Càdlàg Modification in Stochastic Processes

• Càdlàg modification is a version of a stochastic process with right-continuous paths and existing left limits, ensuring alignment with the original process in finite-dimensional distributions.
• Existence theorems like Kolmogorov–Chentsov provide sufficient conditions, using moment and modulus bounds, to guarantee modifications with desired path regularity.
• These modifications underpin rigorous pathwise analysis in Markov, Lévy, and stable processes while enabling measurable spaces that capture key functionals such as stopping times and jump counts.

A càdlàg modification (continue à droite, limites à gauche: right-continuous with left limits) is a process or version of a stochastic process whose sample paths are right-continuous and possess left limits at every point in the indexing set, and which is indistinguishable in finite-dimensional distributions from a given process. Càdlàg modifications are central in probability theory, particularly in the construction and pathwise analysis of Markov processes, infinitely divisible processes, martingales, and other stochastic processes in both classical separable metric spaces and more general topological vector spaces.

1. Formal Definitions and Path Properties

A function $\omega: T \to E$ (with $T \subseteq \mathbb{R}$ and $E$ a metric or topological space) is called càdlàg if, for all $t \in T$,

1. The right-limit $\lim_{s \downarrow t} \omega(s)$ exists,
2. The left-limit $\lim_{s \uparrow t} \omega(s)$ exists (if $t$ is not the minimal element),
3. $\omega$ is right-continuous at $t$, i.e., $\omega(t) = \lim_{s \downarrow t} \omega(s)$.

The set $D(E)$ denotes the space of all càdlàg paths, typically equipped with the cylindrical $\sigma$-algebra $\mathcal{D}$ generated by coordinate projections $X_t(\omega) = \omega(t)$. Two $E$-valued processes $(X_t)_{t \in T}$ and $(\widetilde{X}_t)_{t \in T}$ defined on the same probability space are modifications of each other if $P[X_t = \widetilde{X}_t] = 1$ for all $t \in T$. A càdlàg modification is such a process with càdlàg sample paths almost surely (Erreygers et al., 2023).

2. Classical Existence Theorems

The existence of càdlàg modifications is foundational for “good” pathwise theory in stochastic processes. The classical Kolmogorov–Chentsov theorem provides sufficient conditions for the existence of versions with continuous (and thus also càdlàg) paths: for an $E$-valued process $(X_t)_{t \in [0,\infty)}$, if there exist $\alpha > 0$, $\beta > 1$ such that

$$\mathbb{E}[d(X_t, X_s)^\alpha] \leq C |t-s|^{1+\beta}$$

for all $s, t$, then there is a modification with Hölder-continuous trajectories. Analogously, a modulus-of-continuity bound of the form $P(d(X_t,X_s) > \varepsilon) \leq K |t-s|$ implies the existence of a càdlàg modification (Erreygers et al., 2023).

For Markov processes, if $(E,d)$ is locally compact, $\sigma$-compact, Hausdorff, and the transition semigroup $(Q_t)_{t \geq 0}$ is Feller (i.e., maps $C_0(E) \to C_0(E)$ and is strongly continuous), then there exists a càdlàg modification $\widetilde{X}$ of any $E$-valued Markov process $X$ with semigroup $Q$:

• $\widetilde{X}$ is adapted to the completed & right-continuous filtration,
• $\widetilde{X}$ is a modification of $X$,
• $\widetilde{X}$ is almost surely càdlàg,
• $\widetilde{X}$ retains the Markov property with semigroup $Q$ (Edwin, 3 Sep 2025).

3. Constructive and Generalized Frameworks

Several direct constructions bypass classical product-measure limitations for uncountable index sets. For countable $E$ and arbitrary $T \subset \mathbb{R}$, consistent finite-dimensional distributions $\{\mu_u\}$ are regular if they satisfy:

• (R1) Stochastic right-continuity: for all $t$, $$\lim_{r \downarrow t} \mu_{(t, r)}(\{(x, y): x = y\}) = 1$$,
• (R2) Tightness of jump-number: for each $n$, $$\lim_{k \to \infty} \sup_{u \subset [-n, n]} \mu_u(\{\#\{j: x_{t_{j-1}} \neq x_{t_j}\} \geq k\}) = 0$$.

Under these, there exists a unique probability measure $\mu$ on $(D(E), \mathcal{D})$ realizing the desired finite-dimensional marginals, and such that the canonical process is automatically càdlàg almost surely (Erreygers et al., 2023). This direct approach ensures the sample space and $\sigma$-algebra are sufficiently rich for path functionals such as stopping times, suprema, and jump counts, circumventing the need for subsequent modification.

For processes in the dual $\Phi'$ of a nuclear space $\Phi$, a cylindrical process $X = \{X_t\}$ is shown to have a $\Phi'_\beta$-valued càdlàg modification $Y = \{Y_t\}$ if each coordinate process $\{X_t[\phi]\}$ is càdlàg and certain equicontinuity holds. When additional uniform moment bounds are present,

$$\sup_{t \in [0,T]} \mathbb{E}|X_t[\phi]|^n \leq C_T p(\phi)^n,$$

for a fixed seminorm $p$, one can find a single Hilbert space $\Phi_q'$ such that $Y$ takes values in $\Phi_q'$ and

$$\mathbb{E}\bigl[\sup_{t\in[0,T]} q'(Y_t)^n\bigr]<\infty$$

(Fonseca-Mora, 2015).

4. Càdlàg Modifications for Infinitely Divisible and Series-Represented Processes

Basse-O’Connor–Rosiński establish that for independent random elements $X_j$ in $D([0,1];E)$ (with separable Banach $E$), if the partial sums $S_n = \sum_{j=1}^n X_j$ converge in finite-dimensional distributions to $Y$ in $D([0,1];E)$, then $S_n + y_n \to Y$ almost surely and uniformly for some deterministic centering $y_n$. For symmetric cases, $y_n$ is not needed. This extends the Itô–Nisio theorem to $D([0,1];E)$, despite the space’s non-separability under the sup norm and the discontinuity of addition under the Skorokhod topology (Basse-O'Connor et al., 2011).

For infinitely divisible processes with Lévy–Khintchine characteristics, explicit Poissonian series representations lead to càdlàg modifications via uniform convergence in norm, even in non-Markovian contexts. Symmetric $\alpha$-stable processes with appropriate kernel regularity and moment bounds possess càdlàg modifications, and LePage-type series converge uniformly. Jump functionals such as total jump variation and maximal jump admit explicit stable or Fréchet laws in this framework (Basse-O'Connor et al., 2011).

5. Methodological Proof Techniques

The main technical strategies for constructing càdlàg modifications include:

• Moment and variation control: Bounding expected (truncated) metric increment or variation of the process to rule out pathological trajectories.
• Diagonalization and cylinder measures: Using consistent finite-dimensional laws to extend (via Carathéodory’s theorem) to measures on $D(E)$ (Erreygers et al., 2023).
• Equicontinuity arguments: Verifying continuity with respect to coordinates in topological or nuclear spaces, then employing closed graph or regularization theorems (Fonseca-Mora, 2015).
• Series and coupling arguments: Employing point process representations and uniform convergence results for infinitely divisible or stable processes (Basse-O'Connor et al., 2011).
• Fatou’s lemma and null set arguments: Excluding high-variation or exceptional non-càdlàg paths via integrability arguments (Edwin, 3 Sep 2025).
• Measurability and filtration enlargement: Ensuring that adaptation, Markov properties, and pathwise properties are retained upon modification (Edwin, 3 Sep 2025).

6. Applications and Pathwise Structure

Càdlàg modifications underpin the construction and pathwise analysis of diverse classes of stochastic processes:

• Piecewise-constant sample paths in finite Markov chains,
• Diffusions and Lévy processes in Euclidean spaces and manifolds,
• Infinite-dimensional processes in Banach spaces, nuclear spaces, and their duals,
• Infinitely divisible and stable processes, including shot noise and series representations,
• Martingales and cylindrical processes in locally convex vector spaces.

In all these settings, the existence of a càdlàg modification is both a regularity result and a technical necessity for measure-theoretic and functional analytic arguments. Importantly, these constructions provide a sample space whose $\sigma$-algebra is rich enough to capture stopping times, level sets, excursions, and other functionals critical to stochastic analysis and filtration theory (Erreygers et al., 2023, Basse-O'Connor et al., 2011, Fonseca-Mora, 2015, Edwin, 3 Sep 2025).

7. Comparative and Advanced Perspectives

Direct methods working on $D(E)$ offer significant advantages over classical constructions based on $E^T$ with the product $\sigma$-algebra, particularly for uncountable $T$. They avoid measure-theoretic pathologies and post hoc modification arguments, immediately yielding a measurable space adapted to the intended filtrations and pathwise analyses. They also allow sharper results in complex topological settings (e.g., nonseparable Banach spaces, nuclear spaces) and in the construction of regular process versions under minimal or natural regularity and moment conditions.

Results for Hilbert-space valued, Banach-space valued, and nuclear-space-valued processes generalize classical theorems and highlight the fundamental role of equicontinuity, moment estimates, and the interplay between cylindrical measures and genuine modifications in functional stochastic analysis (Fonseca-Mora, 2015). The extension from Markov and diffusion processes to general Lévy and stable processes via explicit series and coupling constructions underlines the breadth and flexibility of the càdlàg modification paradigm (Basse-O'Connor et al., 2011).