# Markovian lifts of positive semidefinite affine Volterra-type processes

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## Abstract

We consider stochastic partial differential equations appearing as Markovian lifts of matrix-valued (affine) Volterra-type processes from the point of view of the generalized Feller property (see, e.g., Dörsek and Teichmann in A semigroup point of view on splitting schemes for stochastic (partial) differential equations, 2010. arXiv:1011.2651). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein–Uhlenbeck processes whose state space is the set of matrix-valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes-type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston-type model.

## Keywords

Stochastic partial differential equations Affine processes Wishart processes Hawkes processes Stochastic Volterra processes Rough volatility models## Mathematics Subject Classification

60H15 60J25## JEL Classification

C.5 G.1## 1 Introduction

It is the goal of this article to investigate the results of Cuchiero and Teichmann (2018) on infinite dimensional Markovian lifts of stochastic Volterra processes in a multivariate setup: We are mainly interested in the case where the stochastic Volterra processes take values in the cone of positive semidefinite matrices \(\mathbb {S}^d_+\). We shall concentrate on the affine case due to its relevance for tractable *rough covariance modeling*, extending rough volatility (see, e.g., Alòs et al. 2007; Gatheral et al. 2018; Bayer et al. 2016) to a setting of *d* “roughly correlated” assets.

Viewing stochastic Volterra processes from an infinite dimensional perspective allows to dissolve a generic non-Markovianity of the at first sight naturally low-dimensional volatility process. Indeed, this approach makes it actually possible to go beyond the univariate case considered so far and treat the problem of multivariate rough covariance models for more than one asset. Moreover, the considered Markovian lifts allow to apply the full machinery of affine processes. We refer to the introduction of Cuchiero and Teichmann (2018) for an overview of theoretical and practical advantages of Markovian lifts in the context of Volterra-type processes.

*W*a \(d \times d\) the matrix of Brownian motions. The (necessary) presence of the dimension

*d*in the drift is an obvious obstruction to infinite dimensional versions of this equation, which could be projected to obtain Volterra-type equations by the variation of constants formula; see Cuchiero and Teichmann (2018) for such a projection on \(\mathbb {R}_+\). In order to circumvent this difficulty, we present two approaches in this paper:

We develop a theory of infinite dimensional affine Markovian lifts of pure jump positive semidefinite Volterra processes.

We develop a theory of squares of Gaussian processes in a general setting to construct infinite dimensional analogs of Wishart processes. Their finite dimensional projections, however, look different from naively conjectured Volterra Wishart processes following the role model of Volterra Cox–Ingersoll–Ross processes. They are also different in dimension one, as outlined below.

*K*a (potentially fractional) kernel in \(L^2(\mathbb {R}_+, \mathbb {S}_+^d)\) and

*N*a pure jump process of finite variation with jump sizes in \(\mathbb {S}^d_+\), whose compensator is a linear function in

*V*. This allows, for instance, to define a multivariate Hawkes process \(\widehat{N}\)

^{1}with values in \(\mathbb {N}_0^d\) given by the diagonal entries of

*N*, i.e., \({\text {diag}}(N)=\widehat{N}\), and the compensator of \(\widehat{N}_i\) is given by \(\int _0^{\cdot }V_{s,ii} \mathrm{d}s\) (see Example 4.16). By means of the affine transform formula for the infinite dimensional lift of (1.2), we are able to derive an expression for the Laplace transform of \(V_t\) which can be computed by means of matrix Riccati–Volterra equations.

The difficulty of the continuous part arises from geometric constraints, which can, however, be circumvent by building squares of unconstrained processes. Let us illustrate the idea in a finite dimensional setting: Let *W* be an \(n \times d \) matrix of Brownian motions and let \(\nu \) be a matrix in \(\mathbb {R}^{d \times \mathrm{d}k}\) consisting of *k* submatrixes \(\nu _i \in \mathbb {R}^{d \times d}\), \(i=1, \ldots , k\), i.e., \(\nu =(\nu _1, \ldots , \nu _k)\).

*B*satisfying

*h*and

*K*are as in (1.2),

*W*an \(n \times d \) matrix of Brownian motions and \(Y(t,s)= \int _0^{\infty } e^{-x (t-s)}\gamma _s(\mathrm{d}x)\). As explained in Remark 5.4, \(V_t\) corresponds to the matrix square of a Volterra Ornstein–Uhlenbeck process \(X_t\), obtained as finite dimensional projection of \(\gamma (\mathrm{d}x)\). The Volterra Wishart process (1.6) can then also be written in terms of the forward process of \(X_t\), i.e., \((\mathbb {E}[X_t|{\mathcal {F}}_s])_{s\le t}\), namely

*Y*(

*t*,

*s*) or \({\mathbb {E}}[X_t|\mathcal {F}_s]\), respectively, cannot be expressed as a function of \(V_t\). By moving to a Brownian field analogous to (1.4), it could, however, be expressed as a path functional of \((V_s)_{s \le t}\). For \(n=d=1\), it also gives rise to a different equation than the Volterra CIR process. We explain the connection between (1.6) and (1.3)–(1.5) in detail in Sect. 5.

Note that by choosing *K* to be a matrix of fractional kernels, the trajectories of (1.6) become rough, whence *V* qualifies for rough covariance modeling with potentially different roughness regimes for different assets and their covariances. This is in accordance with econometric observations. In Sect. 6, we show how such models can be defined: We introduce a (rough) multivariate Volterra Heston-type model with jumps and show that it can again be cast in the affine framework. This is particularly relevant for pricing basket or spread options using the Fourier pricing approach.

The remainder of the article is organized as follows: In Sect. 1.1, we introduce some notation and review certain functional analytic concepts. In Sects. 2 and 3, we recall and extend results on generalized Feller processes as outlined in Cuchiero and Teichmann (2018). In particular, Theorem 2.8 provides a result on invariant (sub)spaces for generalized Feller processes that is crucial for the square construction as outlined above. In Sect. 4, we apply the presented theory to SPDEs which are lifts of matrix-valued stochastic Volterra jump processes of type (1.2). Section 5 is devoted to present a theory of infinite dimensional Wishart processes which in turn give rise to (rough) Volterra Wishart processes. In Sect. 6, we apply these processes for multivariate (rough) volatility modeling.

### 1.1 Notation and some functional analytic notions

For the background in functional analysis, we refer to the excellent textbook of Schaefer and Wolff (1999) as main reference and to the equally excellent books of Engel and Nagel (2000) and Pazy (1983) for the background in strongly continuous semigroups.

*Y*be a Banach space and \( Y^* \) its dual space, i.e., the space of linear continuous functionals with the strong dual norm

*all*linear functionals \( \langle y , \cdot \rangle \) on \( Y^* \) continuous. Let us recall the following facts:

The weak-\(*\)-topology is metrizable if and only if

*Y*is finite dimensional: This is due to Baire’s category theorem since \(Y^*\) can be written as a countable union of closed sets, whence at least one has to contain an open set, which in turn means that compact neighborhoods exist, i.e., a strictly finite dimensional phenomenon.Norm balls \( K_R \) of any radius

*R*in \( Y^* \) are compact with respect to the weak-\(*\)-topology, which is the Banach–Alaoglu theorem.These balls are metrizable if and only if

*Y*is separable: This is true since*Y*can be isometrically embedded into \( C(K_1) \), where \( y \mapsto \langle y,\cdot \rangle \), for \( y \in Y \). Since*Y*is separable, its embedded image is separable, too, which means—by looking at the algebra generated by*Y*in \( C(K_1) \)—that \( C(K_1) \) is separable, which is the case if and only if \( K_1 \) is metrizable.

*Y*with \( P_t P_s = P_{t+s} \) for \( s,t \ge 0 \) and with \( P_0 =I \) where

*I*denotes the identity is called strongly continuous semigroup if \( \lim _{t \rightarrow 0} P_t y = y \) holds true for every \( y \in Y \). We denote its generator usually by

*A*which is defined as \( \lim _{t \rightarrow 0} \frac{P_t y - y}{t} \) for all \( y \in {\text {dom}}(A) \), i.e., the set of elements where the limit exists. Notice that \( {\text {dom}}(A) \) is left invariant by the semigroup

*P*and that its restriction on the domain equipped with the operator norm

Moreover, as already used in the introduction, \(\mathbb {S}^d\) denotes the vector space of symmetric \(d \times d\) matrices and \(\mathbb {S}^d_+\) the cone of positive semidefinite ones. Furthermore, we denote by \({\text {diag}}(A)\) the vector consisting of the diagonal elements of a matrix *A*.

## 2 Generalized Feller semigroups and processes

In the context of Markovian lifts of stochastic Volterra processes (signed), measure-valued processes appear in a natural way. The generalized Feller framework is taylor-made for such processes, as it allows to consider non-locally compact state spaces, going beyond the standard theory of Feller processes as provided e.g. in Ethier and Kurtz (1986). This is explicitly needed in Sect. 5 for Ornstein-Uhlenbeck processes which take values in the space of matrix-valued measures. Beyond that jump processes with unbounded, but finite activity can be easily constructed in this setting, see Proposition 3.4 and Sect. 4. We shall first collect some results from Cuchiero and Teichmann (2018) and generalize accordingly for the purposes of this article.

### 2.1 Definitions and results

First, we introduce weighted spaces and state a central Riesz–Markov–Kakutani representation result. The underlying space *X* here is a completely regular Hausdorff topological space.

### Definition 2.1

A function \(\varrho :X\rightarrow (0,\infty )\) is called *admissible weight function* if the sets \(K_R:=\left\{ x\in X:\varrho (x)\le R \right\} \) are compact and separable for all \(R>0\).

An admissible weight function \(\varrho \) is necessarily lower semicontinuous and bounded from below by a positive constant. We call the pair *X* together with an admissible weight function \(\varrho \) a *weighted space*. A weighted space is \(\sigma \)-compact. In the following remark, we clarify the question of local compactness of convex subsets \({\mathcal {E}} \subset X\) when *X* is a locally convex topological space and \(\varrho \) convex.

### Remark 2.2

Let *X* be a separable locally convex topological space and \({\mathcal {E}}\) a convex subset. Moreover, let \(\varrho \) be a *convex* admissible weight function. Then, \( \varrho \) is continuous on \( {\mathcal {E}} \) if and only if \(\mathcal {E}\) is locally compact. Indeed, if \( \varrho \) is continuous on \(\mathcal {E}\), then of course, the topology on \( {\mathcal {E}} \) is locally compact since every point has a compact neighborhood of type \( \{ \varrho \le R \} \) for some \( R > 0 \). On the other hand, if the topology on \( {\mathcal {E}} \) is locally compact, then for every point \( \lambda _0 \in {\mathcal {E}} \), there is a convex, compact neighborhood \( V \subset \mathcal {E}\) such that \( \varrho (\lambda )-\varrho (\lambda _0) \) is bounded on *V* by a number \( k > 0 \), whence by convexity \( |\varrho (s(\lambda -\lambda _0)+\lambda _0)-\varrho (\lambda _0)| \le s k \) for \( \lambda - \lambda _0 \in s(V-\lambda _0) \) and \( s \in ]0,1] \). This in turn means that \( \varrho \) is continuous at \( \lambda _0 \).

*Z*be a Banach space with norm \({||\cdot ||}_Z \). The vector space

*Z*-valued functions

*f*equipped with the norm

*Z*-valued bounded continuous functions, the continuous embedding \(\mathrm {C}_b(X;Z)\subset \mathrm {B}^\varrho (X;Z)\) holds true, where we consider the supremum norm on bounded continuous functions, i.e., \(\sup _{x \in X}\Vert f(x)\Vert \).

### Definition 2.3

We define \(\mathcal {B}^{\varrho }(X;Z)\) as the closure of \(\mathrm {C}_b(X;Z)\) in \(\mathrm {B}^{\varrho }(X;Z)\). The normed space \(\mathcal {B}^{\varrho }(X;Z)\) is a Banach space.

If the range space \(Z=\mathbb {R}\), which from now on will be the case, we shall write \( \mathcal {B}^\varrho (X) \) for \(\mathcal {B}^{\varrho }(X; \mathbb {R})\) and analogously \(B^{\varrho }(X)\).

Let us now state the following crucial representation theorem of Riesz type:

### Theorem 2.4

*X*such that

We shall next consider strongly continuous semigroups on \( \mathcal {B}^\varrho (X) \) spaces and recover very similar structures as well known for Feller semigroups on the space of continuous functions vanishing at \( \infty \) on locally compact spaces.

### Definition 2.5

*generalized Feller semigroup*if

- (i)
\(P_0=I\), the identity on \(\mathcal {B}^{\varrho }(X)\),

- (ii)
\(P_{t+s}=P_tP_s\) for all

*t*, \(s\ge 0\), - (iii)
for all \(f \in {\mathcal {B}}^{\varrho }(X)\) and \(x\in X\), \(\lim _{t\rightarrow 0}P_t f(x)=f(x)\),

- (iv)
there exist a constant \(C\in \mathbb {R}\) and \(\varepsilon >0\) such that for all \(t\in [0,\varepsilon ]\), \(||P_t||_{L(\mathcal {B}^{\varrho }(X))}\le C \).

- (v)
\(P_t\) is positive for all \(t\ge 0\), that is, for \(f \in {\mathcal {B}}^{\varrho }(X)\), \(f\ge 0\), we have \(P_t f\ge 0\).

We obtain due to the Riesz representation property the following key theorem:

### Theorem 2.6

One can also establish a positive maximum principle in case that the semigroup \( P_t \) grows around 0 like \( \exp (\omega t) \) for some \(\omega \in \mathbb {R}\) with respect to the operator norm on \( \mathcal {B}^{\varrho }(X) \). Indeed, the following theorem proved in Dörsek and Teichmann (2010, Theorem 3.3) is a reformulation of the Lumer–Phillips theorem for pseudo-contraction semigroups using a *generalized positive maximum principle* which is formulated in the sequel.

### Theorem 2.7

*A*be an operator on \(\mathcal {B}^{\varrho }(X)\) with domain

*D*, and \(\omega \in \mathbb {R}\).

*A*is closable with its closure \(\overline{A}\) generating a generalized Feller semigroup \((P_t)_{t\ge 0}\) with \(||P_t||_{L(\mathcal {B}^{\varrho }(X))}\le \exp (\omega t)\) for all \(t\ge 0\) if and only if

- (i)
*D*is dense, - (ii)
\(A-\omega _0\) has dense image for some \(\omega _0>\omega \), and

- (iii)
*A*satisfies the generalized positive maximum principle, that is, for \(f\in D\) with \((\varrho ^{-1}f)\vee 0\le \varrho (z)^{-1}f(z)\) for some \(z\in X\), \(Af(z)\le \omega f(z)\).

As a new contribution to the general theorems, we shall work out a statement on invariant subspaces which will be crucial for constructing squares of infinite dimensional OU processes.

### Theorem 2.8

*X*be a weighted space with weight \( \varrho _1\) and \( q : X \rightarrow q(X) \) be a (surjective) continuous map from \( (X,\varrho _1) \) to the weighted space \( (q(X),\varrho _2) \). Let \( P^{(1)} \) be a generalized Feller semigroup acting on \( \mathcal {B}^{\varrho _1}(X) \). Assume that \( \varrho _2 \circ q \le \varrho _1 \) on

*X*. Let

*D*be a dense subspace of \(\mathcal {B}^{\varrho _2}(q(X)) \). Furthermore, for every \( f \in D \subset \mathcal {B}^{\varrho _2}(q(X)) \) and for every \( t \ge 0 \), there is some \( g \in \mathcal {B}^{\varrho _2}(q(X)) \) such that

### Proof

*q*defines a linear operator

*M*from \( \mathcal {B}^{\varrho _2}(q(X)) \) to \( \mathcal {B}^{\varrho _1}(X) \) via \( f \mapsto f \circ q \). Notice that

*M*is bounded, since

*M*is continuous, its graph is closed, whence \( P^{(2)}_t \) is a bounded linear operator by the closed graph theorem. Moreover, property (iv) of Definition 2.5 holds true due to Assumption (2.9). Positivity is also preserved, since for \(f \ge 0\), we have due to Assumption (2.8) and the fact that \(P^{(1)}\) is a generalized Feller semigroup,

*g*is nonnegative due the positivity of \(P^{(1)}_t (f \circ q)\). By (2.8) and the definition of \(P^{(2)}\), (2.10) clearly holds true. Hence,

\(\square \)

### Remark 2.9

In the setting of general semigroups, it is not clear that restrictions of semigroups to (not even closed) subspaces preserve strong continuity.

### Remark 2.10

There are several methods to show that (2.8) is satisfied. In general, it is not sufficient to assume that the generator of \( P^{(1)} \) has this property.

### Corollary 2.11

We restate from Cuchiero and Teichmann (2018) assertions on existence of generalized Feller processes and path properties. It is remarkable that in this very general context, càg versions exist for countably many test functions.

### Theorem 2.12

### Theorem 2.13

Let \( (P_t)_{t\ge 0} \) be a generalized Feller semigroup, and let \( (\lambda _t)_{t \ge 0} \) be a generalized Feller process on a filtered probability space. Then, for every countable family \( {(f_n)}_{n \ge 0} \) of functions in \( \mathcal {B}^\varrho (X) \), we can choose a version of the processes \( {\left( \frac{f_n(\lambda _t)}{\varrho (\lambda _t)} \right) }_{t \ge 0} \), such that the trajectories are càglàd for all \( n \ge 0 \). If additionally \( P_t \varrho \le \exp (\omega t) \varrho \) holds true, then \( (\exp (- \omega t) \varrho (\lambda _t))_{t \ge 0} \) is a super-martingale and can be chosen to have càglàd trajectories. In this case, we obtain that the processes \( {\big ( f_n(\lambda _t) \big )}_{t \ge 0} \) can be chosen to have càglàd trajectories.

### Remark 2.14

In the general case, when \( P_t \varrho \le M \exp (\omega t) \varrho \) for \(M >1\), we obtain for \( {\big ( f_n(\lambda _t) \big )}_{t \ge 0} \) only càg trajectories. To see this, consider the measurable set of sample events \( \{ \sup _{0 \le t \le 1} \varrho (\lambda _t) \le R \} \). Then, we can construct on the metrizable compact set \( \{ \varrho \le R \} \) a càglàd version of the processes \( {\left( \frac{f_n(\lambda _t)}{\varrho (\lambda _t)} \right) }_{t \le 1} \) and \( \left( {\frac{1}{\varrho (\lambda _t)}}\right) _{t \le 1} \) and in turn also of \( {\big ( f_n(\lambda _t) \big )}_{t \ge 0}\). The limit \( R \rightarrow \infty \), however, only leads to a càg version since we cannot control the right limits.

### 2.2 Dual spaces of Banach spaces

*Y*where \(Y^{*}\) is equipped with its weak-\(*\)-topology. Consider a lower semicontinuous function \(\varrho :\mathcal {E} \rightarrow (0,\infty )\) and denote by \((\mathcal {E},\varrho )\) the corresponding weighted space. We have the following approximation result (see Döorsek and Teichmann (2010, Theorem 4.2)) for functions in \(\mathcal {B}^{\varrho }(\mathcal {E})\) by cylindrical functions. Set

*Y*. We denote by \({\text {Cyl}}:=\bigcup _{N\in \mathbb {N}}{\text {Cyl}}_N\) the set of bounded smooth continuous cylinder functions on \(\mathcal {E}\).

### Theorem 2.15

The closure of \({\text {Cyl}}\) in \(\mathrm {B}^\varrho (\mathcal {E})\) coincides with \(\mathcal {B}^\varrho (\mathcal {E})\), whose elements appear to be precisely the functions \(f\in \mathcal {B}^{\varrho }(\mathcal {E})\) which satisfy (2.3) and that \(f|_{K_R}\) is weak-\(*\)-continuous for any \(R>0\).

### Proof

See Cuchiero and Teichmann (2018).

\(\square \)

### Assumption 2.16

Let \((\lambda _t)_{t\ge 0}\) denote a time homogeneous Markov process on some stochastic basis \((\Omega ,\mathcal {F}, (\mathcal {F}_t)_{t\ge 0}, \mathbb {P}^{\lambda _0})\) with values in \(\mathcal {E}\).

- (i)there are constants
*C*and \(\varepsilon >0\) such that$$\begin{aligned} \mathbb {E}_{\lambda _0}[\varrho (\lambda _t)]\le C\varrho (\lambda _0) \quad \text {for all } \lambda _0\in \mathcal {E} \text { and } t\in [0,\varepsilon ]; \end{aligned}$$(2.12) - (ii)$$\begin{aligned} \lim _{t\rightarrow 0} \mathbb {E}_{\lambda _0}[f(\lambda _t))] = f(\lambda _0) \quad \text {for any } f\in \mathcal {B}^{\varrho }(\mathcal {E}) \text { and } \lambda _0\in \mathcal {E}; \end{aligned}$$(2.13)
- (iii)
for all

*f*in a dense subset of \( \mathcal {B}^\varrho (\mathcal {E}) \), the map \( \lambda _0 \mapsto \mathbb {E}_{\lambda _0}[f(\lambda _t)] \) lies in \( \mathcal {B}^\varrho (\mathcal {E}) \).

### Remark 2.17

Of course inequality (2.12) implies that \( |\mathbb {E}_{\lambda _0}[f(\lambda _t)]|\le C \varrho (\lambda _0) \) for all \( f \in \mathcal {B}^{\varrho }(\mathcal {E}) \), \( \lambda _0 \in \mathcal {E} \) and \( t \in [0,\varepsilon ]\).

### Theorem 2.18

Suppose Assumptions 2.16 hold true. Then, \(P_t f(\lambda _0):=\mathbb {E}_{\lambda _0}[f(\lambda _t)]\) satisfies the generalized Feller property and is therefore a strongly continuous semigroup on \(\mathcal {B}^\varrho (\mathcal {E})\).

### Proof

This follows from the arguments of Dörsek and Teichmann (2010, Section 5).

\(\square \)

## 3 Approximation theorems

In order to establish existence of Markovian solutions for general generators *A*, we could at least in the pseudo-contractive case either directly apply Theorem 2.7, where we have to assume that the generator *A* satisfies on a dense domain *D* a generalized positive maximum principle and that for at least one \( \omega _0 > \omega \) the range of \( A - \omega _0 \) is dense, or we approximate a general generator *A* by (finite activity pure jump) generators \(A^n \) and apply the following (well known) approximation theorems. They also work in the general context when the constant \(M >1\).

### Theorem 3.1

*Z*with generators \( (A^n)_{n \in \mathbb {N}} \) such that there are uniform (in

*n*) growth bounds \( M \ge 1 \) and \( \omega \in \mathbb {R} \) with

- (i)
*D*is an invariant subspace for all \( P^n \), i.e., for all \( f \in D \), we have \( P^n_t f \in D \), for \( n \ge 0 \) and \( t \ge 0 \). - (ii)There is a norm \( {\Vert .\Vert }_D \) on
*D*such that there are uniform growth bounds with respect to \( {\Vert .\Vert }_D \), i.e., there are \( M_D \ge 1 \) and \( \omega _D \in \mathbb {R} \) withfor \( t \ge 0 \) and for \( n \ge 0 \).$$\begin{aligned} {\Vert P^n_t f \Vert }_D \le M_D \exp (\omega _Dt) {\Vert f\Vert }_D \end{aligned}$$ - (iii)The sequence \( A^n f \) converges as \( n \rightarrow \infty \) for each \( f \in D \), in the following sense: There exists a sequence of numbers \( a_{nm} \rightarrow 0 \) as \( n,m \rightarrow \infty \) such thatholds true for every \( f \in D \) and for all$$\begin{aligned} \Vert A^n f - A^m f \Vert \le a_{nm} {\Vert f \Vert }_D \end{aligned}$$
*n*,*m*.

*Z*such that \( \lim _{n \rightarrow \infty } P^n_t f = P^\infty _t f \) for all \( f \in Z \) uniformly on compacts in time and on bounded sets in

*D*. Furthermore on

*D*, the convergence is of order \( O(a_{nm}) \). If in addition for each \(n \in \mathbb {N}\), \((P_t^n)_{t \ge 0}\) is a generalized Feller semigroup, then this property transfers also to the limiting semigroup.

### Proof

See Cuchiero and Teichmann (2018). \(\square \)

For the purposes of affine processes, a slightly more general version of the approximation theorem is needed, which we state in the sequel:

### Theorem 3.2

*Z*with generators \( (A^n)_{n \in \mathbb {N}} \) such that there are uniform (in

*n*) growth bounds \( M \ge 1 \) and \( \omega \in \mathbb {R} \) with

*subset*with the following two properties:

- (i)
The linear span \({\text {span}}(D)\) is dense.

- (ii)There is a norm \( {\Vert .\Vert }_D \) on \( {\text {span}}(D) \) such that for each \( f \in D \) and for \( t > 0 \), there exists a sequence \( a^{f,t}_{nm} \), possibly depending on
*f*and*t*,holds true for$$\begin{aligned} \Vert A^n P^m_u f - A^m P^m_u f \Vert \le a^{f,t}_{nm} {\Vert f \Vert }_D \end{aligned}$$*n*,*m*and for \( 0 \le u \le t\), with \( a^{f,t}_{nm} \rightarrow 0 \) as \( n,m \rightarrow \infty \).

*Z*such that \( \lim _{n \rightarrow \infty } P^n_t f = P^\infty _t f \) for all \( f \in Z \) uniformly on compacts in time. If in addition for each \(n \in \mathbb {N}\), \((P_t^n)_{t \ge 0}\) is a generalized Feller semigroup, then this property transfers also to the limiting semigroup.

### Proof

See Cuchiero and Teichmann (2018).\(\square \)

Our first application of Theorem 3.1 is the next proposition that extends well-known results on bounded generators toward unbounded limits.

We repeat here a remark from Cuchiero and Teichmann (2018) since it helps to understand the fourth condition on the measures:

### Remark 3.3

### Proposition 3.4

Let \( (X,\varrho ) \) be a weighted space with weight function \( \varrho \ge 1 \). Consider an operator *A* on \(\mathcal {B}^{\varrho }(X)\) with dense domain \({\text {dom}}(A)\) generating on \( \mathcal {B}^\varrho (X) \) a generalized Feller semigroup \( (P_t)_{t\ge 0} \) of transport type as in (3.2), such that for all \(t \ge 0\), we have \( \Vert P_t\Vert _{L(B^{\varrho }(X))} \le M_1 \exp (\omega t)\) for some \(M_1\) and \( \omega \) and such that \( \mathcal {B}^{\sqrt{\varrho }}(X) \subset \mathcal {B}^\varrho (X) \) is left invariant.

*X*such that the operator

*B*acts on \(\mathcal {B}^{\varrho }(X)\) by

- For all \( x \in X \)as well as$$\begin{aligned} \int \varrho (y) \mu (x,\mathrm{d}y) \le M \varrho ^2 (x), \end{aligned}$$(3.3)and$$\begin{aligned} \int \sqrt{\varrho (y)} \mu (x,\mathrm{d}y) \le M \varrho (x), \end{aligned}$$(3.4)hold true for some constant$$\begin{aligned} \int \mu (x,\mathrm{d}y) \le M \sqrt{\varrho (x)}, \end{aligned}$$(3.5)
*M*. - For some constant \( \widetilde{\omega } \in \mathbb {R} \),for all \( x \in X \). In particular, \( y \mapsto \sup _{t \ge 0} \exp (-\omega t) P_t \varrho (y) \) should be integrable with respect to \( \mu (x,.) \)$$\begin{aligned} \int \Big | \frac{\sup _{t \ge 0} \exp (-\omega t) P_t \varrho (y) -\sup _{t \ge 0} \exp (- \omega t ) P_t \varrho (x)}{\sup _{t \ge 0} \exp (-\omega t) P_t \varrho (x)} \Big | \mu (x,\mathrm{d}y) \le \widetilde{\omega } , \end{aligned}$$(3.6)

### Proof

See Cuchiero and Teichmann (2018).\(\square \)

### Remark 3.5

In contrast to classical Feller theory, also processes with unbounded jump intensities can be constructed easily if \( \varrho \) is unbounded on *X*. The general character of the proposition allows to build general processes from simple ones by perturbation.

## 4 Lifting stochastic Volterra jump processes with values in \(\mathbb {S}^d_+ \)

We consider here the setting of Sect. 2.2. The underlying Banach space \(Y^*\) is here the space of finite \(\mathbb {S}^d\)-valued regular Borel measures on the extended half real line \(\overline{\mathbb {R}}_+:=\mathbb {R}_+ \cup \{\infty \}\), and \(\mathcal {E}\) denotes a (positive definite) subset of \(Y^*\). Moreover, \(\mathcal {A}^*\) is the generator of a strongly continuous semigroup \(\mathcal {S}^*\) on \(Y^*\), \(\nu \in Y^*\) (or in a slightly larger space denoted by \(Z^*\) in the sequel). The pre-dual space *Y* is given by \(C_{b}(\overline{\mathbb {R}}_+, \mathbb {S}^d)\) functions. Note that since \(\overline{\mathbb {R}}_+\) is compact, \(Y=C_{b}(\overline{\mathbb {R}}_+, \mathbb {S}^d)\) is separable. The driving process *X* is an \(\mathbb {S}^d\)-valued pure jump Itô-semimartingale, whose differential characteristics depend linearly on \(\lambda \), precisely specified below. Let us remark that other forms of differential characteristics of *X*, in particular beyond the linear case, can be easily incorporated in this setting.

*Y*and \(Y^*\), denoted by \(\langle \cdot , \cdot \rangle \), is specified via:

### Assumption 4.1

- (i)We are given an admissible weight function \( \varrho \) on \( Y^* \) (in the sense of Sect. 2) such thatwhere \(\Vert \cdot \Vert _{Y^*}\) denotes the norm on \(Y^*\), which is the total variation norm of \( \lambda \).$$\begin{aligned} \varrho (\lambda ) = 1+ {\Vert \lambda \Vert }_{Y^*}^2, \quad \lambda \in Y^*, \end{aligned}$$
- (ii)
We are given a closed convex cone \( \mathcal {E} \subset Y^* \) (in the sequel the cone of \(\mathbb {S}^d_+\) valued measures) such that \( (\mathcal {E},\varrho ) \) is a weighted space in the sense of Sect. 2. This will serve as state space of (4.1).

- (iii)
Let \( Z \subset Y \) be a continuously embedded subspace.

- (iv)We assume that a semigroup \( \mathcal {S}^* \) with generator \( \mathcal {A}^* \) acts in a strongly continuous way on \( Y^* \) and \( Z^* \), with respect to the respective norm topologies. Moreover, we suppose that for any matrix \(A \in \mathbb {S}^d\), it holds that$$\begin{aligned} \mathcal {S}^*_t(\lambda (\cdot ) A+ A \lambda (\cdot ))= (\mathcal {S}^*_t\lambda (\cdot )) A+ A (\mathcal {S}^*_t \lambda (\cdot )). \end{aligned}$$(4.3)
- (v)
We assume that \( \lambda \mapsto \mathcal {S}^*_t\lambda \) is weak-\(*\)-continuous on \( Y^* \) and on \( Z^* \) for every \( t \ge 0 \) (considering the weak-\(*\)-topology on both the domain and the image space).

- (vi)
We suppose that the (pre-) adjoint operator of \( \mathcal {A}^* \), denoted by \(\mathcal {A}\) and domain \( {\text {dom}}(\mathcal {A}) \subset Z \subset Y \), generates a strongly continuous semigroup on

*Z*with respect to the respective norm topology (but*not*necessarily on*Y*).

### Remark 4.2

Notice that drift specifications could be more general here, but for the sake or readability, we leave this direction for the interested reader.

For notational convenience, we shall often leave the \(\mathrm{d}x\) argument away when writing an (S)PDE of type (4.4) subsequently. Under the following assumptions on \( \mathcal {S}^* \) and \( \nu \in Z^* \), we can guarantee that (4.4) can be solved on the space \(Y^*\) for all times in the mild sense with respect to the dual norm \(\Vert \cdot \Vert _{Y^*}\) by a standard Picard iteration method.

### Assumption 4.3

- (i)
\( \mathcal {S}^*_t \nu \in Y^* \) for all \( t > 0 \) even though \( \nu \) does not necessarily lie in \( Y^* \) itself, but only in \( Z^* \);

- (ii)
\( \int _0^t \Vert \mathcal {S}^*_s \nu \Vert ^2_{Y^*} \mathrm{d}s < \infty \) for all \( t > 0 \).

### Example 4.4

*K*, are the following specifications:

### Remark 4.5

*Z*: Let \(Z \subset Y\) such that for all \(y \in Y\) the map

*Z*equipped with the operator norm, i.e.,

### Remark 4.6

In this article, we choose to work with state spaces of matrix-valued measures using the representation of the kernel *K* as Laplace transform of a matrix-valued measure \(\nu \) as specified in Example 4.4. We could, however, perform the same analysis on a Hilbert space of forward covariance curves. This corresponds then to a multivariate analogon of Cuchiero and Teichmann (2018, Section 5.2).

### Proposition 4.7

*C*and \( \varepsilon \).

### Remark 4.8

### Proof

*t*on compact intervals. For details on strongly continuous semigroups and mild solutions, see Pazy (1983).

### Assumption 4.9

- (i)
\(\mathcal {S}^*_t (\mathcal {E}) \subseteq \mathcal {E},\quad \forall t \ge 0\);

- (ii)
\(\nu \) is an \(\mathbb {S}^d_+\)-valued measure;

- (iii)
\(\beta (\mathcal {E}) \subseteq \mathbb {S}^d_+\).

### Proposition 4.10

Let Assumptions 4.3 and 4.9 be in force. Then, the solution of (4.4) leaves \( \mathcal {E} \) invariant and it defines a generalized Feller semigroup on \( (\mathcal {E},\varrho ) \) by \( P_t f(\lambda _0) := f(\lambda _t) \) for all \( f \in \mathcal {B}^\varrho (\mathcal {E}) \) and \( t \ge 0 \).

### Proof

Since by Proposition , the solution operator is weak-\(*\)-continuous, we can conclude that \(\lambda _0 \mapsto f(\lambda _t)\) lies in \( \mathcal {B}^\varrho (\mathcal {E}) \) for a dense set of \( \mathcal {B}^\varrho (\mathcal {E}) \) by Theorem 2.15. Moreover, it satisfies the necessary bound (2.12) for \( \varrho \) and (2.13) is satisfied by (norm)-continuity of \(t \mapsto \lambda _t\). Hence, all the conditions of Assumption 2.16 are satisfied and the solution operator therefore defines a generalized Feller semigroup \( (P_t) \) on \( \mathcal {B}^\varrho (\mathcal {E}) \) by Theorem 2.18. This generalized Feller semigroup of course coincides with the previously constructed limit. \(\square \)

### Proposition 4.11

- (i)
Then, for every \( \lambda _0 \in \mathcal {E} \) and \( \varepsilon > 0 \) , the SPDE (4.12) has a solution in \( \mathcal {E} \) given by a generalized Feller process associated with the generator of (4.12).

- (ii)This generalized Feller process is
*also*a probabilistically weak and analytically mild solution of (4.12), i.e.,which justifies Eq. (4.12). In particular for every initial value the process$$\begin{aligned} \lambda _t&= \mathcal {S}^*_t \lambda _0 \mathrm{d}s +\int _0^t \mathcal {S}^*_{t-s}\nu \beta (\lambda _s) \mathrm{d}s + \int _0^t\beta (\lambda _s) \mathcal {S}_{t-s}^*\nu \mathrm{d}s + \\&\quad +\int _0^t\mathcal {S}^*_{t-s+\varepsilon } \nu \mathrm{d}N_s+ \int _0^t \mathrm{d}N_s \mathcal {S}^*_{t-s+\varepsilon } \nu \, , \end{aligned}$$*N*can be constructed on an appropriate probabilistic basis. The stochastic integral is defined in a pathwise way along finite variation paths. Moreover, for every family \((f_n)_n \in \mathcal {B}^{\varrho }(\mathcal {E})\), \(t \mapsto f_n(\lambda _t)\) can be chosen to be càglàd for all*n*. - (iii)For every \( \varepsilon > 0 \), the corresponding Riccati equation \(\partial _t y_t=R(y_t)\) with \(R: \mathcal {D} \cap \mathcal {E}_* \rightarrow Y\) given byadmits a unique global solution in the mild sense for all initial values \( y_0 \in \mathcal {E}_* \).$$\begin{aligned} R(y)= & {} \mathcal {A} y + \beta _*\left( \int _0^{\infty } y(x) \nu (\mathrm{d}x) + \nu (\mathrm{d}x) y(x)\right) \nonumber \\&+ \beta _*\left( \int _{\mathbb {S}^d_+} \left( \exp ( \langle y , \mathcal {S}^*_{\varepsilon } \nu \xi +\xi \mathcal {S}^*_{\varepsilon } \nu \rangle )-1 \right) \mu (\mathrm{d}\xi )\right) , \end{aligned}$$(4.13)
- (iv)The affine transform formula holds true, i.e.,where \(y_t\) solves \(\partial _t y_t=R(y_t)\) for all \(y_0 \in \mathcal {E}_*\) in the mild sense with$$\begin{aligned} \mathbb {E}_{\lambda _0}\left[ \exp ( \langle y_0, \lambda _t \rangle )\right] =\exp (\langle y_t, \lambda _0 \rangle ), \end{aligned}$$
*R*given by (4.13). Moreover, \(y_t \in \mathcal {E}_*\) for all \(t \ge 0\).

### Proof

We assume that \( \nu \ne 0 \), otherwise there is nothing to prove. To prove the first assertion, we apply Proposition 3.4. By Propositions and 4.10, the deterministic equation (4.4) has a mild solution on \(\mathcal {E}\) which—by Assumption 4.3—defines a generalized Feller semigroup \((P_t)_{t\ge 0}\) on \( \mathcal {B}^\varrho (\mathcal {E}) \). The operator *A* in Proposition 3.4 then corresponds to the generator of \((P_t)_{t\ge 0}\), i.e., the semigroup associated with the purely deterministic part of (4.12). This is a transport semigroup, and in view of Remark 3.3, we can have an equivalent norm with respect to a new weight function \( \tilde{\varrho }\) on \(\mathcal {B}^{\varrho }(\mathcal {E})\), such that \( \Vert P_t \Vert _{L(B^{\tilde{\varrho }}(\mathcal {E}))} \le \exp (\omega t) \). Therefore, we find ourselves in the conditions of Proposition 3.4.

Note that by the same arguments as in Proposition 4.10 and by applying Theorem 2.18, we can prove that \((P_t)_{t \ge 0}\) also defines a generalized Feller semigroup on \( \mathcal {B}^{\sqrt{\varrho }}(\mathcal {E}) \). For the detailed proof which translates literally to the present setting, we refer to Cuchiero and Teichmann (2018).

*B*is given by

*y*be as above with the additional property that \( \langle \langle y , \mathcal {S}^*_{\varepsilon }\nu \xi +\xi \mathcal {S}^*_{\varepsilon } \nu \rangle \rangle = \pi \xi + \xi \pi \) for all \(\xi \in \mathbb {S}^d_+\) and some fixed \( \pi \in \mathbb {S}^d_+\). For such

*y*, define

*y*by construction. Indeed, for all \(y_i\) with \( \langle \langle y_i , \mathcal {S}^*_{\varepsilon }\nu \xi + \xi \mathcal {S}^*_{\varepsilon } \nu \rangle \rangle = \pi \xi + \xi \pi \) for all \(\xi \), \(i=1,2\), we clearly have

Moreover, by the definition of \(N^\pi \) in (4.15), its compensator is given by \(\int _0^t \int (\pi \xi + \xi \pi ) {{\,\mathrm{Tr}\,}}(\beta (\lambda _s) \mu (\mathrm{d}\xi ))\mathrm{d}s\). Since it is sufficient to perform the previous construction for finitely many \( \pi \) to obtain all necessary projections, a process *N* can be defined such that \( N^\pi = \pi N + N \pi \), as suggested by the notation.

*A*denotes the generator associated with (4.12). Setting \(u(t,\lambda )=\exp (\langle y_t, \lambda \rangle )\), we have

### Assumption 4.12

### Theorem 4.13

- (i)Then, the stochastic partial differential equation (4.17) admits a unique Markovian solution \((\lambda _t)_{t\ge 0} \) in \( \mathcal {E} \) given by a generalized Feller semigroup on \( \mathcal {B}^\varrho (\mathcal {E}) \) whose generator takes on the set of Fourier elementsfor \(y \in \mathcal {D} \cap \mathcal {E}_*\) where \(\mathcal {D}\) is defined in (4.11) the form$$\begin{aligned} f_y: \mathcal {E} \rightarrow [0,1]; \lambda \mapsto \exp ( \langle y , \lambda \rangle ) \end{aligned}$$with \(\mathcal {R}: \mathbb {S}^d_- \rightarrow Y\) given by$$\begin{aligned} Af_y(\lambda )=f_y(\lambda ) (\langle \mathcal {A}y, \lambda \rangle + \langle \mathcal {R}(\langle \langle y, \nu \rangle \rangle ), \lambda \rangle ), \end{aligned}$$(4.20)$$\begin{aligned} \mathcal {R}(u)= & {} \beta _*(u)+ \beta _*\left( \int _{\mathbb {S}^d_+} \left( \exp ( {{\,\mathrm{Tr}\,}}(u \xi ) -1 \right) \frac{\mu (\mathrm{d}\xi )}{\Vert \xi \Vert \wedge 1} \right) . \end{aligned}$$(4.21)
- (ii)This generalized Feller process is
*also*a probabilistically weak and analytically mild solution of (4.17), i.e.,This justifies Eq. (4.17); in particular, for every initial value, the process$$\begin{aligned} \lambda _t = \mathcal {S}^*_t \lambda _0 \mathrm{d}s +\int _0^t \mathcal {S}^*_{t-s}\nu \mathrm{d}X_s + \int _0^t \mathrm{d}X_s \mathcal {S}_{t-s}^*\nu , \end{aligned}$$*X*can be constructed on an appropriate probabilistic basis. The stochastic integral is defined in a pathwise way along finite variation paths. Moreover, for every family \((f_n)_n \in \mathcal {B}^{\varrho }(\mathcal {E})\), \(t \mapsto f_n(\lambda _t)\) can be chosen to be càg for all*n*. - (iii)The affine transform formula is satisfied, i.e.,where \(y_t\) solves \(\partial _t y_t=R(y_t)\) for all \(y_0 \in \mathcal {E}_*\) and \(t >0\) in the mild sense with \(R: \mathcal {D} \cap \mathcal {E}_* \rightarrow Y\) given bywith \(\mathcal {R}\) defined in (4.21). Furthermore, \(y_t \in \mathcal {E}_*\) for all \(t \ge 0\).$$\begin{aligned} R(y) = \mathcal {A} y + \mathcal {R}( \langle \langle y , \nu \rangle \rangle ) \end{aligned}$$(4.22)
- (iv)For all \(\lambda _0 \in \mathcal {E}\), the corresponding stochastic Volterra equation, \(V_t:= \beta (\lambda _t )\), given byadmits a probabilistically weak solution with càg trajectories. Here, \(h(t):=\beta (\mathcal {S}^*_t \lambda _0)\).$$\begin{aligned} V_t = \beta ( \lambda _t )= & {} \beta (\mathcal {S}_t^* \lambda _0) + \int _0^t\beta (\mathcal {S}_{t-s}^*\nu ) \mathrm{d}X_s + \int _0^t \mathrm{d}X_s \beta (\mathcal {S}_{t-s}^*\nu )\nonumber \\= & {} h(t) + \int _0^t K(t-s)\mathrm{d}X_s +\int _0^t \mathrm{d}X_s K(t-s) \end{aligned}$$(4.23)
- (v)The Laplace transform of the Volterra equation \(V_t\) is given bywhere \(h(t)=\beta (\mathcal {S}_t^*\lambda _0 )\), \(\mathfrak {R}: \mathbb {S}^d_- \rightarrow \mathbb {S}^d_-,\, u \mapsto \mathfrak {R}(u)= u + \int _{\mathbb {S}_+^d} (e^{{{\,\mathrm{Tr}\,}}(u\xi )}-1 ) \frac{\mu (\mathrm{d}\xi )}{\Vert \xi \Vert \wedge 1}\) and \(\psi _t\) solves the matrix Riccati–Volterra equation$$\begin{aligned} \mathbb {E}_{\lambda _0}\left[ \exp \left( {{\,\mathrm{Tr}\,}}(u V_t)\right) \right] =\exp \left( {{\,\mathrm{Tr}\,}}(u h(t))+ \int _0^t {{\,\mathrm{Tr}\,}}(\mathfrak {R}(\psi _s) h(t-s) ) \mathrm{d}s\right) , \end{aligned}$$(4.24)Hence, the solution of the stochastic Volterra equation in (4.23) is unique in law.$$\begin{aligned} \psi _t=u K(t)+\int \mathfrak {R}(\psi _s) K(t-s) \mathrm{d}s, \quad t >0. \end{aligned}$$

### Remark 4.14

One essential point here is that we loose the càglàd property as stated in Proposition 4.11 (ii) when we let \( \varepsilon \) of \(\mathcal {S}_{\varepsilon }\) tend to zero. As long as the kernel *K* has a singularity at \( t = 0 \), it is impossible to preserve finite growth bounds *with*\( M = 1 \), as \( \varepsilon \rightarrow 0 \), but we get càg versions (compare with the second conclusion in Theorem 2.13 and Remark 2.14).

### Remark 4.15

Note that for \(\beta \) as of Example 4.4, the above equations simplify considerably. In particular, \(\beta _*\) in (4.21) is simply the identity.

### Proof

*K*some other constant. Moreover, for the last terms, we have

*T*. We use \(\Vert S_t^*\lambda _0\Vert ^2 \le C_0 \Vert \lambda _0\Vert ^2\) for \(t \in [0,T]\), as well as \( \Vert \mathcal {S}^*_{t-s+\frac{1}{n}} \nu \Vert _{Y^*} \le C \Vert \mathcal {S}^*_{t-s} \nu \Vert _{Y^*}\) for some constant

*C*and all \(n \in \mathbb {N}\) due to strong continuity. Exactly by the same arguments as in the proof of Proposition , we thus obtain for \(t \in [0,T]\) for some fixed

*T*

*D*as of Theorem 3.2, we here choose Fourier basis elements of the form

*y*. Note that

*D*invariant for all \(n \in \mathbb {N}\) by Proposition 4.11 (iv), we have

*u*with \(y_0=y\). Moreover, \(\widetilde{a}^1_{nm}(\xi )\) and \(\widetilde{a}^2_{nm}\) can be chosen uniformly for all \(u \le t\) and tend to 0 as \(n,m \rightarrow \infty \). This is possible since for the chosen initial values

*y*we obtain that \(y^m_u\) is bounded on compact intervals in time uniformly in

*m*(see Cuchiero and Teichmann 2018 for details). This together with dominated convergence for the first term (note that \(b_{nm}(\xi ) \widetilde{a}^1_{nm}(\xi ) \) can be bounded by \(\Vert \xi \Vert \wedge 1\)) we thus infer (4.26). The conditions of Theorem 3.2 are therefore satisfied, and we obtain a generalized Feller semigroup whose generator is given by (4.20).

For the second assertion, we proceed as in the proof of Proposition 4.11, the proof of the existence of *X* can be transferred verbatim. However, one looses the existence of càglàd paths of \(f_n(\lambda )\) due to the possible lack of finite mass of \( \nu \). Here, we only obtain càg trajectories (compare with Remarks 2.14 and 4.14).

*A*denotes the generator (4.20), we infer that \(y_t\) satisfies \(\partial _t y_t=R(y_t)\) with

*R*given by (4.22). This is because \(A\exp (\langle y_t, \lambda \rangle )=\exp (\langle y_t, \lambda \rangle )R(y_t)\).

The fourth claim follows from statement (ii), property (4.5) and the definition of *K* in (4.6).

*h*, we find

*h*is symmetric. This proves the assertion.\(\square \)

The following example illustrates how a multivariate Hawkes process can easily be defined by means of (4.18).

### Example 4.16

*N*jump, and we can define \(\widehat{N}:= {\text {diag}}(N)\) which is a process with values in \(\mathbb {N}_0^d\). Its components jump by one, and the compensator of \(N_{ii}=\widehat{N}_i \) is given by \(\int _0^{\cdot }V_{s,ii} \mathrm{d}s\), which justifies the name multivariate Hawkes process. Note that the components of

*V*are not independent if \(\nu \) and in turn

*K*are not diagonal.

## 5 Squares of matrix-valued Volterra OU processes

*continuous affine Volterra-type processes*on \(\mathbb {S}_+^d\), which we construct as squares of matrix-valued Volterra Ornstein–Uhlenbeck (OU) processes (see Remark 5.4). Following the finite dimensional analogon (Bru 1991), we start by considering matrix measure-valued OU processes of the form

*W*is a \(n\times d\) matrix of Brownian motions and \(\nu \in Y^*=:Y^*(\mathbb {S}^d)\) or \(Z^*\), as defined in Sect. 4 such that Assumption 4.3 holds true. The pre-dual space denoted by \(Y(\mathbb {R}^{n \times d})\) is given by \(C_{b}(\overline{\mathbb {R}}_+, \mathbb {R}^{n \times d})\) functions, where we fix the pairing \(\langle \cdot , \cdot \rangle \) as follows

### Remark 5.1

Observe the analogy to the process \(\gamma \) defined in the introduction. If \(\mathcal {A}^*=0\) and \(\nu \) is supported on a finite space with *k* points, then (5.1) is exactly the process from the introduction.

### Proposition 5.2

### Proof

The construction of the generalized Feller process can be done by jump approximation of the Brownian motion similarly as in Cuchiero and Teichmann (2018, Theorem 4.16). Notice here that we consider the process on the whole space \(Y^*(\mathbb {R}^{n \times d})\). So no issues with state space constraints occur.

\(\square \)

*contracted*, i.e., one matrix multiplication is performed, algebraic tensor product is denoted by \(Y^{*}(\mathbb {R}^{n\times d}) \widehat{\otimes }Y^{*}(\mathbb {R}^{n\times d})\), and we set

*n*, product measures on \(\overline{\mathbb {R}}_+ \times \overline{\mathbb {R}}_+\). We shall introduce a particular dual topology on \( \widehat{\mathcal {E}} \), namely \( \sigma ( \widehat{\mathcal {E}},Y \otimes Y) \), where the corresponding pairing is given by

*F*where

*F*stands here for \(\mathbb {R}^{n\times d}\), or \(\mathbb {S}^d\) with the property that for a constant matrix

*A*with appropriate matrix dimensions, we have

*Volterra Wishart process*in the following definition.

### Definition 5.3

For \(\beta \), \(\widehat{\beta }\) as given in (5.8)–(5.9) and an \(\mathbb {S}^d\)-valued kernel *K*(*t*) defined by \(K(t) =\beta (\mathcal {S}_t^*\nu )\), we call the process defined in (5.10), *Volterra Wishart process*.

### Remark 5.4

- (i)Note that \(\beta (\gamma _t)\) defines an \(\mathbb {R}^{n\times d}\)-valued Volterra OU process, that is,By the definition of \(\widehat{\beta }\), the Volterra Wishart process$$\begin{aligned} X_t:=\beta (\gamma _t)=\beta (\mathcal {S}^*_t \gamma _0)+ \int _0^t \mathrm{d}W_s K(t-s). \end{aligned}$$(5.11)is thus the matrix square of a Volterra OU process, which justifies the terminology.$$\begin{aligned} V_t=\widehat{\beta }(\lambda _t)= \beta (\gamma _t (\cdot ))^{\top } \beta (\gamma _t(\cdot ))=X^{\top }_t X_t \end{aligned}$$
- (ii)Note that different lifts of the Volterra OU process given in (5.11) are possible, e.g., the forward process lift \(f_t(x):=\mathbb {E}[X_{t+x}|\mathcal {F}_t]\). Then, \(f_t(0)=X_t\), and similarly as in Cuchiero and Teichmann (2018, Section 5.2), it can be shown that
*f*is an infinite dimensional OU process that solves the following SPDE (in the mild sense)on a Hilbert space$$\begin{aligned} \mathrm{d}f_t(x)=\frac{\mathrm{d}}{\mathrm{d}x} f_t(x) \mathrm{d}t + \mathrm{d}W_t K(x), \quad f_0(x) = \beta (\mathcal {S}^*_x \gamma _0), \end{aligned}$$*H*of absolutely continuous functions (AC) with values in \(\mathbb {R}^{n \times d}\), precisely \( H=\left\{ f \in AC(\mathbb {R}_+, \mathbb {R}^{n \times d}) \, | \, \int _0^{\infty } \Vert f'(x)\Vert ^2 \alpha (x) \mathrm{d}x < \infty \right\} \) where \( \alpha > 0 \) denotes a weight function (compare Filipović 2001). We can then set \(\lambda _t(x,y)=f_t^{\top }(x)f_t(y)\) and define the same Volterra Wishart process as in (5.10) by \(V_t:=\lambda _t(0,0)=X_t^{\top }X_t\). By Itô’s formula and variation of constants, its dynamics can then equivalently be expressed viaComparing (5.12) and (5.10) yields$$\begin{aligned} V_t:=\lambda _t(0,0)= & {} f^{\top }_0(t)f_0(t)+ n\int _0^t K(t-s)K(t-s)\mathrm{d}s\nonumber \\&+\int _0^t K(t-s)\mathrm{d}W^{\top }_sf_s(t-s) + \int _0^t f^{\top }_s(t-s)\mathrm{d}W_s K(t-s).\nonumber \\ \end{aligned}$$(5.12)$$\begin{aligned} \beta (\mathcal {S}_x^*\gamma _t)=f_t(x)=\mathbb {E}[X_{t+x}| \mathcal {F}_t], \quad x,t \ge 0. \end{aligned}$$(5.13) - (iii)In the case when \(\beta \) and \(\mathcal {S}^*\) are as in Example 4.4, (5.10) reads asHence by (5.13), \(\int _0^{\infty } e^{-x(t-s)}\gamma _s(\mathrm{d}x)=\mathbb {E}[X_t | \mathcal {F}_s]\). This yields exactly Eq. (1.6) considered in the introduction. Note that if \(\nu \) and in turn$$\begin{aligned} \int _{\mathbb {R}^2} \lambda (\mathrm{d}x_1, \mathrm{d}x_2)&= \int _{\mathbb {R}^2}e^{-(x_1+x_2)t}\lambda _0(\mathrm{d}x_1,\mathrm{d}x_2) + n \int _0^t K(t-s) K(t-s)\mathrm{d}s\\&\quad + \int _0^t\int _0^{\infty } K(t-s)\mathrm{d}W^{\top }_s e^{-x(t-s)} \gamma _s(\mathrm{d}x)\\&\quad + \int _0^t \int _0^{\infty } e^{-x(t-s)} \gamma ^{\top }_s(\mathrm{d}x)\mathrm{d}W_s K(t-s). \end{aligned}$$
*K*are chosen as in Remark 4.5, this Volterra Wishart process has exactly the roughness properties desired in rough covariance modeling.

In the following remark, we list several properties of Volterra Wishart processes.

### Remark 5.5

- (i)
Note that the marginals of

*V*are Wishart distributed as they arise from squares of Gaussians. - (ii)
In order to bring (5.6) in a “standard” Wishart form (with the matrix square root) as in (1.1) by replacing \(\gamma (\mathrm{d}x)\) by \(\sqrt{\lambda }(\mathrm{d}x,\mathrm{d}y)\), new notation has to be introduced, compare with (5.7).

- (iii)Nevertheless, both the drift and the diffusion characteristics of \(\lambda \) depend linearly only on \(\lambda \), e.g.,which indicates that \((\lambda _t)_{t\ge 0}\) is Markovian on its own. This is shown rigorously below.$$\begin{aligned} \frac{\mathrm{d}[\lambda _{ij}(\mathrm{d}x_1,\mathrm{d}x_2), \lambda _{kl}(\mathrm{d}y_1,\mathrm{d}y_2)]_t}{\mathrm{d}t}&= (K(x_1) K(y_1))_{ik} \lambda _{t,jl}(\mathrm{d}x_2,\mathrm{d}y_2)\\&\quad +(K(x_1) K(y_2))_{il} \lambda _{t,jk}(\mathrm{d}x_2,\mathrm{d}y_1)\\&\quad + (K(x_2)K(y_1))_{jk}\lambda _{t,il}(\mathrm{d}x_1,\mathrm{d}y_2)\\&\quad + (K(x_2)K(y_2))_{jl}\lambda _{t,ik}(\mathrm{d}x_1,\mathrm{d}y_1) \, , \end{aligned}$$

*affine*, in the sense that its Laplace transform is exponentially affine in the initial value. The process \(\lambda \) can therefore be viewed as an

*infinite dimensional Wishart process*on \(\widehat{\mathcal {E}}\) analogously to Bru (1991), Cuchiero et al. (2011).

### Theorem 5.6

### Proof

*g*such that

The following lemma provides an explicit expression for the Laplace transform of \(\gamma _t \widehat{\otimes }\gamma _t\). This resembles not surprisingly the Laplace transform of a non-central Wishart distribution with *n* degrees of freedom.

### Lemma 5.7

### Proof

*n*, note that we can write

*W*and thus take values in \(\mathbb {R}^{1 \times d }\). Similarly,

*bt*by

## 6 (Rough) Volterra-type affine covariance models

*d*assets. We exemplify this with the Volterra Wishart process of Sect. 5 and define a (rough) multivariate Volterra Heston-type model with possible jumps in the price process. Roughness can be achieved by specifying \(\nu \) and in turn the kernel of the Volterra Wishart process as in Remark 4.5. The log-price process denoted by

*P*and taking values in \(\mathbb {R}^d\) evolves according to

*W*appearing in (5.1) as follows

*W*and \(\varrho \in \mathbb {R}^d\). Moreover, \(\mu ^P\) denotes the random measure of the jumps with compensator \({{\,\mathrm{Tr}\,}}(V m(\mathrm{d}\xi ))\), where

*V*is the Volterra Wishart process of (5.10) and

*m*a positive semidefinite measure supported on \(\mathbb {R}^d\).

As a corollary of Section 5 and Cuchiero (2011, Section 5), we obtain the following result, namely that the log-price process together with the infinite dimensional Wishart process \(\lambda \) given in (5.5) is an affine Markov process.

^{2}\(\langle P_{i}, \lambda _{kl}(\mathrm{d}x_1, \mathrm{d}x_2)\rangle _t \) is given by

### Corollary 6.1

*P*defined in (6.1) is Markovian with state space \((\widehat{\mathcal {E}}, \mathbb {R}^d)\). It is affine in the sense that for \((y, v) \in Y(\mathbb {R}^{n \times d}) \times \mathbb {R}^d\), we have

### Remark 6.2

*V*given in (4.23). The log-price process (under some risk neutral measure) evolves then according to

*B*is a

*d*-dimensional Brownian motion and the jump measure

*m*of

*P*and \(\mu \) of the Markovian lift \(\lambda \) as given in (4.17) can be the marginals of some common measure supported on \(\mathbb {S}^d_+ \times \mathbb {R}^d \).

## Footnotes

## Notes

### Acknowledgements

Open access funding provided by Vienna University of Economics and Business (WU).

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