Lévy-Driven Multivariate OU Process

• Lévy-driven multivariate OU process is a multidimensional Markov model with linear drift and jump noise, enabling analysis of ergodicity and heavy-tailed behavior.
• Its spectral theory and invariant measures facilitate explicit simulation and calibration in finance, extremes, and stochastic volatility modeling.
• Advanced estimation methods such as GMM and MLE, along with efficient simulation techniques, offer robust tools for model inference.

A Lévy-driven multivariate Ornstein–Uhlenbeck (OU) process is a multidimensional Markov process governed by a linear stochastic differential equation in which the driving noise is an $\mathbb{R}^d$-valued Lévy process. These processes generalize classical OU diffusions by incorporating jump behavior via the Lévy–Itô decomposition. Specific instances include CARMA(1,0), supOU, and heavy-tailed models, with applications in extremes, finance, and statistical inference. The mathematical structure and spectral theory allow for extensive analysis of ergodicity, regular variation, explicit simulation, and statistical calibration.

1. Stochastic Differential Equation, Lévy–Itô Structure, and Solution Properties

A Lévy-driven multivariate OU process $(X_t)_{t\ge 0}$ solves the SDE: $dX_t = B X_t\,dt + dL_t,$ where $B \in \mathbb{R}^{d\times d}$ is the drift (mean-reversion) matrix with $\sigma(B)\subset\{\Re \lambda < 0\}$, and $L_t$ is a $\mathbb{R}^d$-valued Lévy process with characteristic triplet $(Q, b, \nu)$, i.e., covariance $Q$, drift $b$, and Lévy measure $(X_t)_{t\ge 0}$0 (Bartosz et al., 2018, Grabchak et al., 2024).

The explicit solution is

$(X_t)_{t\ge 0}$1

For $(X_t)_{t\ge 0}$2 and $(X_t)_{t\ge 0}$3, $(X_t)_{t\ge 0}$4 converges in law to the stationary distribution

$(X_t)_{t\ge 0}$5

with a characteristic function

$(X_t)_{t\ge 0}$6

where $(X_t)_{t\ge 0}$7 is the Lévy–Khintchine exponent of $(X_t)_{t\ge 0}$8 (Bartosz et al., 2018, Grabchak et al., 2024, Lu, 2020).

2. Spectral and Invariant Measures, Semigroup Theory

The generator of the Lévy-OU semigroup is a non-local operator on $(X_t)_{t\ge 0}$9: $dX_t = B X_t\,dt + dL_t,$0 (Sarkar, 21 Feb 2025).

Under hypoellipticity and integrability conditions, the process admits a unique invariant probability measure $dX_t = B X_t\,dt + dL_t,$1 with characteristic function

$dX_t = B X_t\,dt + dL_t,$2

where $dX_t = B X_t\,dt + dL_t,$3 is the Lévy exponent, and stationary Lévy measure

$dX_t = B X_t\,dt + dL_t,$4

A central result is the isospectrality of the OU semigroup for Lévy processes and its Gaussian counterpart: For $dX_t = B X_t\,dt + dL_t,$5, the spectrum on $dX_t = B X_t\,dt + dL_t,$6 of the Lévy-driven OU semigroup coincides with that of the Gaussian OU semigroup (Sarkar, 21 Feb 2025). The intertwining relation via a Markov operator $dX_t = B X_t\,dt + dL_t,$7 shows

$dX_t = B X_t\,dt + dL_t,$8

and enables explicit identification of eigenfunctions and multiplicities.

3. Regular Variation, Extremes, and Function Space Limit Theorems

If the Lévy measure $dX_t = B X_t\,dt + dL_t,$9 is regularly varying with index $B \in \mathbb{R}^{d\times d}$0, the OU process inherits heavy-tailed behavior. Regular variation propagates from Lévy increments to finite-dimensional marginals and also to entire sample paths in $B \in \mathbb{R}^{d\times d}$1 (Stelzer et al., 2012).

For a mixed moving average (MMA) representation

$B \in \mathbb{R}^{d\times d}$2

with Lévy basis $B \in \mathbb{R}^{d\times d}$3, sufficient kernel regularity implies that $B \in \mathbb{R}^{d\times d}$4 is functionally regularly varying with index $B \in \mathbb{R}^{d\times d}$5. In the special case $B \in \mathbb{R}^{d\times d}$6 (supOU), the criterion for a càdlàg version and regular variation in $B \in \mathbb{R}^{d\times d}$7 is concrete and model-independent: $B \in \mathbb{R}^{d\times d}$8 This describes full extremal behavior including cluster indices and weak convergence of exceedance point processes (Stelzer et al., 2012).

4. Estimation, Inference, and Simulation

Given discrete-time observations $B \in \mathbb{R}^{d\times d}$9 of an OU process, the increments of the driving Lévy process can be recovered exactly or approximately: $\sigma(B)\subset\{\Re \lambda < 0\}$0 allowing one to reconstruct i.i.d. samples from the underlying Lévy increment law (Brockwell et al., 2012). For grid increments, finite difference and numerical integration approximations yield errors of order $\sigma(B)\subset\{\Re \lambda < 0\}$1.

Generalized Method of Moments (GMM) and Maximum Likelihood Estimation (MLE) performed on approximate increments retain the same asymptotic distribution as with true increments, provided the mesh size $\sigma(B)\subset\{\Re \lambda < 0\}$2 and sample size $\sigma(B)\subset\{\Re \lambda < 0\}$3 satisfy $\sigma(B)\subset\{\Re \lambda < 0\}$4 (Brockwell et al., 2012).

Efficient simulation algorithms are available for heavy-tailed and tempered stable noise. For $\sigma(B)\subset\{\Re \lambda < 0\}$5-tempered $\sigma(B)\subset\{\Re \lambda < 0\}$6-stable OU processes, explicit series representations and scale-mixture (GGSM, IGa, IBGM, DGGa) simulation methods yield accurate transition law generation in multivariate settings (Grabchak et al., 2022).

5. Law Equivalence, Calibration, and Weak Subordination

Law equivalence results (Girsanov-type) assert that, for two OU processes driven by the same Lévy process with strictly positive-definite Gaussian part, the laws on path space are equivalent if the drift matrices are different (Bartosz et al., 2018). In the pure-jump (non-diffusive) case, absolutely continuous change of measure does not alter the process unless the solutions coincide pathwise.

Calibration of Lévy-driven OU models, especially when involving parametric families such as the weak variance alpha–gamma process, is possible via AR(1)-likelihoods and explicit computation of innovation densities (by Fourier inversion when necessary). This supports both exact and approximate simulation as well as robust parametric inference (Lu, 2020).

6. Structural and Spectral Properties

The spectral representation of Lévy-driven OU processes generalizes the classical theory by identifying explicit eigenfunctions, co-eigenfunctions, and spectral multiplicities. When the drift matrix is diagonalizable with real eigenvalues, eigenfunctions may be given by explicit Hermite-type polynomials or generating integrals incorporating analytic jump-encodings $\sigma(B)\subset\{\Re \lambda < 0\}$7: $\sigma(B)\subset\{\Re \lambda < 0\}$8 The spectrum of the semigroup is governed solely by the drift, not by the jump structure, provided mild moment and regularity assumptions hold (Sarkar, 21 Feb 2025).

Compactness properties are governed by the decay of the invariant law: if the law decays sufficiently rapidly, $\sigma(B)\subset\{\Re \lambda < 0\}$9 is compact; for stable jump processes ($L_t$0), $L_t$1 is non-compact (Sarkar, 21 Feb 2025).

7. Applications and Examples

Lévy-driven multivariate OU processes are the canonical model for stochastic volatility (e.g., the Barndorff-Nielsen–Shephard model), heavy-tailed noise in CARMA and MMA settings, statistical modeling of jumps in finance and insurance, and as building blocks for more general state-space and stochastic filtering models (Grabchak et al., 2024, Stelzer et al., 2012, Lu, 2020).

Table: Core Mathematical Ingredients

Mathematical Object	Definition/Role	Reference
Drift matrix $L_t$2	Generator of mean-reversion; $L_t$3 for stationarity	(Sarkar, 21 Feb 2025)
Lévy Process $L_t$4	Driving noise, with triplet $L_t$5; spectrum of jumps and possible diffusion	(Bartosz et al., 2018, Grabchak et al., 2024)
Invariant law $L_t$6	Infinitely divisible law; specified via stationary characteristic function	(Sarkar, 21 Feb 2025)
Generator $L_t$7	Integro-differential operator on $L_t$8	(Sarkar, 21 Feb 2025)
Regular variation index $L_t$9	Tail index governing extremes and finite/infinite variance regime	(Stelzer et al., 2012)

A plausible implication is that Ornstein–Uhlenbeck dynamics driven by Lévy noise are robust as a modeling class for both finite- and infinite-variance regimes, supporting explicit solution theory, tractable inference, and detailed probabilistic description of both clustering and pathwise extremes. The isospectrality of Lévy-OU semigroups with classical diffusions implies that essential spectral properties of the linear drift persist under a wide class of jump perturbations.