Lévy-driven CARMA Processes

• Lévy-driven CARMA processes are multivariate, stationary stochastic models defined via continuous-time state-space formulations driven by Lévy noise.
• They establish a direct link between continuous-time ARMA equations and discrete VARMA representations through state-space realization and exponential mixing properties.
• The strong mixing of sampled innovations under regularity conditions ensures reliable parameter estimation and asymptotically normal inference in practical applications.

A Lévy-driven CARMA (Continuous-time Autoregressive Moving Average) process is a multivariate, stationary stochastic process that generalizes discrete-time vector ARMA models to continuous time, with dynamics driven by an underlying multivariate Lévy process. CARMA processes are characterized by their equivalence to linear, continuous-time state-space models, and under mild regularity, their sampled innovations exhibit strong mixing properties that are essential for reliable statistical inference (Schlemm et al., 2012).

1. Mathematical Definition and State-Space Realization

Formally, given a multivariate two-sided Lévy process $L=(L(t))_{t\in\mathbb{R}}$ taking values in $\mathbb{R}^m$, with characteristic function

$$E[e^{i\langle u, L(t)\rangle}] = \exp \{ t \psi^L(u) \},$$

and parameters $\gamma \in \mathbb{R}^m$ (drift), $\Sigma^{\mathcal G} \succeq 0$ (Gaussian covariance), and $\nu$ (Lévy measure), the CARMA$(p,q)$ process $Y(t) \in \mathbb{R}^d$ is defined by

$P(D) Y(t) = Q(D) D L(t), \qquad D := \tfrac{d}{dt}$

where $$P(z) = I_d z^p + A_1 z^{p-1} + \cdots + A_p \in M_d(\mathbb{R}[z])$$ and $$Q(z) = B_0 z^q + B_1 z^{q-1} + \cdots + B_q \in M_{d,m}(\mathbb{R}[z])$$. As $L$ has non-smooth paths, the equation is interpreted via its state-space realization: \begin{align*} d\mathbf{G}(t) &= \mathcal{A} \mathbf{G}(t)\,dt + \mathcal{B}\,dL(t), \ Y(t) &= \mathcal{C} \mathbf{G}(t), \end{align*} where $\mathcal{A}$ is the companion-block matrix of size $pd \times pd$, $\mathcal{B}$ encodes the moving average structure, and $\mathcal{C}$ projects state to output. Under the stability condition $\Re(\sigma(\mathcal{A})) < 0$, the unique strictly stationary causal solution is

$$\mathbf{G}(t) = \int_{-\infty}^{t} e^{\mathcal{A}(t-u)} \mathcal{B} dL(u), \qquad Y(t) = \mathcal{C} \mathbf{G}(t).$$

2. Equivalence with Continuous-Time State-Space Models

Any causal linear stochastic differential equation (SDE) of the form

$$dX(t) = A X(t)\,dt + B\,dL(t), \qquad Y(t) = C X(t),$$

with $\Re(\sigma(A)) < 0$, admits transfer function $H(z) = C (zI_N - A)^{-1} B$. One can always construct polynomials $P(z)$, $Q(z)$ such that $H(z) = P(z)^{-1} Q(z)$, with appropriate degrees and minimality conditions. Thus, CARMA$(p,q)$ processes and causal linear state-space models are precisely equivalent under stationary solutions; minimal realization requires that $P$ and $Q$ share no zeros and that $P$ has simple roots, yielding the minimal state dimension $pd$.

3. Mixing Properties of Sampled Innovations

Consider regular sampling at interval $h>0$: $Y_n^{(h)} = Y(nh)$. Decomposing the process into a sum of Ornstein-Uhlenbeck modes,

$Y(t) = \sum_{\nu=1}^N Y_\nu(t)$

with eigenvalues $\lambda_\nu$ of $\mathcal{A}$, one shows (for distinct $\lambda_\nu$) that $Y^{(h)}$ admits a vector ARMA$(N, \leq N-1)$ representation: $\varphi(B) Y_n^{(h)} = \Theta(B) \varepsilon_n,$ where $\varphi(z) = \prod_\nu (1 - e^{\lambda_\nu h} z)$, $\varepsilon_n$ are linear innovations, and $\Theta(z)$ is Schur-stable. The associated innovations are driven by the i.i.d. sequence of block-integrals

$M_\nu^{(h)} = \int_0^h e^{\lambda_\nu(h-u)} dL(u).$

Under a mild continuity condition on $L$—satisfied in most practical cases including when $L$ has a nonsingular Gaussian component, compound Poisson jumps with an absolutely continuous jump-size distribution, or infinite activity with a Lévy measure admitting a density—the law of $M^{(h)}$ has a nontrivial absolutely continuous component. Consequently, the innovations $\{\varepsilon_n\}$ are exponentially completely regular ($\beta$-mixing), as per Mokkadem’s theorem.

4. Sampling, Inference, and Statistical Estimation

The mixing property is fundamental for consistent and asymptotically normal estimation procedures for CARMA processes sampled on grids. In practice, for any permissible Lévy driver,

• The sampled process can be modeled via its discrete-time VARMA structure.
• The innovations estimation enables use of classical methods (e.g., Kalman filter, quasi-maximum likelihood), exploiting the strong mixing properties.
• Asymptotic results hold under the stated regularity, such as finite second moment of $L(1)$ and mixing, ensuring reliability of inference on the underlying parameters and Lévy characteristics.

5. Sufficient Conditions for Mixing

The continuity assumption for exponential $\beta$-mixing is met when

• $L$ has a nonsingular Gaussian part ($\Sigma^{\mathcal G} > 0$),
• $L$ is compound Poisson with absolutely continuous jump-size law,
• $L$ possesses infinite activity with Lévy measure admitting a density in a ball around zero,
• More generally, $\nu$ does not concentrate on any hyperplane.

These draw a comprehensive boundary for practical applicability and encompass most empirically relevant Lévy processes in financial econometrics, signal processing, and spatial statistics.

6. Theoretical Foundations and Main Theorems

The main structural results established are:

• Equivalence Theorem: Every Lévy-driven MCARMA$(p,q)$ process coincides exactly with the output of a causal linear SDE of dimension $pd$; conversely, every such SDE corresponds to an MCARMA process (Schlemm et al., 2012).
• Weak VARMA Sampling Theorem: For distinct eigenvalues, the regularly sampled process admits a stable ARMA representation with i.i.d. innovations (Schlemm et al., 2012).
• Mixing of Innovations Theorem: Absolute continuity of the block-integral law guarantees exponential $\beta$-mixing of the innovations, crucial for parametric inference and estimation (Schlemm et al., 2012).

7. Practical Relevance and Scope

These results underpin model fitting, simulation, and inference in multivariate CARMA models, justifying the broad use of these processes for modeling temporal dependence with jumps, and supporting the use of conventional ARMA-type estimation techniques for continuous-time data sampled on fixed grids.

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Reference: "Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes" (Schlemm et al., 2012).