Multivariate Càdlàg Stochastic Processes

• Multivariate càdlàg stochastic processes are vector- or matrix-valued models with right-continuous paths and left limits, capturing jumps and discontinuities in complex systems.
• They underpin key frameworks such as functional limit theorems, stochastic integration, and extremes analysis, often leveraging the weak M1 topology for multidimensional convergence.
• Advanced representations like stochastic delay equations, multivariate CARMA, and MUCOGARCH structures demonstrate their practical applications in financial volatility, time series, and physical system modeling.

A multivariate càdlàg stochastic process is a vector- or matrix-valued process whose sample paths possess right-continuity with left limits (“càdlàg” from the French ‘continue à droite, limites à gauche’), generalizing the classical concept of regular processes in probability theory to multidimensional and functional settings. These processes are fundamental in the theory of multivariate extremes, stochastic integration, time series, limit theorems, and applications to financial and physical systems where jumps and non-smooth dynamics are essential.

1. Formal Definition and the Space of Multivariate Càdlàg Functions

Let $d \ge 1$ and define $\mathbb{R}_+^d = [0,\infty)^d$. The canonical space of multivariate càdlàg paths is

$$D([0,1], \mathbb{R}_+^d) = \Bigl\{\, x:[0,1]\to\mathbb{R}_+^d \;\Big|\; x \text{ is right-continuous with left limits} \,\Bigr\}.$$

Each coordinate $x^{(k)}$ has, for all $t \in (0,1]$, a left limit $x^{(k)}(t-) := \lim_{s\uparrow t} x^{(k)}(s)$ and no discontinuity of the second kind. Processes with sample paths in this space arise as limits of appropriately normalized maxima, Lévy processes, stochastic delay equations, and volatility models. The topology imposed on $D([0,1], \mathbb{R}_+^d)$ is tailored to the specific continuity properties relevant for functional limit theorems and stochastic analysis, particularly jump processes (Krizmanić, 2016).

2. Topological Structures: Skorohod Weak $M_1$ Topology and Its Significance

The standard Skorohod $M_1$ topology is suitable for processes with jumps in the univariate case. Multivariate extension is nontrivial due to the absence of a natural total order. The completed graph

$$\Gamma_x = \Bigl\{\, (t, z) \in [0,1]\times \mathbb{R}_+^d : z \in [x(t-), x(t)] \,\Bigr\}$$

equipped with a partial ordering on time and coordinate-wise increments, together with weak parametric representations

$$(r, u) : [0,1] \to \Gamma_x, \quad r(0) = 0,\; r(1) = 1,\; u(1) = x(1),$$

induces the weak $M_1$ pseudometric $d_w$ via infima over synchronized paths.

Convergence under the weak $M_1$ topology is strictly weaker than strong $M_1$ for $d \ge 2$, crucially allowing coordinate-wise extremal functionals to be continuous and supporting the development of multivariate maxima process limit theory. Notably, functionals such as the partial maxima process fail to be continuous under the strong $M_1$ topology, as demonstrated via $m$-dependent sequences (Krizmanić, 2016).

3. Stochastic Delay Equations, Càdlàg Solutions, and Multivariate CARMA Representations

Multivariate stochastic delay differential equations (MSDDEs) generalize autoregressive moving average models to continuous time. Given a matrix-valued measure $\eta$ and a Lévy-type càdlàg noise $Z$, the MSDDE

$dX_t = (\eta * X)(t)\, dt + dZ_t$

has unique stationary càdlàg solutions if $\det h(iy) \neq 0$ for all $y$ (where $h(z) = -zI - \mathcal{L}\{\eta\}(z)$). The solution admits both an explicit convolution (moving-average) form,

$X_t = \int_{(-\infty,t]} g(t-s)\, dZ_s,$

and a higher-order SDDE ("CAR($\infty$)") representation for multivariate CARMA$(p,q)$ processes. The pathwise càdlàg property is ensured by the jump structure of $g$ and the regularity of $Z$. This permits unified analysis of Markovian (finite-memory) and fractional/infinite-memory models (Basse-O'Connor et al., 2018).

4. Multivariate COGARCH(1,1) Processes: Structure, Moments, and Stationarity

The MUCOGARCH(1,1) process consists of an observable returns process $G_t$ and latent covariance matrix $V_t$ driven by a $d$-dimensional Lévy process $L_t$: $$dG_t = V_{t-}^{1/2} \, dL_t, \qquad dV_t = [B(V_t - C) + (V_t - C)B^*] dt + A V_{t-}^{1/2} d[L,L]_t^{\mathrm{d}} V_{t-}^{1/2} A^*,$$ where $B, A$ are $d \times d$ matrices and $C$ is positive-definite. Positivity, strong Markov property, and càdlàg paths are preserved through locally Lipschitz drift and jump integrands. Stationarity criteria involve spectral properties of a block operator $\mathbb{B}$ and log-moment conditions on the Lévy measure. Explicit formulas for first and second moments, as well as covariance structure and autocovariance decay, are available under general conditions (Stelzer, 2010).

5. Lévy Processes, Dickman Approximations, and Path Properties

Every multivariate Lévy process admits a càdlàg version in $D([0,\infty),\mathbb{R}^d)$ by construction of the Lévy–Khintchine triplet $(A, \nu, b)$. Decomposition into small and large jumps yields compensated finite-variation and compound-Poisson components. Scaling small jumps with regularity conditions leads to convergence to a Dickman-type infinitely divisible law, both in characteristic functions and in the pathwise sense: $$\epsilon^{-1} X^{(\epsilon)} \Rightarrow Y^1 \;\text{in}\; D([0,\infty),\mathbb{R}^d),$$ with $Y^1$ a pure-jump process of Dickman type. Uniqueness arises via zero-stable functional equations for the scaling property. Dickman-type processes have càdlàg sample paths with self-similar structure and Hurst exponent $H=1$ (Grabchak et al., 2024).

6. Applications and Illustrative Examples in Multivariate Càdlàg Process Theory

Multivariate càdlàg processes underpin the theory of functional extremes, volatility modeling, and continuous-time time series. Key examples include:

• m-dependent maxima process convergence, requiring weak $M_1$ topology for functional limit theorems (Krizmanić, 2016).
• Stochastic recurrence equations with Markov or non-Markov dynamics, extending Kesten-Goldie theory (Krizmanić, 2016).
• Multivariate CARMA processes, captured via MSDDEs with Lévy-driven noise, allowing moving-average and SDDE representations (Basse-O'Connor et al., 2018).
• The CCC-GARCH$(p,q)$ process and MUCOGARCH(1,1), characterizing multivariate volatility under a càdlàg covariance matrix (Stelzer, 2010).
• Non-Gaussian OU processes, generalized multivariate gamma distributions, and BNS stochastic volatility models, leveraging Dickman approximations for small jumps and simulation schemes (Grabchak et al., 2024).

These processes form the analytical and probabilistic backbone of modern multivariate stochastic modeling, supporting both rigorous theoretical investigation and robust application in complex systems.