Càdlàg Modifications in Stochastic Processes

• Càdlàg modifications are versions of stochastic processes with paths that are right-continuous and have left limits, ensuring rigorous treatment of discontinuities.
• They underpin existence and uniqueness results in various contexts such as Markov, Lévy, and infinite-dimensional processes using series representations and jump-control techniques.
• Construction methods like Kolmogorov–Chentsov arguments and Skorohod topology facilitate practical applications in rough paths, operator-valued martingales, and cylindrical process regularizations.

A càdlàg modification of a stochastic process is a version with sample paths that are right-continuous and possess left limits (“càdlàg:” continue à droite, limites à gauche). Existence, uniqueness, construction, and application of such modifications play a foundational role in modern probability theory and stochastic analysis, especially in the study of Markov and Lévy processes, rough paths, infinite-dimensional dynamics, and the abstract theory of stochastic processes on function spaces.

1. Definitions and Basic Framework

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $T$ a time index (typically $[0,\infty)$ or $[0,1]$), and $E$ a state space (often a Polish space, a separable Banach space, or a locally compact Hausdorff space). An $E$-valued stochastic process $(X_t)_{t\in T}$ is said to admit a càdlàg modification if there exists another process $(\widetilde X_t)_{t\in T}$ such that $P(X_t = \widetilde X_t) = 1$ for all $t$, and for almost all $\omega$, the path $t\mapsto \widetilde X_t(\omega)$ is right-continuous with left limits:

• For every $t>0$, $\lim_{s\uparrow t} \widetilde X_s(\omega)$ exists,
• For every $t$, $$\lim_{s\downarrow t} \widetilde X_s(\omega) = \widetilde X_t(\omega)$$.

In path space $D([0,1];E)$ (Skorohod space), this is formalized via the $J_1$-topology: $$d_{J_1}(x, y) = \inf_{\lambda \in \Lambda} \max \left( \sup_t |x(t) - y(\lambda(t))|_E,\, \sup_t |t - \lambda(t)| \right)$$ where $\Lambda$ consists of strictly increasing continuous bijections of $[0,1]$ onto itself (Basse-O'Connor et al., 2011).

Càdlàg modifications are central to stochastic process theory because many path properties (semimartingale calculus, Itô integration, Markov property in strong form) require such regularity.

2. Key Existence Theorems and Construction Principles

Feller-Markov Setting

For time-homogeneous Markov processes with a Feller transition semigroup $(Q_t)$ on a metrizable, locally compact, $\sigma$-compact Hausdorff space $E$ and natural filtration, a standard result holds (Edwin, 3 Sep 2025):

• There exists a càdlàg modification $\widetilde X_t$ such that $(\widetilde X_t)$ is adapted to the right-continuous completed filtration, is a modification of $X_t$ for each $t$, and, almost surely, $t \mapsto \widetilde X_t(\omega)$ is càdlàg.
• Any two such modifications coincide up to indistinguishability.
• The proof employs estimates on the expectation of increments via the truncated metric $\tilde\rho(x, y)$ and boundedness of total variation, extended by a rational approximation and tightness/Borel–Cantelli argument.

Countable-State Spaces and the Daniell-Kolmogorov Problem

The classical Daniell-Kolmogorov extension for constructing measures on $X^T$ for countable-state spaces $X$ is insufficient because the corresponding $\sigma$-algebra is too coarse. The existence of a càdlàg modification “enriches” the path-space so that hitting times and similar events become measurable.

More generally, if the consistent family of finite-dimensional distributions $\{\mu_u\}_{u}$ satisfies:

• (R1) Stochastic right-continuity: For all $t$, $$\lim_{r\searrow t} \mu_{(t,r)}(\{(x,x)\colon x\in X\})=1$$.
• (R2) Control of jumps: For any interval $I$, $$\sup_{u\in U_I}\mu_u\{\text{number of jumps} \geq K\} \to 0$$ as $K \to \infty$,

then the process admits a unique extension to a probability on $\Cad(X,T)$; i.e., no post hoc modification is needed (Erreygers et al., 2023).

Banach-Lie Group Valued Processes

For multiplicative processes $x = (x^s_t)$ valued in a Banach–Lie group $G$, under mere stochastic continuity and multiplicativity, there always exists a càdlàg modification that remains multiplicative and adapted, with all group structural properties retained. The key is to count “large oscillations” on a countable dense time set using exponential chart neighborhoods and show their number is almost surely finite, so the sample path is regulated (Behme et al., 21 Nov 2025).

3. Criteria and Techniques for Construction

Kolmogorov–Chentsov Type Arguments

For general infinitely divisible processes and Lévy-driven models, right-continuity in probability plus tight modulus-of-continuity estimates (with exponent $>\frac{1}{2}$) suffice for a càdlàg modification, using generalized Kolmogorov–Chentsov criteria on path space $D$ (Basse-O'Connor et al., 2011).

Series Representation Methods

Whenever the process admits a shot-noise (Poisson) or similar series expansion

$$Y^{u}(t) = b(t) + \sum_{j: \Gamma_j \leq u} H(t, \Gamma_j, V_j) - A^u(t),$$

with $H(\cdot, r, v) \in D([0,1];E)$ for all $(r,v)$ and $r \mapsto \|H(\cdot, r, v)\|$ nonincreasing, the (truncated) sum converges almost surely uniformly in $t$ to a càdlàg process. This construction ensures the process has càdlàg sample paths and, under proper coupling, is indistinguishable from the “raw” process (Basse-O'Connor et al., 2011).

For symmetric $\alpha$-stable processes, the analogous criterion (moment-modulus conditions (4.14)-(4.15)) ensures existence of càdlàg modifications; LePage series representation establishes pathwise regularity.

Infinite-Dimensional and Cylindrical Processes

For cylindrical processes $X = \{X_t: \Phi \to L^0(\Omega)\}_{t \geq 0}$ in the strong dual of a nuclear space $\Phi'_\beta$, the existence of a càdlàg version $Y$ follows if

• each $X(\phi)$ admits a càdlàg version,
• and $\{ X_t: \Phi \to L^0(\Omega) \}_{0 \leq t \leq T}$ is equicontinuous for every $T>0$.

The resulting $Y$ is $\Phi'_\theta$-valued for a suitable countably Hilbertian topology $\theta$ and is unique up to indistinguishability (Fonseca-Mora, 2015).

4. Applications Across Mathematical Domains

Context	Core Result or Application	Reference
Markov processes (general)	Existence/uniqueness of càdlàg modifications	(Edwin, 3 Sep 2025)
Countable-state processes	Kolmogorov extension + regularity = càdlàg directly	(Erreygers et al., 2023)
Infinite-dimensional Lie groups	Multiplicative stochastic processes admit càdlàg modifications	(Behme et al., 21 Nov 2025)
Cylindrical/stochastic processes in nuclear spaces	Càdlàg modifications of cylindrical processes	(Fonseca-Mora, 2015)
Infinitely divisible/stable processes	Uniformly convergent series representations yield càdlàg modifications	(Basse-O'Connor et al., 2011)
Rough paths (semimartingale, Gaussian, model-free)	Existence of càdlàg rough path lifts via dyadic-Itô approximation	(Liu et al., 2017)

Applications include:

• Solutions of linear SDEs on matrix and gauge groups, via multiplicative exponentials (Behme et al., 21 Nov 2025).
• Lévy processes in infinite-dimensions: path-regularity enables potential theory constructions (Behme et al., 21 Nov 2025).
• Operator-valued martingales in duals of nuclear spaces: regularization extends Mitoma–Martias–Itô–Nawata theory (Fonseca-Mora, 2015).
• Rough path theory with jumps: SDEs driven by jump processes require càdlàg rough path lifts for well-posedness (Liu et al., 2017).
• Construction of explicit shot-noise representations for non-Markovian or stable processes, facilitating analysis of jump functionals and extremes (Basse-O'Connor et al., 2011).

5. Uniqueness and Counterexamples

Càdlàg modifications, when they exist, are unique up to indistinguishability—any two càdlàg versions coincide on a dense time set almost surely, hence everywhere by right-continuity (Behme et al., 21 Nov 2025). However, stochastic continuity alone does not guarantee existence: a process can be continuous in probability but fail to have a càdlàg version if it accumulates infinitely many jumps in finite time. Explicit tightness or jump-control conditions are necessary (e.g., (R2) in (Erreygers et al., 2023)).

6. Path Space Topologies and Measurability

Most results on càdlàg modifications exploit the structure of the Skorohod space $D([0,T];E)$:

• The $J_1$-topology accommodates the lack of uniform continuity and discontinuous addition, via “elastic” time reparametrizations (Basse-O'Connor et al., 2011).
• Cylinder $\sigma$-algebras and projective limit constructions facilitate extension theorems for consistent finite-dimensional distributions, under regularity (tightness and continuity) conditions (Erreygers et al., 2023).
• The map associating process values at each time to sample-path-valued random elements is measurable with respect to these structures, critically used in Carathéodory and Minlos extension arguments.

7. Construction Recipes and Practitioner Guidelines

A synthesis for verifying or constructing càdlàg modifications (Basse-O'Connor et al., 2011):

1. Finite-dimensional convergence: Check that process laws converge appropriately.
2. Series representations: Identify a kernel $H$ for a Poisson or LePage-type expansion with càdlàg path regularity.
3. Path-regularity: Ensure jump size and frequency are appropriately controlled, with modulus of continuity exponents $>1/2$ if using Kolmogorov–Chentsov reasoning.
4. Apply uniform convergence/approximation theorem: Verify that truncated partial sums converge uniformly in probability or almost surely to a càdlàg process.
5. Stable process specifics: For symmetric $\alpha$-stable processes, verify modulus-of-moment conditions to deduce existence of modification.
6. Explicit jump functionals: Once a version is constructed, study pathwise properties (e.g., jump amplitudes, variation norms) via the series representation.

A plausible implication is that, in practice, stochastic continuity, tightness/modulus-of-continuity or jump-regularity, and pathwise construction are the universal tools to obtain càdlàg regularizations across modern stochastic process theory.