UvA-DARE (Digital Academic Repository) Affine pure-jump processes on positive Hilbert–Schmidt operators

We show the existence of a broad class of affine Markov processes on the cone of positive self-adjoint Hilbert–Schmidt operators. Such processes are well-suited as infinite-dimensional stochastic covariance models. The class of processes we consider is an infinite-dimensional analogue of the affine processes on the cone of positive semi-definite and symmetric matrices studied in Cuchiero et al. (2011). As in the finite-dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well as jumps governed by a jump measure that depends affine linearly on the state. The fact that the cone of positive self-adjoint Hilbert–Schmidt operators has empty interior calls for a new approach to proving existence: instead of using standard localization techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann (2010) and further developed in Cuchiero and Teichmann (2020). Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process.


Introduction
In this article we show the existence of time-homogeneous affine Markov processes on the cone of positive self-adjoint Hilbert-Schmidt operators. The affine class is known for its tractability and flexibility. It is tractable because the Fourier-Laplace transform of such processes depends in an exponentially affine way on the initial state vector of the process. More specifically, denote by (H, ·, · ) the Hilbert space of self-adjoint Hilbert-Schmidt operators on a Hilbert space (H, ·, · H ) and by H + ⊆ H the cone of positive self-adjoint Hilbert-Schmidt operators. A H + -valued time-homogeneous Markov process (X t ) t≥0 is affine, if there exist functions φ : R + × H + → R + , ψ : R + × H + → H + such that E e − Xt,u |X 0 = x = e −φ(t,u)− x,ψ (t,u) , for all u ∈ H + . The functions φ and ψ are typically solutions of ordinary differential equations given in terms of the parameters of the model. The affine class is flexible because the parameters of the model satisfy certain assumptions that allow for desired features such as constant and bounded linear drifts and constant and affine state-dependent jumps of infinite-variation. Our motivation for studying affine processes in the state space H + lies in the fact that such processes are well-qualified as models for infinite dimensional covariance processes, i.e., they can be used for the modeling of stochastic volatility in, for example, bond and commodity markets. See e.g. [17,6,2,3] for the modeling of forward price dynamics in bond and commodity markets as a process with values in a Hilbert space. In particular, in [4] a stochastic volatility model is constructed that involves a covariance process driven by Lévy noise and taking values in the positive Hilbert-Schmidt operators. Our model extends the covariance model in [4] from The authors gratefully acknowledge Christa Cuchiero for fruitful discussions. Moreover, this research is partially funded by The Dutch Research Council (NWO).
Lévy driven processes to processes allowing for state-dependent jumps (see also [7,Section 4.1]). More specifically, the affine processes we consider in this paper are of pure-jump type where the jumps can be state-dependent and of infinite variation. Let us state our main result in an abbreviated form, see also Theorem 2.8 below and its proof: with initial values φ(0, u) = 0 and ψ(0, u) = u for u ∈ H + .
More specifically, the processes we consider have a constant drift vector b, a linear drift term B, a constant jump measure m, and a state-dependent jump measure µ. In addition to Theorem 1.1, and as a by-product of our method of proof, we establish explicit formulas for the first and second moments of the affine processes, see Proposition 4.17.
Note that equation (1.3) is a non-linear differential equation on the cone of positive self-adjoint Hilbert-Schmidt operators which, in general, cannot be solved explicitly. Numerical methods for approximating solutions to infinite-dimensional Riccati equations are considered in e.g. [15] and [34]. A numerical approximation method tailored for this specific equation will be analysed in forthcoming work [22]. There is a vast number of articles dealing with affine processes in several state spaces in finite dimensions, we mention, for example, [8,14,25,24,36,20,9]. In [14] and [9], the authors considered affine processes respectively on the canonical state space R d + × R m , d, m ∈ N, and on the cone of positive semi-definite symmetric matrices. Both articles give sufficient and necessary admissible parameter conditions and characterize the class of stochastically continuous affine processes by means of their Markovian generator. The literature on affine processes in infinite-dimensional state spaces is more sparse. Existence of affine diffusion processes on Hilbert spaces was investigated in [35]. In [19], the author investigated affine processes in general locally convex vector spaces and in [10], existence of affine Markovian lifts of finitedimensional Volterra processes was shown. The Markovian lift process takes values in a certain cone in a space of measures and shares many features of the affine processes which we consider. The biggest challenge we face is that like many infinite-dimensional cones, the cone of positive self-adjoint Hilbert-Schmidt operators has empty interior. One consequence is that one cannot employ classical localisation arguments to establish existence of the desired processes; we take a different approach outlined below. Another consequence is that it is difficult to incorporate a diffusion term. Indeed, although formally this involves a non-commutative version of the superprocesses studied in e.g. [27], the methods in [27] break down in the non-commutative setting. Thus it remains an open question whether and under what conditions infinite-dimensional affine processes on positive Hilbert-Schmidt operators allow for a diffusion term. Our new approach involves approximating the transition semigroup associated with our Markov process by simpler transition semigroups corresponding to affine finiteactivity jump processes. We then exploit the generalized Feller theory introduced in [13] and the approximation results [10, Proposition 3.3 and Theorem 3.2] as well as a version of the Kolmogorov extension theorem proven in [10,Theorem 2.11] to show that the limiting semigroup gives rise to a generalized Feller process. Note that the idea of showing the existence of affine processes with jumps of infinite variation through an approximation with simpler affine processes was already used on e.g. convex sets in finite dimensions, where it is known that affine processes are (classical) Feller processes (see [14] and [9]). However, our approach is somewhat different, and a considerable amount of effort goes into verifying that the approximating generalized Feller semigroups satisfy all necessary conditions to ensure convergence. In particular, a subtle analysis of the regularity of φ and ψ is conducted and we derive a uniform growth bound for the approximating semigroups.
1.1. Layout of the article. In Section 2 we provide the definition of admissible parameter sets and we state our main result (Theorem 2.8) on the existence of affine pure-jump processes on the cone of positive self-adjoint Hilbert-Schmidt operators. Moreover, we specify the exact form of the weak generator of these Markov affine processes on the linear span of the Fourier basis elements in terms of the introduced admissible parameter set. A brief outline of the proof of Theorem 2.8 is presented in Section 2 and the full proof is left to Section 4. In Section 3 we show the existence and uniqueness of the solution to the generalized Riccati equations (1.2) and (1.3) and we study the regularity of this solution with respect to its initial value. We recall the generalized Feller setting in Section 4.1. Then in Section 4.2 and 4.3 we making use of the results in Section 3 and some intricate approximation techniques for generalized Feller semigroups to complete the proof of Theorem 2.8. In Appendices A, B and C, we, respectively, add a comparison theorem that we need in our derivations, collect some 'standard' results on integration with respect to vector-valued measures, and provide a regularity result of the solution to our considered generalized Riccati equations.
1.2. Notation. We set N = {1, 2, . . .} and N 0 = {0, 1, . . .}. For a vector space X and U ⊆ X we denote the linear span of U by lin(U ). For (X, τ ) a topological space and S ⊆ X we let B(S) denote the Borel-σ-algebra generated by the relative topology on S. Let (H, ·, · H ) be a Hilbert space. Then we denote by C(S, H) the space of H-valued functions on S that are continuous with respect to the relative topology and we denote by C b (S, H) the space of bounded H-valued continuous functions on S. This is a Banach space when endowed with the supremum norm · C(S) . Notice that when H = R, we typically omit H in the notation: C(S) := C(S, R). Let L(X) denote the space of bounded linear operators from a Banach space X to X. This is a Banach space when equipped with the operator norm · L(X) . If G is a linear operator on a Banach space X, we denote its domain by dom(G) and denote by I the identity in L(X). We denote unbounded operators by a calligraphic font and bounded ones by the standard font, e.g., G versus G. Let L (2)  where (e n ) n∈N is an orthonormal basis for H and ·, · L2(H) is independent of the choice of the orthonormal basis (see, e.g., [37,Section VI.6]). A nonempty subset K of a vector space is called a wedge if K + K ⊆ K and αK ⊆ K for all α ≥ 0, if moreover K ∩ (−K) = {0} then we call K a cone. A cone K in a vector space X induces a partial ordering: and we say that We say that a cone K is regular if for all y, [21]. In finite dimensions, self-dual normal cones have non-empty interior. However, in infinite dimensions, the property H = K − K does in general not imply that K has non-empty interior, see [26].

Affine processes on H + and statement of main result
In this section we give a detailed definition of affine processes on the state space H + and introduce the notion of admissible parameter sets. We compare our admissible parameter conditions with the matrix valued case, this is done in Remark 2.4. Given an admissible parameter set we deduce first properties of the right-hand side functions of the differential equations in (1.2)-(1.3). At the end of this section we state our main result of this article in Theorem 2.8, which guarantees the existence of affine Markov processes on H + associated with a given admissible parameter set and specifies the form of their weak generator on the Fourier-basis elements. However, we postpone the proof to Section 4.3 and only give a brief outline at the end of this section. We consider a time-homogeneous Markov process X with state space H + and transition semigroup (P t ) t≥0 acting on functions f ∈ C b (H + ), where p t (x, ·), t ≥ 0, x ∈ H + , is the transition kernel of X. Moreover for x ∈ H + , we denote the law of X given X 0 = x by P x .
Definition 2.1. The Markov process (X, (P x ) x∈H + ) is called affine if its Laplace transform has exponential-affine dependence on the initial state, i.e., if for all t ≥ 0, and u, x ∈ H + , for some functions φ : R + × H + → R + and ψ : We follow the approach in [9] and consider the Laplace transform instead of the characteristic function which is justified by the non-negativity of X. Note, that we do not require stochastic continuity of the affine process here, as in this work we are not aiming to provide a characterization of affine processes. As discussed in the introduction, our existence result requires an analysis of the corresponding generalized Riccati equations. In particular, a direct consequence of our approach (see Theorem 2.8 below) is that the processes we consider are regular in the sense of [9, Def. 2.2]. We recall this concept for the reader's convenience: We call the affine process regular, whenever the functions ∂φ(t, u) ∂t | t=0+ and ∂ψ(t, u) ∂t | t=0+ , exist and are continuous at u = 0.
As we will see, the established class of affine processes satisfy an even stronger regularity condition, see Section 3.2. In finite dimensions stochastically continuous affine processes are always regular (see [25]), however, there exist finite-dimensional affine processes that are not stochastically continuous. Arguably, such processes are of minor interest in applications. In infinite dimensions the regularity condition is somewhat more restrictive, as it implies e.g. that the operator B in Definition 2.3 must be bounded. We refer to [22,Section 3] for a construction of an infinitedimensional affine process involving unbounded B.
In order to identify pure-jump affine processes, we introduce an admissible parameter set in the following definition. We think of b as the constant drift vector, B the linear term in the drift, m the constant jump measure, and µ the state-dependent jump measure.
Recall that Appendix B summarizes theory on integration with respect to a Hilbert space valued measure.
for all u, x ∈ H + satisfying u, x = 0 ; iv) an operator B ∈ L(H) with adjoint B * satisfying for all x, u ∈ H + satisfying u, x = 0.
Remark 2.4 (Comparison to the finite-dimensional case). Definition 2.3 above is analogous to the definition of an admissible parameter set for R d + -valued processes see [14,Def. 2.6]) and the case of positive semi-definite and symmetric matrices, see [9,Def. 2.3]. However, as mentioned in the introduction, we do not consider any diffusion terms in this work. A more subtle difference is that we require second moment conditions on the measures m(dξ) and µ(dξ) ξ 2 , whereas no moment conditions are needed in the finite-dimensional setting. These second moment conditions are a consequence of our generalized Feller approach, for which we take the weight function ρ = · 2 + 1. See Remark 4.18 for a detailed discussion regarding the necessity of these moment conditions to our approach.
In what follows we will frequently use the following observation: Given admissible parameters (b, B, m, µ), we define F : H + → R and R : H + → H, respectively, by  Inspired by the finite-dimensional theory, we consider a system of ordinary differential equations associated with the admissible parameter set (b, B, m, µ) as introduced in the equations (1.2)-(1.3). The equations are commonly known as the associated generalized Riccati equations which is due to the typically quadratic growth of F and R and by using the formulas for F and R in (2.4), we write: Definition 2.7. Let u ∈ H + . We say that (φ(·, u), ψ(·, u)) : [0, ∞) → R × H is a solution to (2.7) if (φ(·, u), ψ(·, u)) is continuously differentiable, takes values in R + × H + , and satisfies (2.7).
For a transition semigroup (P t ) t≥0 defined on bounded measurable functions on H + we recall the notion of a weak generator (A, dom(A)) of (P t ) t≥0 (see [32,Definition 9.36] and The following theorem is our main result, it asserts the existence of affine pure-jump processes on the cone of positive self-adjoint Hilbert-Schmidt operators admitting for state-dependent jumps of infinite variation and it specifies the form of the weak generator on a space of functions containing the Fourier basis elements. For the proof see Section 4.3, which relies on Section 3 and Section 4. and for all t ≥ 0 and u, x ∈ H + , where (φ(·, u), ψ(·, u)) is the unique solution to the associated generalized Riccati equations in (2.7). Moreover let (A, dom(A)) be the weak generator of (P t ) t≥0 , then lin e − ·,u : u ∈ H + ⊆ dom(A) and for every f ∈ lin e − ·,u : u ∈ H + we have: Outline of the proof. The proof is based on the approximation procedure that we conduct in detail in Section 4.2, where we work in the realm of generalized Feller semigroups, see the preliminaries given in Section 4.1. Here we limit ourselves to give a brief outline of the proof that shall give a rough guidance for the upcoming sections and condensing the main ideas therein. The detailed proof is then given in Section 4.3. Inspired by [10], we approximate the Kolmogorov type operator A in (2.9) by operators (A (k) ) k∈N corresponding to processes of pure-jump type with finite activity, i.e. for every k ∈ N we replace the constant jump measure m(dξ) in formula (2.9) by 1 {ξ≥1/k} m(dξ) and the linear jump measures µ(dξ) by 1 {ξ≥1/k} µ(dξ). The approximation operators A (k) generate strongly continuous semigroups (P (k) t ) t≥0 on a space of functions, being weakly continuous with subquadratic growth, see Proposition 4.13. Having established the existence of affine processes of pure-jump type associated with the strongly continuous semigroups (P (k) t ) t≥0 , we next apply a Trotter-Kato type result from [10] to obtain the limiting semigroup (P t ) t≥0 , see Proposition 4.16. To this end we first need to establish growth bounds on (P (k) t ) t≥0 , that are uniform in k, see Proposition 4.15. This requires understanding the associated generalized Riccati equations (1.2)-(1.3). We provide global existence and uniqueness results in Section 3. The crucial importance of the associated ODEs is that they substitute for the Kolmogorov equations, hence semigroup theoretic arguments involving the Kolmogorov type operators or the abstract Cauchy problem can be reduced to ODE theoretic arguments. Lastly, we apply a version of Kolmogorov's extension theorem (see Theorem 4.5) to the limiting semigroup (P t ) t≥0 , which then yields the existence of an underlying Markovian process. This process associated via the semigroup to the operator (A, dom(A)) is the desired affine process identified by the admissible parameter set (b, B, m, µ).
The second equation for ψ(·, u) in the generalized Riccati equations (2.7) is a nonlinear differential equation on the cone of positive self-adjoint Hilbert-Schmidt operators. This type of infinite-dimensional differential equations has been of interest in the literature as they also show up e.g. in optimal control problems and stochastic filtering theory [11,18,29]. Hence several articles deal with the problem of numerical tractability of this type of equations. See, e.g. [34] where Galerkin approximation and convergence theory was developed for operator-valued Riccati differential equations formulated in the space of Hilbert-Schmidt operators and [15] where the author studied a backward Euler approximation scheme and convergence results for this type of equations. In a subsequent article [22], we investigate the Galerkin approximation further and draw a connection to matrix-valued affine processes. An example of a stochastic volatility model where the covariance process is an affine Markov process on H + is the infinite-dimensional lift of the BNS model constructed in [4] to model forward rates in commodity markets. In [7, Section 4] we constructed several other examples to model stochastic volatility in this context of forward rates in commodity markets and we showed that our model class allow multiple modeling options for the instantaneous covariance process, including state-dependent jump intensity.

Analysis of the generalized Riccati equations
In this section we investigate the generalized Riccati equations given by (2.7). In Subsection 3.1 we introduce Lipschitz continuous approximations of the mappings R and F in (2.4) and use these approximations to show existence and uniqueness of a solution to (2.7). In Subsection 3.2 we establish regularity properties of R and F and use this to show that the solution map depends in a differentiable way on its initial value.
3.1. Solving the generalized Riccati equations (2.7). The goal of this subsection is to prove the existence of a unique solution to the generalized Riccati equations given an admissible parameter set (b, B, m, µ). A common approach in the finite-dimensional case, e.g. in the case of the cone of positive semi-definite and symmetric matrices, is to use a localisation argument exploiting the fact that the function R is analytic on the interior of the cone. Note, however, that in general R fails to be Lipschitz continuous on the boundary of the cone. The cone of positive self-adjoint Hilbert-Schmidt operators has an empty interior, a property that is shared by many cones in infinite dimensions. This has the consequence that localisation arguments for solving equations (2.7) on the interior of R + × H + are not valid anymore. Instead, for every k ∈ N we introduce approximations F (k) of F in equation (3.2) and R (k) of R in equation (3.3), which involve only finite-activity jump-measures, see (3.1) below. These approximations are Lipschitz continuous on H + , and in Proposition 3.7 we show that the solution to the generalized Riccati equations associated with (b, B, m (k) , µ (k) ) converges to the (unique) solution to equation (2.7). We begin by introducing the approximating functions for F and R: for k ∈ N we set and we introduce the functions F (k) : H + → R and R (k) : H + → H defined respectively as follows We denote the generalized Riccati equations associated to (b, B, m (k) , µ (k) ) by: The notion of quasi-monotonicity will be needed to guarantee that the solution to (3.4) stays in R + × H + .
Definition 3.1. Let (V, · V ) be a Hilbert space and let K ⊂ V be a self-dual cone.
In addition, let D ⊆ V and let f : Intuitively, quasi-monotone functions are pointing 'inwards' at the boundary points, which ensures that solutions stay in a cone (see Theorem A.1). For details on quasimonotone functions on Banach spaces and their connection to differential equations see [12,Section 5.3].
The following lemma states that the admissibility of parameters implies that R (k) , k ∈ N, is quasi-monotone with respect to H + . The proof is analogous to the proof of [9, Lemma 5.1], we present an abridged version.

Lemma 3.2. Let B and µ satisfy the admissibility conditions iii) and iv) in Definition 2.3. Then for all
Proof. The admissibility condition iv) in Definition 2.3 (which makes sense thanks to condition iii) in Definition 2.3) and the monotonicity of the exponential function imply the quasi-monotonicity of R (k) .
By removing the small jumps and since m and µ have finite first moment, we obtain Lipschitz continuous mappings on H + : Lemma 3.3. Let B and µ satisfy the admissibility conditions iii) and iv) in Definition 2.3. Let k ∈ N and R (k) given by (3.3). Then for all u, v ∈ H + we have Thus, (B.4) and (B.7) imply that Note that R is typically not Lipschitz continuous on the whole H + : , k ∈ N, be given by equation (3.3). Then for every k ∈ N and u ∈ H + there exists a solution (φ (k) (·, u), ψ (k) (·, u)) to (3.4). Moreover, for all t ≥ 0 and for all t ≥ 0 and u, v ∈ H + .
Proof. Let k ∈ N. By Lemma 3.3 the function R (k) is Lipschitz continuous on H + , by (3.5) with v = 0 the function R (k) satisfies the linear growth condition This and Gronwall's lemma implies the second inequality (3.8).
The next proposition guarantees the existence of a unique solution to the original generalized Riccati equations (2.7) on [0, ∞). First, we prove the following lemma: Lemma 3.6. Let B and µ satisfy the admissibility conditions iii) and iv) in Definition 2.3, let R (k) and R be respectively given by equation (3.3) and (2.7). Then for every M > 0 we have Proof. It follows immediately from (B.7) and (2.3) that The assertion follows from the above and the continuity of µ, see (B.2).
and ψ(t, u) = lim k→∞ ψ (k) (t, u) for all t ≥ 0 and u ∈ H + , as well as and Proof Now fix u ∈ H + . By Proposition 3.5 we know that there exists a unique global solution ψ (k) (·, u) to equation (3.4) for every k ∈ N. This combined with (3.13) implies that for all k ∈ N and t ≥ 0 we have It follows from Lemma 3.3 and Theorem A.

Regularity with respect to the initial value of the solution.
Having established the existence of a unique solution to (2.7), we now turn to the regularity of the solution with respect to the initial value. To this end we first must introduce a fitting concept of differentiability: Definition 3.8. Let X and Y be Banach spaces and D ⊆ X a convex subset. We say that a function f : D ⊆ X → Y has a one-sided derivative at x ∈ D in the direction v ∈ X, whenever x + λv ∈ D for all λ sufficiently small and the limit exists for all λ sufficiently small and moreover the limit exists in Y . We denote the second one-sided derivative of f at Let (b, B, m, µ) be an admissible parameter set conform Definition 2.3 and let F and R be given by (2.4). For u ∈ H + define dR(u) ∈ L(H) by

19)
and dF (u) ∈ L(H, R) by and Then the operator dR(u) is quasi-monotone for all u ∈ H + , and for all u 0 , u 1 ∈ H + and v, w ∈ H we have Proof. The quasi-monotonicity of dR follows directly from the admissibility assumption. As for all u, ξ ∈ H + and all v ∈ H, we obtain (3.23). Estimate (3.24) is obtained similarly, estimate (3.25) is immediate from the definition, and the continuity of u → d 2 R(u)(v, w) follows from the dominated convergence theorem (Theorem B.5). We next confirm the asserted differentiability of the map u → R(u). Let u, v ∈ H + then where the interchange of the integral and the limit in equation (3.30) is justified, since λ → e − u+λv,ξ is a convex mapping, hence its differential quotient is nondecreasing in λ and non-negative and thus we can apply the monotone convergence theorem to obtain that the one-sided derivative of R exists in u in the direction v and (3. 26) holds. An analogous derivation for F leads to equation (3.28).
The proof that the second one-sided directional derivative of both F and R exist and that (3.27)-(3.29) hold is again analogous. Note in particular that for the existence of the second derivatives we use that the measures m(dξ) and µ(dξ) ξ 2 have finite second moments. Proposition 3.11 below states that the solution (φ(·, u), ψ(·, u)) to (2.7) is such that the mappings u → ψ(t, u) and u → φ(t, u) are twice one-sided differentiable in 0 in all directions. The techniques to prove this are well-known, however, as we are dealing with a non-standard concept of differentiability we provide the details of the proof in Appendix C.
Remark 3.10. In fact, one can prove that u → ψ(t, u) and u → φ(t, u) are twice one-sided differentiable in u for every u ∈ H + , in every direction (v, w) ∈ H + × H + . We do not need this, but we do need the existence of the first derivative in u ∈ H + for u sufficiently small in order to obtain the second derivative. See also Appendix C.
For u = 0 we derive explicit formulas for the solutions to the pairs of differential equations in (3.32) and (3.34) of Proposition 3.11, as those will be needed for proving Lemma 4.14 in the approximating case and for Proposition 4.17 below. First, note that Recall the definition of dR(0) from (3.19). The solution of equation (3.32) is then given by 4. Existence of affine pure-jump processes in H + In this section we use the well-posedness and regularity results of the generalized Riccati equations (2.7) from Section 3 to show the existence of an affine process in H + associated to a given admissible parameter set (b, B, m, µ) conform Definition 2.3. Due to the lack of local compactness of the underlying state space, standard Feller theory cannot be employed in our context and we use the theory of generalized Feller processes as introduced in [13]. The existence proof is based on the approximation procedure roughly sketched at the end of Section 2. In this section we rigorously build up this approximation procedure in the generalized Feller setting. Essentially, we approximate the transition semigroup (P t ) t≥0 , that can be associated to an affine process in H + with infinite-activity jump behavior, by simpler transition semigroups corresponding to affine finite-activity jump processes. The considered semigroups are strongly continuous semigroups on a certain Banach space of real functions being weakly-continuous on compact sets and having at most quadratic growth in the tails. We briefly introduce the generalized Feller setting, that is we define generalized Feller semigroups and processes in Section 4.1 and consequently in Section 4.2 we apply approximation results from the theory of strongly continuous semigroups adapted to the generalized Feller setting by [10].

Preliminaries: generalized Feller semigroups.
We recall the concept of generalized Feller semigroups introduced in [13] and further developed in [10]. Throughout this section let (Y, τ ) be a complete regular Hausdorff space.
The following useful characterization of B ρ (Y ) is proven in [13, Theorem 2.7]: We can now present the definition of a generalized Feller semigroup, as introduced in [13, Section 3]. ii) P t+s = P t P s for all t, s ≥ 0, iii) lim

4)
almost surely with respect to P x .
Let (P t ) t≥0 be a generalized Feller semigroup satisfying P t 1 = 1 for all t ≥ 0. The process (X t ) t≥0 , the existence of which is guaranteed by Theorem 4.5, is called a generalized Feller process with initial value x with respect to the measure P x . From now on we write E x for expectations with respect to the probability measure P x .
for t ≥ 0. If moreover (P t ) t≥0 is associated to a Markov process (X t ) t≥0 such that equation (4.3) holds, we obtain: This can be seen by equation (4.5) and a monotone convergence argument by choosing for every n ∈ N the approximations ρ n = n i=1 ·, e i 2 ∧n ∈ B ρ (Y ), where (e i ) i∈N is an ONB of H, then ρ n → ρ in pointwise as n → ∞ and ρ n ≤ ρ n+1 for all n ∈ N.

Approximation of semigroups associated to affine processes in H + .
We equip the Hilbert space H with its weak topology σ(H, H ′ ) (which, by the Riesz representation theorem, is the weak- * -topology). Note that as H + is self-dual, it is closed in (H, σ(H, H ′ )). For brevity of notation we let H + w denote the complete regular Hausdorff space (H + , σ(H, H ′ ) H + ), where σ(H, H ′ ) H + denotes the relative topology σ(H, H ′ ) on H + . In addition, we define ρ : H + → R by ρ(x) := 1 + x 2 , x ∈ H + , (4.6) and observe that ρ is an admissible weight function on H + w by the Banach-Alaoglu theorem, i.e., (H + w , ρ) is a weighted space. Note that for every R > 0, the pre-image {x ∈ H + : ρ(x) ≤ R} is compact in H + equipped with the norm topology, if and only if H is finite-dimensional. As we assume throughout the article that H is infinite-dimensional, we see that ρ is not an admissible weight function in the norm topology.
The linear span of the set of Fourier basis elements e − ·,u : u ∈ H + is denoted by D := lin e − ·,u : u ∈ H + . (4.7) The relevance of this set lies in the following lemma.
Lemma 4.7. The set D is dense in B ρ (H + w ). Proof. It suffices to prove that for every ε > 0 and every f ∈ C b (H + w ) there exists an f ε ∈ D such that f − f ε ρ < ε. To this end, observe that for every ε > 0 and q j e − ·,uj : n ∈ N, q j ∈ Q, u j ∈ U is dense in B ρ (H + w ). Throughout the remainder of this section let (b, B, m, µ) be an admissible parameter set, see Definition 2.3. First, we define for k ∈ N,B (k) ∈ L(H) andb (k) ∈ H + bỹ where m (k) and µ (k) are as defined in (3.1). Note that the fact that B ∈ L(H) and that µ is an H + -valued measure, as well as (B.7) and (3.1) ensure thatB (k) ∈ L(H) is well-defined. Moreover, i) in Definition 2.3 and (2.2) ensure thatb (k) ∈ H + is welldefined. For x ∈ H + and k ∈ N we consider the following deterministic equation in differential form: Standard infinite-dimensional ODE theory ensures that for all x ∈ H + and k ∈ N the unique classical solution to (4.8) is given by The following lemma provides some properties of x (x,k) , x ∈ H + , k ∈ N. For x ∈ H + and k ∈ N let x (x,k) be given by (4.9). Then for all k ∈ N, x ∈ H + , and t ≥ 0.
Proof. It follows immediately from Definition 2.3 iv) that H ∋ x →b (k) +B (k) (x) ∈ H is quasi-monotone with respect to H + . Asb (k) ∈ H + , Theorem A.1 with K = H + , F (·) =b (k) +B (k) (·), f ≡ 0, and g(·) = x Moreover, for all k ∈ N and x ∈ H + we havẽ This implies that for every x ∈ H + , k ∈ N, and t ≥ 0 we have Again applying Theorem A. Then P and Proof. For every t ≥ 0 the operator e tB (k) is strong-to-strong continuous, hence it is also weak-to-weak continuous, and thus P (det,k) t f ∈ C b (H + w ). Next note that Lemma 4.9 implies that for all x ∈ H + . Using the above estimate and (4.11) we obtain Recall that if (A, dom(A)) is the generator of a strongly continuous semigroup S = (S t ) t≥0 on a Banach space X, then a subspace D ⊆ dom(A) is a core for A if D is dense in dom(A) for the graph norm · dom(A) = · X + A · X (see [ ) and Proof. Let k ∈ N. It follows from Lemma 4.10 that (P where we used Lemma C.1 twice, which is applicable as the one-sided derivatives of f , considered as a function on H + , exist. Observe that Moreover we have This, the linearity of G and define the operator G Note that for all k ∈ N the measure jump be as defined in (4.23) and (4.24).
We will prove that g f ∈ B ρ (H + w ) using Theorem 4.3. All other terms in the definition of G To see that g f is continuous on K R := {ρ ≤ R} for all R > 0 it suffices to show that g f is sequentially continuous on K R for every R > 0 as the weak topology restricted to K R is metrizable. Fix R > 0 and let (x n ) n∈N be a sequence in K R converging (weakly) to an x ∈ K R . By the dominated convergence theorem (Theorem B.5) and the fact that sup n∈N x n ≤ √ R we obtain Finally, observe that lim R→∞ sup x∈H + :ρ(x)≥R |ρ(x)| −1 |g f (x)| = 0 as f is bounded recall (B.7)). By Theorem 4.3 this ensures that g f ∈ B ρ (H + w ), which completes the proof of the lemma. In the next proposition we achieve an important intermediate stage, that allows us to conclude the existence of generalized Feller processes in H + admitting for bounded drifts and finite-activity jump behavior, as well as satisfying the exponential affine formula (1.1): Let (b, B, m, µ) be an admissible parameter set conform Definition 2.3. Let k ∈ N, and let (φ (k) (·, u), ψ (k) (·, u)) be the unique solution to (3.4) (cf. Proposition 3.5). Let D ⊆ B ρ (H + w ) be given by (4.7) and G (k) det and G (k) jump be as defined in (4.17), respectively (4.24). Consider the operator G  , and iv). In order to obtain these statements we need to dig into the proof of [10,Proposition 3.3], which makes this proof somewhat technical and tricky. To enhance the readability, we split the proof in to several parts.
Step 1: Verifying the assumptions of [10,Proposition 3.3]. We consider, in the notation of that Proposition, (X,

(4.28) and
(4.29) Next, observe that by Lemma 4.9 and the fact that (0, B, 0, µ) is also an admissible parameter set, we have e tB (k) ξ ∈ H + whenever ξ ∈ H + . Thus for all x, ξ ∈ H + . This together with estimates similar to (4.16) yields (note that This implies that u k is differentiable and u ′ k (t) = v k (t), which implies that f ∈ dom(G (k) ) and Step 4: Proof of iii). In order to verify that P (k) t 1 = 1 for all t ≥ 0, observe that G (k,n) jump 1 = 0 (whence e tG (k,n) jump 1 = 1 for all t ≥ 0), whence the Trotter product formula (see, e.g., [16, Chapter III, Corollary 5.8]) implies that P (k,n) t 1 = 1 for all t ≥ 0. It follows that P (k) t 1 = 1 for all t ≥ 0.
Step 5: Proof of iv). Recall the definition of R (k) and F (k) from (3.2) and (3.3). Recall from Lemmas 4.11 and 4.12 that e − ·,u ∈ D ⊆ dom(G jump ) for all u ∈ H + , and that for all u, x ∈ H + . On the other hand, Proposition 3.5 implies that for all u, x ∈ H + . Therefore for all u ∈ H + it holds that the function [0, ) is a classical solution to the following abstract Cauchy problem: By the uniqueness of the classical solution we conclude (4.26).
From Proposition 4.13 on the existence of the generalized Feller semigroup (P (k) t ) with P (k) t 1 = 1, together with the version of Kolmogorov's extension Theorem 4.5, we conclude that there exists a generalized Feller process associated to (P . Item a) and equation (4.33) in the proof of Proposition 4.13 result in exponential bounds on P (k) t L(Bρ(H + w )) that depend on k ∈ N. In order to proceed, we need to establish bounds that are uniform in k. We begin with a lemma that builds on top of the results in Proposition 3.11: Lemma 4.14. Let (b, B, m, µ) be an admissible parameter set conform Definition 2.3. Moreover for every k ∈ N, let (φ (k) (·, u), ψ (k) (·, u)) be the solution of (3.4), the existence of which is established in Proposition 3.5, and the mappings d + φ(·, 0), d + ψ(·, 0), d 2 + φ (k) (·, 0) and d 2 + ψ (k) (·, 0) be as in Proposition 3.11 for the admissible parameter set (b, B, m (k) , µ (k) ). Moreover, let (X (k) t ) t≥0 be the generalized Feller process associated to (P (k) t ) t≥0 . Then for every v, w ∈ H and t ≥ 0 the following formulas hold true: and (4.37) Proof. Let k ∈ N arbitrary, but fixed. Recall from Remark 4.6 that for all t ≥ 0: We first show that the formulas (4.36) and (4.37) holds for v, w ∈ H + and subsequently extend these to v, w ∈ H. Let u ∈ H + , x ∈ H + and t ≥ 0, then we set and by the affine property of (X By Proposition 3.11 the right-hand side of equation (4.39) is one-sided differentiable in u ∈ H + in the direction v for every v ∈ H + . In particular, by applying the chainrule at u = 0 we have: where d + φ (k) (t, 0) = d + φ(t, 0) and d + ψ (k) (t, 0) = d + ψ(t, 0) for all t ≥ 0 and k ∈ N, see Lemma 3.9. Moreover, note that for θ ∈ R + the random variable e − X (k) t ,θv is integrable and for P x -almost all ω ∈ Ω the mapping θ → e − X (k) is integrable. Hence, all the requirements for switching the derivative with respect to θ and the expectation with respect to P x are fulfilled, thus the left-hand side of equation (4.39) together with equation (4.40) yields: Again due to equation (4.38) we obtain by differentiating both sides of equation (4.39) at u = 0 twice in the direction v and w the formula in (4.37). Note that for  Proof. Recall from Remark 4.6, that in order to show the existence of a M ≥ 1 and w ∈ R + such that equation (4.44) holds, it suffices to show the existence of a ǫ > 0 and C ≥ 0, independent of k ∈ N, such that

By introducing the linear functional
Let k ∈ N be arbitrary, but fixed and denote by (e n ) n∈N an ONB of H, then by Parseval's identity and monotone convergence we have: for every t ≥ 0 and x ∈ H + . By equation (4.37), in particular using the notation in equation (4.43), we have for all n ∈ N: We show separately for the first and second terms on the right-hand side of equation (4.46) that, when summing over all n ∈ N, we find a ǫ > 0 and C ≥ 0 such that equation (4.45) holds. Since we deduce for the second term ion the right hand side of (4.46): for The terms d + φ(t, 0) and d + ψ(t, 0) * L(H) are bounded for all t ≥ 0. Therefore, we deduce the existence of ǫ > 0 and C ≥ 0, independent of k ∈ N, such that for all t ∈ [0, ǫ] and x ∈ H + . We continue with the first term on the right hand side of (4.46). Recall formulas (3.20), (3.22), (3.33), (3.35) and (3.36), from which we obtain:  Hence the two terms on the right hand side of equation (4.46) can be estimated by where we used that for all k ∈ N: Therefore there exist ǫ > 0 andC ≥ 0 such that for all t ∈ [0, ǫ] and x ∈ H + . Taking the sum of the latter constantC and the constant C found in equation (4.47) yields (4.45).
In the next step we show that the family (P t ) t≥0 , defined by P t := lim k→∞ P (k) t for t ≥ 0, gives rise to a generalized Feller semigroup and deduce the existence of a generalized Feller process (X t ) t≥0 with generator G as in formula (2.9).
for all t ≥ 0 and x, u ∈ H + , where (φ(·, u), ψ(·, u)) is the unique solution to the generalized Riccati equation (2.7). The semigroup (P t ) t≥0 gives rise to a generalized Feller process and the generator G of (P t ) t≥0 is of the form in equation (2.9) on D.
Observing that φ (k) (s, u) ∈ R + and ψ (k) (s, u) ∈ H + for all s ≥ 0, we can bound e −φ (k) (s,u)− x,ψ (k) (s,u) by 1 for all x ∈ H + and get from equation (4.52), that for all s > 0: where C u = sup x∈H + ( x + 1)/( x 2 + 1). Thus condition (ii) in Theorem 3.2 in [10] is satisfied with · D = · ∞ and we deduce the existence of a generalized Feller semigroup (P t ) t≥0 with the same growth bound as the semigroup (P (k) t ) t≥0 and such that P t f = lim k→∞ P (k) t f , for all f ∈ B ρ (H + w ), uniformly on compacts in time. Since P t 1 = 1, for all t ≥ 0, we deduce from Theorem 4.5 that there exists a generalized Feller process (X t ) t≥0 such that P t f (x) = E x [f (X t )] for all t ≥ 0 and x ∈ H + . The exponential affine formula (4.50) follows from formula (4.26) and the fact that lim k→∞ φ (k) (t, u) = φ(t, u) and lim k→∞ ψ (k) (t, u) = ψ(t, u) for all t ≥ 0 and u ∈ H + . From this we further derive the particular form of the generator G on the space D by noting that t → P t e − ·,u (x) uniquely solves the abstract Cauchy problem associated to (G, dom(G)) and hence by mimicking the proof of the approximation case in Proposition 4.13, we conclude formula (2.9).
Analogous to the approximating processes (X (k) t ) t≥0 , for k ∈ N in Lemma 4.14, we now deduce explicit formulas for the expressions E x [ X t , v ] as well as for E x X t , v 2 , where x ∈ H + , t ≥ 0 and v ∈ H + .
Moreover, for v ∈ H + , ·, v ∈ dom(G) and Proof. Formulas (4.53) and (4.54) can be obtained analogous to the computation of the formulas (4.36) and (4.37) derived for the approximating case, combined with the explicit formulas (3.35)-(3.36). As in the proof of Lemma 4.14 we use Proposition 3.11 and the finite second moments of the process (X t ) t≥0 to interchange the operations of the expectation and the one-sided derivatives. To obtain more explicit formulas, we consider the analogous of the formulas (4.36) and (4.37) and recall that d can be expressed in terms of dF (0), d 2 F (0), d + ψ(t, 0)(v), and d 2 + ψ(t, 0)(v, w), see (3.31) and (3.33). Then, we recall the expressions (3.35) and (3.36) for d + ψ(t, 0)(v), and d 2 + ψ(t, 0)(v, w). To prove (4.55), observe that using the analogue of (4.36), we get ξ, v m(dξ) The latter together with formulas (3.31) and (3.32), yield and recalling the formulas for dR(0) and dF (0) respectively in (3.19) and (3.20), we conclude that ·, v ∈ dom(G), for v ∈ H + and that (4.55) holds. to ask is whether one could perform the analysis with a different (weaker) weight function. However, in the proof of Lemma 4.11 we consider the square root of the weight function √ ρ, more specifically, we need that ρ(x) ≥ c x , x ∈ H + , for some constant c ∈ (0, ∞). Naturally, the second moments of m and µ are also used to derive the explicit formulas for the first and second moments of the affine process in Proposition 4.17. Finally, we note that the existence of a first moment of µ(dξ) ξ 2 is already used in Lemma 3.3 to ensure that the approximating mappings R (k) are Lipschitz continuous.
In general we do not obtain a version of the process X in Proposition 4. 16  Proof. By [10, Theorem 2.13] it in fact suffices to prove that the generalized Feller semigroup (P t ) t≥0 associated to X is quasi-contractive on Bρ(H + w ), whereρ : H + → [0, ∞) is an admissible weight function such that its associated norm · ρ is equivalent to · ρ . Note that in the finite activity setting we can apply Proposition 4.13 with k = ∞ (with the understanding that m (∞) := m and µ (∞) := µ) to directly obtain (P t ) t≥0 (i.e., no approximation over k is necessary). In particularω ∞ < ∞, whereω ∞ is defined by taking k = ∞ in (4.32). It then follows from statement a) on page 22 that (P t ) t≥0 is quasi-contractive on Bρ ∞ (H + w ) whereρ ∞ is an admissible weight function with associated norm equivalent to · ρ .
In the next section we give the proof of Theorem 2.8. The proof is based on collecting the results from this section and transferring from a generalized Feller setting to the classical setting that we used for presenting the results in Section 2.
4.3. Proof of Theorem 2.8. Let (b, B, m, µ) be an admissible parameter set. Then by Proposition 4.16 there exists a generalized Feller semigroup (P t ) t≥0 and the associated generalized Feller process (X t ) t≥0 in H + such that and the Markov property (4.4) holds. The existence of constants M, ω ∈ [1, ∞) such that (2.8) is satisfied follows from Remark 4.6. The space H is a separable Hilbert space and hence the Borel-σ-algebras B(H + ) and B(H + w ) coincide. This means that the transition kernels (p t (x, dy)) t≥0 defining the semigroup (P t ) t≥0 stay unaffected under the change of topology and hence the process (X t ) t≥0 is also a Markov process in H + with the strong topology. The asserted exponential-affine formula in (2.1) is precisely formula (4.50) from Proposition 4.16. By this and Proposition 3.7 we have for all x ∈ H + : (4.56) In particular, we see that A(D) ⊆ C b (H + ) and since (P t ) t≥0 is a strongly continuous semigroup on B ρ (H + w ) we have P t e − ·,u (x) = e − ·,x (x) + t 0 P s A e − ·,u (x) ds. Consequently, we have shown that D ⊆ dom(A) and from formula (4.56) we see that formula (2.9) holds true on D.

Conclusions and Outlook
With Theorem 2.8 we have proven the existence of affine Markov processes in the cone of positive self-adjoint Hilbert-Schmidt operators by a novel approach inspired by [10]. In particular, our approach relies on the theory of generalized Feller processes, taking the weight function ρ = · 2 + 1. This approach requires the existence of first and second moments of the jump measures m and µ. A beneficial by-product is that we obtain explicit formulas for the first and second moments of the affine Markov process, see Proposition 4.17. See Remark 4.18 for a discussion regarding the necessity of the second-moment condition. Below, we discuss and motivate three further directions of research.
On relaxing the condition on existence of moments. A possible direction of further research is to investigate whether one can adapt the proof in such a way to allow for the weight function ρ = · + 1. In this case a first moment conditions on m and µ should suffice. On a more abstract level, the question arises whether it is possible to establish existence without any moment conditions, as can be done in the finite dimensional setting where the cone of interest does not have empty interior. Another tantalizing question is to what degree an infinite dimensional affine process on the cone of positive self-adjoint Hilbert-Schmidt operators allows for diffusion. It is clear from [5] that certain constructions are possible.
On the construction of stochastic volatility models. Our main motivation for considering affine processes on the space of positive self-adjoint Hilbert-Schmidt operators is that such processes qualify as infinite dimensional stochastic covariance processes. Hence we consider in [7] stochastic volatility models in Hilbert spaces, where the introduced class of affine pure-jump processes will be used for modeling the operator-valued instantaneous variance process. Specifically, we will consider a process (Y t ) t≥0 in a Hilbert space (H, ·, · ) given by where A : dom(A) ⊆ H → H is a possibly unbounded operator with dense domain dom(A), (W t ) t≥0 is a cylindrical Brownian motion in H, Q ∈ H + , and (σ t ) t≥0 is an operator valued stochastic process given by the square-root of an affine pure-jump process, the existence of which is guaranteed by our main result Theorem 2.8.
On considering a different state space for the covariance process. Note that we take σ in (5.1) to be the square root of an affine process in order to obtain that Y is again affine. However, this means that the 'natural' state space for σ is not the cone of positive self-adjoint Hilbert-Schmidt operators, but the cone of positive self-adjoint trace class operators. Unfortunately, this is no longer a cone in a Hilbert space. As self-duality of the cone was used at various instances in the proof of Theorem 2.8, it is not clear how much can be salvaged if we consider trace class operators. This would be a further interesting direction of research.
We leave it to the reader to now verify that also d 2 + φ(t, u)(v, w) exists and that d 2 + φ(t, u)(v, w) satisfies (3.34).