on the results of the present paper. Actually, we cannot see any reasonable other way to establish consistency and asymptotic normality of estimators for the MUCOGARCH parameters  than using the geometric ergodicity conditions for the volatility of the present paper and to establish that under stationarity it implies that the increments of the MUCOGARCH process G are ergodic and strongly mixing (see 	<cit.> for details).
			
  *  Simulations:

			Geometric ergodicity implies that simulations (of a Markov process)  can be started with an arbitrary initial value and that after a (not too long) burn-in period the simulated path behaves essentially like one following the stationary dynamics. On top V-uniform ergodicity (which is actually obtained above) provides results on the finiteness of certain moments and the convergence of the moments to the stationary case. We illustrate this now in some exemplary simulations.
		
					We consider the set-up of Example <ref> in dimension two and choose α=0.14, β=-0.01, C=I_2, γ=1 and the jumps to be standard normally distributed.  
					
			
	
			Then condition (<ref>) (or (<ref>)) is satisfied (but not conditions (<ref>), (<ref>), (<ref>)) and so we have geometric ergodicity with convergence of the (absolute) first moment. However, this is a rather extreme case as (given all the other parameter choices)  (<ref>) is satisfied if and only if |α|<√(0.02)≈ 0.1414 and thus some convergences are only  to be seen clearly for very long paths. We have simulated a path up to time 20 million with Matlab (see Figure <ref>). The paths themselves look immediately stationary (at this time scale) but extremely heavy tailed. In the long run (and not too fast) the running empirical means are reasonably converging (to their true values of 50 for the variances and 0 for the covariance). In sharp contrast to this there seems to be no convergence for the empirical second moments at all and actually we strongly conjecture that the stationary covariance matrix is infinite in this case. Looking at lower moments, like the 0.25th (absolute) empirical moments there is again convergence to be seen which is faster than the one for the first moment (as is to be expected). Zooming into the paths at the beginning in Figure <ref> (left plot)  shows that the process clearly does not start of like a stationary one, but the behaviour looks stationary to the eye very soon.	
			
			Changing only the parameter α to 0.142 and using the same random numbers we obtain the results depicted in Figure <ref>. The initial part of the paths (see  right plot in Figure <ref>) looks almost unchanged (observe that the scaling of the vertical axes in the plots of Figure  <ref> differs). The paths in total clearly appear to be even more heavy tailed. Now neither the empirical second moments nor the empirical means seem to converge (and we conjecture  that both mean and variance are infinite for the stationary distribution) . Now condition (<ref>)  does not hold any more and so the simulations suggest that this condition implying geometric ergodicity  with convergence of the first moments is pretty sharp, as expected from the theoretical insight. The simulations also seem to clearly indicate that the 1/4th (absolute) moments still converge. Numerical integration shows for  (<ref>) (again the choice of η makes no difference in this special case):
				
    ∫_^2( ( 1+α^2y^2_2)^1/4 - 1)ν_L(dy)
    				+2β/4≈ 0.0098  -0.005   > 0.

				So our sufficient condition  (<ref>) for convergence of the 1/4th moment is not satisfied, but due to the involved inequalities we expected it not to be too sharp.  Actually, numerically not even the logarithmic condition <cit.> for the existence of a stationary distribution seems to be satisfied. This illustrates that these conditions involving norm estimates are not too sharp, as our simulations and the fact that condition   (<ref>) is almost satisfied strongly indicate that geometric ergodicity with convergence of some moments 0<p<1 should hold.
		
		Finally, we consider an example illustrating that our conditions forp≠1are more precise if one coordinate dominates the others.
		
			We consider the same model as in the previous Example <ref> with the only difference that we now assume the jumps to have a two-dimensional normal distribution with covariance matrix Σ_L=[ 1 0; 0 σ ] for σ>0. For σ→ 0 we conclude from the formulae given in Example <ref> that for p=1(<ref>), (<ref>), (<ref>) are asymptotically equivalent to (<ref>) and as in Example  <ref> the choice of η does not matter (they also are equivalent to (<ref>)  for η=I_2, but there the choice of η may now matter). Note that  (<ref>) is not affected by changes in σ as long as σ≤ 1.
			 
			 Choosing σ=1/1000 and α=0.142 as in the second part of Example  <ref> numerical integration shows for  (<ref>)  with p=1/4:
			 
    ∫_^2( ( 1+α^2y^2_2)^1/4 - 1)ν_L(dy)
    			 +2β/4≈ 0.0049  -0.005   < 0.

			 So we obtain geometric ergodicity and the finiteness of the 1/4th absolute moments from our theory.
			 
			 	
			 	
			 	Figure <ref> shows the results of a simulation using the same time horizon and random numbers as in Example <ref>. The plots of the paths, the empirical means and second moments) show that the first variance component extremely dominates the others. One immediately sees that the empirical mean and second moment of V_11 does not converge. In line with our theory all 0.25th absolute moments seem to converge pretty fast.
		

§ IRREDUCIBILITY AND APERIODICITY OF THE MUCOGARCH VOLATILITY PROCESS

So far we have assumed away the issue of irreducibility and aperiodicity focusing on establishing a Foster-Lyapunov drift condition to obtain geometric ergodicity.  As this is intrinsically related to the question of the existence of transition densities and support theorems, this is a very hard problem of its own. In our case the degeneracy of the noise, i.e. that all jumps of the Lévy process are rank one and that thus the jump distribution has no absolutely continuous component, makes our case particularly challenging.

We now establish sufficient conditions for the irreducibility and aperiodicity ofYby combining at leastdjumps to get an absolutely continuous component of the increments. 

 
Let Y be a MUCOGARCH volatility process driven by a compound 
Poisson process L and with A∈ GL_d(), (σ(B))<0. If the jump distribution of L has a non-trivial absolutely 
continuous component equivalent to the Lebesgue measure on ^d, 
then  Y is irreducible with respect to the Lebesgue measure on  
_d^+ and aperiodic. 


We can soften the conditions on the jump distribution of the Compound Poisson process:

 Let Y be a MUCOGARCH volatility process driven by a compound Poisson process L and with A∈ GL_d(), (σ(B))<0.  If the jump distribution of L has a non-trivial absolutely continuous component  equivalent to the Lebesgue measure on ^d restricted to an open neighborhood of zero, then Y is irreducible w.r.t. the Lebesgue measure restricted to an open neighborhood of zero in _d^+ and aperiodic.

(i) If the driving Lévy process is an infinite activity Lévy process, whose Lévy measure has a non-trivial absolutely continuous component and the density of the component w.r.t. the Lebesgue measure on _d^+ is strictly positive in a neighborhood of zero, we can show that Y is open-set irreducible w.r.t. the Lebesgue measure restricted to an open neighborhood of zero in . For strong Feller processes open-set irreducibility provides irreducibility, but the strong Feller property is to the best of our knowledge hard to establish for Lévy-driven SDEs. A classical way to show irreducibility is by using density or support theorems based on  Malliavin Calculus, see e.g. <cit.>. But they all require, that the coefficients of the SDE have bounded derivatives, which is not the case for the MUCOGARCH volatility process. So finding criteria for irreducibility in the infinite activity case appears to be a very challenging question beyond the scope of the present paper. 

(ii) In the univariate case establishing irreducibility and aperiodicity is much easier, as one can easily use the explicit representation of the COGARCH(1,1) volatility process to establish that one has weak convergence to a unique stationary distribution which is self-decomposable and thus has a strictly positive density. This way is blocked in the multivariate case as we do not have the explicit representation and the linear recurrence equation structure.


Note that the condition that the Lévy measure has an absolutely continuous component with a support containing zero is the obvious analogue to the condition on the noise in <cit.>.


Let us return to the SDE (<ref>) driven by a general matrix subordinator which we introduced in Remark <ref>. It is straightforward to see that then the conclusions of Theorem <ref> or Corollary <ref>, respectively, are valid for the process Y, if the matrix subordinator L is a compound Poisson process with the jump distribution of L having a non-trivial absolutely 
continuous component equivalent to the Lebesgue measure on _d^+   (restricted to an open neighborhood of zero).


§ PROOFS

To prove our results we use the stability concepts for Markov processes of <cit.>.


 §.§ Markov processes and ergodicity

For the convenience of the reader, we first give a short introduction to the definitions and results for general continuous time Markov processes following mainly
 <cit.>.

Given be  a state spaceX, which is assumed to be an open or closed subset of a finite-dimensional real vector spaced equipped with the usual topology and the Borel-σ-algebra(X). We consider a continuous time Markov processΦ=(Φ_t)_t≥0onXwith transition probabilities^t(x,A)=_x(Φ_t ∈A)forx∈X, A ∈(X). 

The operator^tfrom the associated transition semigroup acts on a bounded 
measurable functionfas

    ^t f(x) = ∫_X ^t(x,dy)f(y)

and on aσ-finite measureμonXas

    μ^t(A) = ∫_X μ(dy) ^t(y,A).


To define non-explosivity, we consider a fixed family{ O_n | n ∈_+}of open precompact sets, i.e. the closure ofO_nis a compact subset ofX, 
for whichO_n ↗Xasn →∞. WithT^mwe denote the first 
entrance time toO_m^cand byξthe exit time for the process, defined as
 ξ𝒟= lim_m →∞ T^m.
We call the process Φnon-explosive if _x(ξ = ∞)=1 
for all x∈ X.


 By(X)we denote the set of all continuous and bounded functionsf: X →and 
by(X)those continuous and bounded functions, which vanish at infinity. 


	Let (^t)_t∈_+ be the transition semigroup of a time homogeneous 
	Markov process Φ. 

	(i) (^t)_t∈_+ or Φ is called 
	stochastically continuous if 
	
    lim_t→ 0 
     t≥ 0^t(x, 𝒩(x)) = 1

	for all x ∈ X and open neighborhoods 𝒩(x) of x. 

	(ii) (^t)_t∈_+ or Φ is a (weak) 
		-Feller semigroup or process if it is stochastically continuous 
	and 
	
    ^t ((X)) ⊆(X)   for all   t≥ 0.

	(iii) If in (ii) we have instead of (<ref>)
	
    ^t ((X)) ⊆(X)   for all   t≥ 0

	we call the semigroup or the process (weak) -Feller.

 Combining the definition of strongly continuous contraction 
 semigroups, Theorem 4.1.1  and the definition of Feller processes in  <cit.>  shows that a -Feller process is a Borel-right process (cf.  <cit.> for a definition).

From now on we  assume thatΦis a non-explosive Borel right process. For the definitions and details of 
the existence and structure see <cit.>. 



A σ-finite measure π on (X) with the property 

    π = π^t,  ∀ t ≥ 0

is called invariant.
 


Notation: Byπwe always denote an invariant measure ofΦ, if it exists. 

Φ is called exponentially ergodic, 
if an invariant measure π exists and satisfies for all x ∈ X
    ^t(x,.) - π_TV≤ M(x) ρ^t,   ∀ t≥0

for some finite M(x), some ρ<1 and where μ_TV:=sup_|g|≤ 1, g  measurable | ∫μ(dy)g(y)|  denotes the total variation norm. 

If this convergence holds for the f-norm μ_f:=sup_|g|≤ f, g  measurable | ∫μ(dy)g(y)| (for any signed measure μ) , 
where f is a measurable function from the state space X to [1,∞), we 
call the process 𝐟-exponentially ergodic.


A seemingly stronger formulation ofV-exponential ergodicity isV-uniform ergodicity: We require thatM(x) = V(x) ·Dwith some finite constantD. 
Φ is called 𝐕-uniformly ergodic, if a measurable 
function V: X → [1,∞) exists such that for all x ∈ X
    ^t(x,.) - π_V ≤ V(x) D ρ^t,   t≥0

holds for some D<∞, ρ <1.



To prove ergodicity we need the notions of irreducibility and aperiodicity. 


 For any σ-finite measure μ on ℬ(X) we call the process 
Φμ-irreducible if for any B ∈ℬ(X) 
with μ(B)>0
    _x (η_B) > 0, ∀ x ∈ X

holds, where η_B := ∫_0^∞_{Φ_t ∈ B} dt is the occupation time.



This is obviously the same as requiring 

    ∫_0^∞^t(x,B) dt >0, ∀ x ∈ X.



IfΦisμ-irreducible, there exists a maximal irreducibility measureψsuch that every other irreducibility measureνis absolutely continuous with respect toψ(see <cit.>). 
We writeℬ^+(X)for the collection of all measurable subsetsA∈ℬ(X)withψ(A)>0. 


In <cit.> it was shown, that if the discrete time h-skeleton of a process, the ^h-chain, 
is ψ-irreducible for some h>0, 
then it holds for the continuous time process. If the ^h-chain  is ψ-irreducible for every h>0, 
we call the process simultaneously ψ-irreducible.


One probabilistic form of stability is the concept of Harris recurrence.

 

(i) Φ is called Harris recurrent, if either
		
  * _x(η_A = ∞)=1 whenever ϕ(A)>0 for some σ-finite measure ϕ, or
		
  * _x(τ_A < ∞)=1 whenever μ(A)>0 for some σ-finite measure μ. τ_A:= inf{t≥0 : Φ_t ∈ A } is the first hitting time of A. 
	
	(ii) Suppose that Φ is Harris recurrent with finite invariant measure π, then Φ is called positive Harris recurrent.


To define the class of subsets ofXcalled petite sets, we suppose thatais a probability distribution 
on_+. We define the Markov transition functionK_afor the process sampled byaas

    K_a(x,A):= ∫_0^∞^t(x,A)a(dt),  ∀ x∈ X, A∈ℬ.

 A nonempty set C∈ℬ is called ν_a-petite, if ν_a is a nontrivial measure on ℬ(X) 
and a is a sampling distribution on (0,∞) satisfying

    K_a(x,.) ≥ν_a(.),  ∀ x∈ C.

  When the sampling distribution a is degenerate, i.e. a single point mass, we call the set Csmall. 
 
 Like in the discrete time Markov chain theory the set C is small, if there exists an m>0 and a nontrivial measure
  ν_m on ℬ(X) such that for all x∈ C,B∈ℬ(X)
    ^m(x,B) ≥ν_m(B)

 holds. 


For discrete time chains there exists a well known concept of periodicity, see for example <cit.>.
For continuous time processes this definition is not adaptable, since there are no fixed time steps. 
But a similar concept is the definition of aperiodicity for continuous time Markov processes as introduced in <cit.>.


 A ψ-irreducible Markov process is called aperiodic if for some small set C ∈ℬ^+(X) there exists a T 
 such that 
 ^t(x,C) >0 for all t≥ T and all x∈ C. 

When Φ is simultaneously ψ-irreducible then we know from <cit.> that
 every skeleton chain is aperiodic in the sense of a discrete time Markov chain.
 
 
 For discrete time Markov processes there exist conditions such that every compact set is petite and every petite set is small:
  
 
(i) If Φ, a discrete time Markov process, is a Ψ-irreducible Feller chain with supp(Ψ) having non-empty interior, every compact set is petite. 

(ii)  If Φ is irreducible and aperiodic, then every petite set is small.  
 
 Proposition <ref> (i) is also true for continuous time 
Markov processes, see <cit.>.


To introduce the Foster-Lyapunov criterion for ergodicity we need the concept of the extended generator of a Markov process.

𝒟 () denotes the set of all functions f: X ×_+→ 
for which a function g: X ×_+→ exists, such that ∀ x 
∈ X, t>0
    _x(f(Φ_t,t)) = f(x,0) + _x( ∫_0^tg(Φ_s,s)ds), 
       _x( ∫_0^t|g(Φ_s,s)|ds) < ∞

holds. We write f := g and call  the extended generator of Φ. 
𝒟() is called the domain of . 

 
The next theorem from <cit.> gives for an irreducible and aperiodic 
Markov process a sufficient criterion to beV-uniformly ergodic. This is a modification of the Foster-Lyapunov drift 
criterion of <cit.>. 


 Let (Φ_t)_t≥ 0 be a μ-irreducible and aperiodic Markov process. If 
there exist 
constants b,c > 0 and a petite set C in (X) as well as a measurable function V: 
X → [1,∞) such that 
    V ≤ -bV + c _C ,

where  is the extended generator, then (Φ_t)_t≥ 0 is 
V-uniformly ergodic.


 §.§ Proofs of Section <ref>

We prove the geometric ergodicity of the MUCOGARCH volatility process by using 
Theorem <ref>. The main task is to show the validity of a Foster-Lyapunov drift condition and that the used function belongs to the domain of the extended generator. For the latter we need the existence of moments given in Lemma <ref>. The different steps require similar inequalities. The most precise estimations are needed for the Foster-Lyapunov drift condition. Therefore we below first prove the Forster-Lyapunov drift condition assuming for a moment that we are in the domain of the extended generator and prove this and the finiteness of moments later on, as the inequalities needed there are often obvious from the proof of the Foster-Lyapunov drift condition.





  §.§.§ Proof of Theorem <ref>


The first and main part of the proof is to show the geometric ergodicity, as the positive Harris recurrence is essentially  a consequence of it.  
To prove the geometric ergodicity, and hence the existence and uniqueness of a stationary distribution, of the MUCOGARCH volatility process, it is enough to show that the Foster-Lyapunov drift condition of Theorem <ref> holds. All other conditions of Theorem <ref> follow from Theorem <ref> or  the assumptions.



  
The extended generator 

Using e.g. the results on SDEs, the stochastic symbol and the infinitesimal generator of  <cit.>, <cit.> and <cit.> and that(XY)is the canonical scalar product in_d, we see that for a sufficiently regular functionu: _d^+→in the domain of the (extended/infinitesimal) generator𝒜we have.





























































    u(x)    = ((Bx+xB^⊤) ∇ u(x)) + ∫_^d( u(x + A(C+x)^1/2 yy^⊤(C+x)^1/2A^⊤) - u(x) )  ν_L(dy) 
       =: 𝒟 u(x) + 𝒥 u(x).

We abbreviate the first addend, the drift part, with𝒟 u(x)and the second, the jump part, with𝒥 u(x). 


  
Foster-Lyapunov drift inequality 

As test function we chooseu(x) = (ηx)^p + 1, thusu(x) ≥1. Note that the gradient ofuis given by∇u(x) = p (ηx)^p-1 η. 

Forp ∈(0,1)the gradient ofuhas a singularity in0, but in the end we look at((Bx+xB^⊤) ∇u(x)), which turns out to be continuous in0. 

Now we need to look at the cases (i) - (v) separately, as the proofs work along similar lines, but differ in important details.


    Case (i)
 

We have

    𝒥 u(x)    = ∫_^d( (η(x + A(C+x)^1/2 yy^⊤(C+x)^1/2A^⊤)) - (η x) )  ν_L(dy) 
       =∫_^d( η^1/2A(C+x)^1/2 yy^⊤(C+x)^1/2A^⊤η^1/2)  ν_L(dy)
       =( η^1/2A(C+x)^1/2∫_^d   yy^⊤ν_L(dy)(C+x)^1/2A^⊤η^1/2)  
       ≤( η^1/2A(C+x)A^⊤η^1/2)∫_^d  yy^⊤ν_L(dy)_2.

In the last step we usedX≼X_2I_d=λ_max(X)I_dforX∈_d^+, thatis monotone in the natural order on_d^+,  and that maps of the form_d→_d, X↦ZXZ^⊤withZ∈M_d()are order preserving.

Hence

    𝒥 u(x)    ≤( A^⊤η A(C+x))∫_^d  yy^⊤ν_L(dy)_2.

For the drift part we get

    𝒟 u(x)     =((Bx+xB^⊤) η) =((η B+B^⊤η)x).


Together we have

    𝒜 u(x)    = ((η B+B^⊤η+ A^⊤η A∫_^d  yy^⊤ν_L(dy)_2  ν_L(dy))x)+  ( A^⊤η AC)∫_^d  yy^⊤ν_L(dy)_2 .

Since ηB+B^⊤η+ A^⊤ηA∫_^d  yy^⊤ν_L(dy)_2∈-_d^++, there exists ac>0, such that 
    ((η B+B^⊤η+ A^⊤η A∫_^d  yy^⊤ν_L(dy)_2)x)≤ -c(η x).


Summarizing we have that there existc,d>0such that

    u(x) ≤ - c (η x) + d.


For (x)> kandkbig enough there exist0 < c_1 < csuch that
u(x)   ≤- c_1 ( (ηx) +1 ).
For (x) ≤kwe have
- c_1 (ηx) + d = -c_1 ((ηx) +1 ) + e,
withe := c_1+d >0.
Altogether we have

    u(x) ≤ -c_1 u(x) + e _D_k,

whereD_k:= { x: (x) ≤k }is a compact set. By Proposition <ref> (i) this is also a petite set. Therefore the Foster-Lyapunov drift condition is proved. 



    Case (ii)
 

Compared to Case (i) we just change the inequality for  the jump part. Using Lemma <ref> we have

    𝒥 u(x) 
       =∫_^d(A^⊤η A(C+x)^1/2 yy^⊤(C+x)^1/2)  ν_L(dy)≤λ_max(A^⊤η A)   ((C+x) ∫_^d yy^⊤ν_L(dy))

and hence 

    𝒜 u(x)    = ((η B+B^⊤η+ λ_max(A^⊤η A)∫_^d yy^⊤ν_L(dy))x)+  λ_max(A^⊤η A) (C∫_^d yy^⊤ν_L(dy)).

Now we can proceed as before.



    Cases (iii) and (iv)
 

We have for the drift part

    𝒟 u(x)     =p((η B+B^⊤η)x)(η x)^p-1≤ K_η,B p (η x)^p.

For the jump part we get using thatyy^⊤_2=y_2^2fory∈^d𝒥 u(x)    = ∫_^d( (η(x + A(C+x)^1/2 yy^⊤(C+x)^1/2A^⊤))^p - (η x)^p )  ν_L(dy) 
       ≤∫_^d(( (η x) + y^2_2( η A(C+x)A^⊤))^p - (η x)^p )ν_L(dy)
       ≤∫_^d(( (η x) + y^2_2(η ACA^⊤)+y^2_2K_η,A(η x))^p - (η x)^p )ν_L(dy).

Using the elementary inequality(x+y)^p ≤max{2^p-1,1} (x^p + y^p)for allx,y≥0we obtain

    𝒥 u(x) 
    ≤   (η x) ^p∫_^d(max{2^p-1,1}( 1+K_η,Ay^2_2)^p - 1)ν_L(dy)
       +max{2^p-1,1}(η ACA^⊤)^p∫_^dy^2_2ν_L(dy).

Putting everything together using (<ref>) or (<ref>), respectively, we have that there existc,d>0such that

    u(x) ≤ - c (η x) ^p+ d.

and we can proceed as in Case (i).


    Case (v)
 

We again only argue differently in the jump part compared to the Cases (iii) and (iv) and can assumep>1due to Remark <ref> (iii). For the jump part we again have

    𝒥 u(x) 
    ≤∫_^d(( (η x) + y^2_2(η ACA^⊤)+y^2_2K_η,A(η x))^p - (η x)^p )ν_L(dy).

Next we use the following immediate consequence of the mean value theorem and the already used elementary inequality forp-th powers of sums.

	For x,y ≥ 0, p ≥ 1 it holds that 
	
    0≤ (x+y)^p - x^p ≤ p y (x+y)^p-1.

This gives (using LandauO(·)notation)

    𝒥 u(x) 
    ≤   ∫_^d(( (η x) + y^2_2(η ACA^⊤)+y^2_2K_η,A(η x))^p - (η x)^p )ν_L(dy)
    ≤    p ((η ACA^⊤)+K_η,A(η x))
       ×∫_^dy^2_2( (η x) + y^2_2(η ACA^⊤)+y^2_2K_η,A(η x))^p-1ν_L(dy)
    
    =    pK_η,A(η x)∫_^dy^2_2( (η x) + y^2_2(η ACA^⊤)+y^2_2K_η,A(η x))^p-1ν_L(dy)
       +O((η x)^p-1)
    ≤    pmax{2^p-2,1}K_η,A(η x)^p∫_^dy^2_2(1 +y^2_2K_η,A)^p-1ν_L(dy)+O((η x)^max{p-1,1}).












Combining this with the estimate on the drift part already given for Cases (iii) and (iv) and using  (<ref>), we have that there existc,d>0such that

    u(x) ≤ - c (η x) ^p+ d.

and we can proceed as in Case (i).


  
Test function belongs to the domain 

We now show that the chosen test functionu(x) = (ηx)^p +1belongs to the domain of the extended generator and thatuindeed has the claimed form. So we have to show that for all initial valuesY_0=xand allt ≥0it holds that

    _x(u(Y_t)) = u(x) + _x( ∫_0^t u(Y_s)ds)

and

    _x( ∫_0^t| u(Y_s)|ds) < ∞.

To show (<ref>) we show thatM_t := u(Y_t) - u(x) - ∫_0^t  u(Y_s-) dsis a martingale. For this we apply Itô's formula for processes of finite variation tou(x) = (ηx)^p +1.  For a moment we ignore the discontinuity of∇uin0forp ∈(0,1)by consideringuon∖{0}. 

    u(Y_t) - u(Y_0) =    ∫_0+^t (∇ u(Y_s-) d Y^c_s) + ∑_0 < s ≤ t(u(Y_s) - u (Y_s-) )
    
    =    ∫_0+^t p Y_s-^p-1((B Y_s-+Y_s-  B^⊤) η) ds 
         + ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )μ_L(dy,ds).

Above we implicitly assumed that∑_0 < s ≤t (u(Y_s) - u (Y_s-) )exists. Noting that(ηY_s )≥(ηY_s- )we easily see from the inequalities obtained for the jump part in the proof of the Foster-Lyapunov drift condition above  that this is the case wheneverLis of finite2p-variation which is ensured by the assumptions of the theorem.

Similarly we see that  
    ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )ν_L(dy) ds
  is finite.
Then we get 

    M_t    = u(Y_t) - u(x) - ∫_0^t u(Y_s-)ds 
       = ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )(μ_L(dy,ds)-ν_L(dy)ds).
 
By the compensation formula (see <cit.> for the version for conditional expectations),(M_t)_t ≥0is a martingale if 
    [ ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )ν_L (dy) ds ] < ∞,
 as the integrand is non-negative.
Just like in the deduction of the Foster-Lyapunov drift condition, we get∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )≤ c Y_s-^p + d,for some constantsc, d >0. By Lemma <ref> it holds that(Y_t^p) < ∞for allt ≥0and t ↦(Y_t^p)is locally bounded. Thus (<ref>) follows by Fubini's theorem.

It remains to prove the validity of Itô's formula on the whole state spaceand forp∈(0,1). For this note that ifY_0 ≠0it follows thatY_t ≠0for allt >0. Thus we only need to consider the caseY_0 = 0. If the jumps are of compound Poisson type we defineτ:= inf{ t ≥0: Y_t ≠0 }, which is the first jump time of the driving Lévy process. Fort < τ,Y_t = 0andt= τit is obviously fulfilled, because Lemma <ref> implies thatC (ηY_s-)≥|((B Y_s-+Y_s-  B^⊤) η) |≥c (ηY_s-)for somec,C>0. Fort > τwe can reduce it to the case∖{0}. Thus Itô's formula stays valid if the driving Lévy process is a compound Poisson process. 

Now we assume that the driving Lévy process has infinite activity. In  <cit.> it has been shown, that we can approximateY_tby approximating the driving Lévy process. We use the same notation as in <cit.>. We fixω∈Ωand someT >0. The proof of <cit.> shows that thenY_n,tfor alln∈ℕandY_n,tare uniformly bounded on[0,T]. It follows that·^pis uniformly continuous on the space of values. We need to show that 

    u(Y_n,t) - u(Y_0) 
    =    ∫_0+^t p Y_n,s-^p-1((B Y_n,s-+Y_n,s-  B^⊤) η) ds 
         + ∫_0+^t ∫_^d(u(Y_n,s- +  A(C+Y_n,s-)^1/2 yy^⊤(C+Y_n,s-)^1/2A^⊤) - u(Y_n,s-) )μ_L_n(dy,ds).

converges uniformly on[0,T]. Sinceω∈ΩandT >0was arbitrary, once established this shows that Itô's formula holds almost surely uniformly on compacts. For the first summand uniform  convergence follows directly by the uniform convergence ofY_n,tand the uniform continuity of the functions applied to it. 

 For the second summand we observe that also the jumps ofLare necessarily bounded on[0,T]and that∫_0+^t ∫_^dy_2^2pμ_L_n(dy,ds) ≤∫_0+^t ∫_^dy_2^2pμ_L(dy,ds).We have that 
 
    |∫_0+^t ∫_^d(u(Y_n,s- +  A(C+Y_n,s-)^1/2 yy^⊤(C+Y_n,s-)^1/2A^⊤) - u(Y_n,s-) )μ_L_n(dy,ds) .
           .-∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )μ_L(dy,ds) |
       ≤∫_0+^T ∫_^d|(u(Y_n,s- +  A(C+Y_n,s-)^1/2 yy^⊤(C+Y_n,s-)^1/2A^⊤) - u(Y_n,s-) ).
           .- (u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) ) |μ_L(dy,ds)
           +∫_0+^T ∫_^d|u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) |(μ_L(dy,ds) -μ_L_n(dy,ds)).

 Now the first integral converges to zero again by uniform continuity arguments for the integrand and the second one due to uniform boundedness of the integrand and the uniform convergence ofL_ntoL.
 
It remains to show (<ref>). 
To show this we first deduce a bound foru(x). Using the triangle inequality we splitu(x)again in the drift part and the jump part. For the absolute value of the jump part we can use the upper bounds from the Foster-Lyapunov drift condition proof since the jumps are non-negative. The absolute value of the drift part is bounded as follows

    𝒟 u(x)   ≤ pC Y_n,s^p.

Adding both parts together we get

    (u(x)) ≤  c_2 u(x)

for some constantc_2 >0.

With that (<ref>) follows by Lemma <ref>.


  
Harris recurrence and finiteness of moments 

To show the positive Harris recurrence of the volatility processYand the finiteness of thep-moments of the stationary distribution we use the skeleton chains. In <cit.> it is shown, that the Foster-Lyapunov condition for the extended generator, as we have shown, implies a Foster-Lyapunov drift condition for the skeleton chains. Further observe that petite sets are small, since by the assumption of irreducibility we can use the same arguments as in the upcoming proof of Theorem <ref>.  With that we can apply <cit.> and get the positive Harris recurrence for every skeleton chain and the finiteness of thep-moments of the stationary distribution. By definition the positive Harris recurrence for every skeleton chain implies it also for the volatility processY.§.§ Proof of Lemma <ref>((Y_0)^p) < ∞implies((ηY_0)^p) < ∞for allη∈_d^++.
 
 We apply Itô's formula tou(Y_t) = Y_t^p. The validity of Itô's formula was shown above in the section  “Test function belongs to the domain”. We fix someT >0. Lett ∈[0,T]. As in the proof before we get with Itô's formula 
 
    u(Y_t) - u(Y_0) 
     =    ∫_0+^t p Y_s-^p-1((B Y_s-+Y_s-  B^⊤) η) ds 
         + ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )μ_L(dy,ds)
      
    ≤   ∫_0+^t c_1 Y_s-^p ds  
       + ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )μ_L(dy,ds)  .

 for somec_1>0.
 So we have 
 
    (Y_t^p) 
     ≤   (Y_0^p) + ∫_0^t c_1 (Y_s^p) ds 
       + ( ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )μ_L(dy,ds) ).

 Using the compensation formula and the bounds of the proof of the Foster-Lyapunov drift condition we get that there existc_2,d>0such that 
 
    ( ∫_0+^t ∫_^d(u(Y_s- +  A(C+Y_s-)^1/2 yy^⊤(C+Y_s-)^1/2A^⊤) - u(Y_s-) )μ_L(dy,ds))
           ≤∫_0^t ( c_2 Y_s^p + d) ds .

 Combined we have 
 
    (Y_t^p)    ≤(Y_0^p) + ∫_0^t (c_1+c_2) (Y_s^p) ds  + d T.

 Applying Gronwall's inequality this shows that(Y_t^p) ≤ ( (Y_0^p) + d T) e^(c_1+c_2) t.SinceT >0was arbitrary(Y_t^p)is finite for allt ≥0andt ↦(Y_t^p)is locally bounded. 




 §.§ Proofs of Section <ref>


  §.§.§ Proof of Theorem <ref>


Letνbe the Lévy measure ofL. We haveν= ν_ac + ν̃, whereν_acis the absolute continuous component andν() < ∞. Moreover we can splitLinto the corresponding processesL𝒟= L_ac+L̃, whereL_acandL̃are independent.  We setB_T={ω∈Ω  |   L̃_t = 0  ∀t ∈[0,T]}. Then∀T >0   (B_T)>0and for any event Ait holds that(A ∩B_T) >0 ⇔(A|B_T) >0and(A∩B_T) >0 ⇒(A) >0. 
So in the following we assume w.l.o.g.L̃ = 0as otherwise the below arguments and the independence ofL_acandL̃imply that(Y_t ∈A | Y_0=x)>0results from(Y_t ∈A  |  Y_0 =x, B_t) >0.



  
*Irreducibility: 

By Remark <ref>, to prove the irreducibility of 
the MUCOGARCH volatility process it is enough to 
show it for a skeleton chain.  

Letδ>0and sett_n := nδ, ∀n∈_0.  We consider the skeleton chain 

    Y_t_n =    e^Bt_n  Y_t_0 e^B^⊤ t_n + ∫_0^t_n e^B(t_n-s) A 
    (C+Y_s-)^1/2 d[L,L]^d_s (C+Y_s-)^1/2 A^⊤ e^B^⊤ 
    (t_n-s).


To show irreducibility w.r.t.λ_we have to show that for any𝒜 ∈(_d^+)withλ__d^+(𝒜) >0and anyy_0 ∈there exists anlsuch that 

    (Y_t_l∈𝒜 | Y_0=y_0) >0.

With

    (Y_t_l∈𝒜 | Y_t_0=y_0) 
       ≥(Y_t_l∈𝒜,  exactly one jump in every 
    time interval (t_0,t_1], ⋯, (t_l-1,t_l] | Y_t_0=y_0) 
       =  (Y_t_l∈𝒜 | Y_t_0=y_0, exactly one 
    jump in every time interval (t_0,t_1], ⋯, (t_l-1,t_l] ) 
       ·( exactly one jump in every time interval 
    (t_0,t_1], ⋯, (t_l-1,t_l]),

and the fact that the last factor is strictly positive we can w.l.o.g. assume, that we have exactly one jump in every time interval(t_k-1, t_k]  ∀k=1, ⋯, l.

We denote byτ_kthe jump time of our Lévy process in(t_k-1, t_k]. With the assumption, that we only have one jump on every time interval, the skeleton chain can be represented by the sum of the jumpsX_i, whereL_t=∑_i=1^N_t X_iis the used representation for the compound Poisson ProcessL. We fix the number of time stepsl≥dand get:

    Y_t_l=     e^Bt_l Y_0 e^B^⊤ t_l
       + ∑_i=1^l e^B(t_l- τ_i) A (C+e^B(τ_i-t_i-1)Y_t_i-1e^B^⊤(τ_i-t_i-1))^1/2 X_iX_i^⊤ (C+e^B(τ_i-t_i-1)Y_t_i-1e^B^⊤(τ_i-t_i-1))^1/2 A^⊤ 
    e^B^⊤(t_l-τ_i).


First we show that the sum of jumps in (<ref>) has a positive density on_d^+. 
Note that every single jump is of rank one. 




We define 

    Z_i^(l) := e^B(t_l- τ_i) A (C+e^B(τ_i-t_i-1)Y_t_i-1e^B^⊤(τ_i-t_i-1))^1/2 X_i

  
and with (<ref>) we have 


    Z_i^(l) = e^B(t_l- τ_i) A ( C    + e^Bt_i-1 Y_0 e^B^⊤ t_i-1 + ∑_j=1^i-1 e^B(t_i-1-t_l) Z_j^(l)Z_j^(l)^⊤ e^B^⊤ (t_i-1-t_l))^1/2 X_i.
 

By assumptionX_1, X_2, ⋯are  are iid and absolutely continuous w.r.t. Lebesgue measure on^dwith a (Lebesgue a.e.) strictly positive density. We see 
immediately that
 Z_1^(l) | Y_0, τ_1
is absolutely continuous with a strictly positive densityf_Z_1^(l) | Y_0, τ_1. Iteratively we get that 
every

    Z_i^(l) | Y_0, Z_1^(l), ⋯, Z_i-1^(l), τ_i

is absolutely continuous with a strictly positive densityf_Z_i^(l) | Y_0, Z_1^(l), …, Z_i-1^(l), τ_i , 
for alli=2, …, l.  

We denote withf_Z^(l)|Y_0, τ_1, …, τ_lthe density ofZ^(l)=(Z_1^(l), ⋯, Z_l^(l))^⊤givenY_0, τ_1, …, τ_l. Note that givenZ_j^(l), j<i,Z_i^(l)is independent ofτ_j. By the rules for conditional densities we get

    f_Z^(l)|Y_0, τ_1, …, τ_l = f_Z_1^(l)| Y_0,τ_1· f_Z_2^(l) | Y_0, Z_1^(l), τ_2⋯ f_Z_l^(l)| Y_0, Z_1^(l), ⋯, Z_l-1^(l), τ_l

is strictly positive on^dl. Thus an equivalent measure,∼, exists such thatZ_1^(l)|Y_0,τ_1,…, τ_l,   ⋯ , Z_l^(l)|Y_0, τ_1,…. τ_lare iid normally distributed. In <cit.> it is shown, that forl ≥dΓ:= ∑_i=1^l Z_i^(l)|_Y_0, τ_1,…. τ_l·Z_i^(l)^⊤ |_Y_0, τ_1,…. τ_l

has a strictly positive density underw.r.t. the Lebesgue measure 
on_d^+. But sinceandare equivalent,Γhas also a 
strictly positive density underw.r.t. Lebesgue measure 
on_d^+. 

This yields 

    (Y_t_l∈𝒜 | Y_0=y_0) =∫__+^l( e^Bt_lY_0e^B^⊤ t_l + Γ∈𝒜|Y_0=y_0, τ_1=k_1, …, τ_l=k_l) d_(τ_1,…, τ_l)(k_1,…, k_l) >0

if( e^Bt_lY_0e^B^⊤t_l + Γ∈𝒜 | Y_0=y_0, τ_1=k_1, …, τ_l=k_l)>0.Here we use that the joint distribution of the jump times_(τ_1,…, τ_l)is not trivial, sinceτ_1, …, τ_kare the jump times of a compound Poisson Process. 
Above we have shown that( e^Bt_lY_0e^B^⊤t_l + Γ∈𝒜 | Y_0=y_0, τ_1=k_1, …, τ_l=k_l)>0if 
    λ__d^+(𝒜∩{x ∈_d^+ | x ≽ e^B t_lY_0 e^B^⊤ t_l}) >0.
 

As we assumedσ(B) ⊂(-∞,0) +i,e^BtY_0e^B^⊤t →0fort →∞. Thus we can chooselbig enough such thatλ__d^+ (𝒜 ∩{x ∈_d^+ | x ≽e^B t_lY_0 e^B^⊤t_l } ) >0for any𝒜 ∈_d^+withλ__d^+(𝒜) >0(noteλ__d^+(∂_d^+) =0). This shows the claimed irreducibility  and asδwas arbitrary even simultaneous irreducibility. 



  
*Aperiodicity: 

The 
simultaneous irreducibility and Proposition 
<ref> show that every skeleton chain is aperiodic. Using Proposition <ref> we know for every skeleton chain, that every compact set is also small. 


We define the set 
    𝒞 := { x ∈^d_+ |  x_2 ≤ K},

with a constantK>0. Obviously𝒞is a compact set and thus a small set for every skeleton chain. By Remark <ref> it is also small for the continuous time Markov Process(Y_t)_t≥0.
To show aperiodicity for(Y_t)_t≥0in the sense of Definition <ref> we prove that 
there exists aT>0such that

    ^t(x,𝒞) >0

holds for allx∈𝒞and allt≥T. 

Using 

    ^t(x,𝒞) ≥^t(x, 𝒞∩{ no jump up to time 
    t})

we considerY_tunder the condition “no jump up to timet”. WithY_0 =x ∈𝒞we have

    Y_t = e^Btxe^B^⊤ t.

Sinceλ= max( (σ(B))) <0there existsδ>0,  andC 
≥1such that  e^Bt _2 ≤C e^- δtand hence we have
        e^Bt x e^B^⊤t _2 ≤C e^-2δt x_2≤C e^-2δt K  ≤K
for allt ≥ln(C)/2δ. Hence,Y_t ∈𝒞for allt ≥ln(C)/2δand thus (<ref>) holds.§.§.§ Proof of Corollary <ref>

The proof is similar to that of Theorem <ref> with the difference that we now assume for the jump sizesX_ithat they have a density, which is strictly positive in a neighborhood of zero, e.g.∃k>0such that 
everyX_ihas a strictly positive density on{ x ∈^d:  x  ≤k}. 
We use the same notation as in the previous proof, but we omit the superscripts^(l). 
By the definition ofZ_iand the same iteration as in the first case we show, thatZ:=(Z̃_1, …, Z̃_l)|Y_0,τ_1, …, τ_lhas a strictly positive density on a suitable neighborhood of the origin.

ForA∈M_d()we definej(A):= min_x ∈^d Ax_2/x_2as the modulus of injectivity, which has the following properties:0 ≤j(A) ≤ A_2and  Ax_2 ≥j(A) x_2as well as forA,B ∈M_d()j(A  B) ≥j(A)j(B). 

With that we get forZ_1Z_1 _2    =  e^B(t_l-τ_1) A (C+e^Bτ_1Y_0e^B^⊤τ_1)^1/2 X_1_2  ≥ j(e^B(t_l-τ_1) A (C+e^Bτ_1Y_0e^B^⊤τ_1)^1/2)  X_1_2 
       ≥ j(e^B(t_l-τ_1))  j( A)  j( (C+e^Bτ_1Y_0e^B^⊤τ_1)^1/2)  X_1_2  ≥ j(e^Bt_l)  j( A)  j( C^1/2)  X_1_2

and thusZ_1|Y_0,τ_1has a strictly positive density on{x∈^d: x ≤k̃}, wherek̃ := j(e^Bt_l)  j( A)  j( C^1/2 )   k. 
Iteratively get that everyZ_i|Y_0, τ_i, Z_1, …, Z_i-1has a strictly positive density on{x∈^d: x ≤k̃}and as in the first case this shows thatZ=( Z_1, …,  Z_l)|Y_0,τ_1, …,τ_lhas a strictly positive density on{ x=(x_1,…,x_l)^⊤∈^d ·l:  x_i  ≤k̃  ∀i=1,…,l }.We fix ank̂,0<k̂ < k̃and set𝒦̂:= {x =(x_1,…,x_l)^⊤∈^d ·l:x_i ≤k̂   ∀i=1, …,l }. Now we can construct random variablesZ̃_i,i=1, …, l, such thatZ̃:=(Z̃_1, …, Z̃_l)|Y_0,τ_1, …, τ_lhas a strictly 
positive 
density on^d ·land 

    _𝒦̂·Z̃ |Y_0,τ_1, …, τ_l  𝒟=_𝒦̂·  Z |Y_0,τ_1, …, τ_l .


Due to the first case we now can choose a measuresuch that theZ̃_i|_Y_0, τ_1, …, τ_l, Z_1, …, Z_i-1are iid normal distributed and the random variableΓ̃:= ∑_i=1^l Z̃_i|_Y_0, τ_1, …, τ_l, Z_1, …, Z_i-1    Z̃_i^⊤|_Y_0, τ_1, …, τ_l, Z_1, …, Z_i-1has a strictly positive density on_d^+. 

With the equivalence ofandalso(Γ̃∈𝒜)>0for every𝒜 ∈ℬ().

Further we defineℰ:={x∈: x=∑_i=1^l z_i z_i^⊤, z_i∈^d, x_2 ≤k̂}and𝒦:= {x∈:   x=∑_i=1^l z_i z_i^⊤ for z_1, …, z_l ∈^d   implies  z_i_2 ≤k̂ ∀i=1,…,l }. Letx= ∑_i=1^l z_i z_i^⊤∈ℰ. Thenx = ∑_i=1^l z_i z_i^⊤≽z_j z_j^⊤for allj=1,…, land thusz_j z_j^⊤_2=z_j_2≤k̂, which means thatx ∈𝒦and therebyℰ ⊆𝒦. 

Now let𝒜 ∈ℬ ()and note that_𝒦 ·Γ̃   𝒟= _ 𝒦 ·Γ. Finally we get 

    (Γ∈𝒜∩ℰ)    = (Γ∈𝒜∩ℰ∩𝒦) = (Γ̃∈𝒜∩ℰ )>0

ifλ__d^+(𝒜 ∩ℰ)>0.  

With the same conditioning argument as in the proof of Theorem <ref> and again using the fact that by assumption there always exists alsuch thatλ__d^+(𝒜 ∩ℰ ∩{x ∈_d^+ | x ≽e^B t_lY_0 e^B^⊤t_l })>0ifλ__d^+(𝒜 ∩ℰ)>0, we get simultaneous irreducibility w.r.t. the measureλ__d^+ ∩ℰdefined byλ__d^+ ∩ℰ(B) := λ__d^+(B ∩ℰ)for allB ∈ℬ (). 

Aperiodicity follows as in the proof of Theorem <ref>, since we only used the compound Poisson structure ofLand not the assumption on the jump distribution.§ ACKNOWLEDGEMENTS

The authors are grateful to the editor and an anonymous referee for their reading and insightful comments which improved the paper considerably.
The second author gratefully acknowledges support by the DFG Graduiertenkolleg 1100.



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