Stochastically Continuous Extensions

• Stochastically Continuous Extensions are constructions that modify stochastic processes to achieve pathwise continuity in probability, ensuring regular sample paths.
• They extend finite-dimensional distributions to path spaces using regularity criteria like right-continuity and uniform jump-tail bounds, along with skeleton-extension techniques.
• Their applications span continuous-time Markov processes, SDE modifications, and multistable Lévy motions, offering practical insights for both theoretical analyses and computational simulations.

A stochastically continuous extension is a construction, typically in the context of stochastic processes or stochastic flows, which upgrades a process or a family of transition laws to a version that possesses pathwise continuity properties in probability or almost surely, typically with respect to time, space, or both. Such extensions underpin the modern theory of stochastic processes on non-trivial state spaces, ensure the regularity of sample paths, and are central to both the foundational and computational aspects of continuous-time stochastic models.

1. Foundational Frameworks and Regularity Criteria

Stochastically continuous extensions arise in several key settings:

• Path-Space Probability Measures: Given a consistent family of finite-dimensional distributions, one seeks a (unique) probability measure on a path space (e.g., functions or càdlàg maps) that realizes these distributions. Regularity conditions ensure that this realization yields processes with “stochastically continuous” sample paths (i.e., for any $t$, $X_{t+h}\to X_t$ in probability as $h\to 0$).
• Stochastic Flows and Semiflows: For SDEs or Markov processes, one seeks modifications of the solution map or flow that are continuous in initial data, time, or both, possibly restricted to sets of controlled Hausdorff dimension.

A central paradigm is the framework for countable-state, càdlàg-path processes developed in "Countable-state stochastic processes with càdlàg sample paths" (Erreygers et al., 2023), which replaces the classical Daniell–Kolmogorov extension with a version where the extension is regular (i.e., stochastically continuous and with non-explosive jump structure) and thus yields a true measure on cadlag path space without requiring further modification.

2. Stochastically Continuous Path-Space Extensions

Formally, let $X$ be a countable state space, and consider the path space $D(T,X)$ of all càdlàg functions $ω : T \to X$, with $T \subset \mathbb{R}$ and appropriate limits on the left and right at all $t \in T$.

Given a collection of finite-dimensional distributions $\{\mu_u\}_{u \in U}$ (indexed by tuples $u = (t_1 < \cdots < t_n)$), the extension problem consists of constructing a probability measure $P$ on the cylinder $\sigma$-algebra of $D(T,X)$ whose marginals agree with $\{\mu_u\}$.

The essential regularity (i.e., stochastic continuity) requirements are:

• (R-1) Right-continuity in probability: For all $t \in T \cap \text{rlims}(T)$,

$\lim_{r\downarrow t} \mu_{(t, r)}(X^2_{=}) = 1,$

where $X^2_{=} = \{(x, y) \in X^2: x = y\}$.

• (R-2) Uniform jump-tail bound: For each $n$ with $[-n, n] \cap T \neq \emptyset$,

$$\lim_{K \to \infty} \sup_{u \subset [-n, n]} \mu_u(\{x_u : \# \text{jump}(x_u) \geq K\}) = 0.$$

These guarantee both local stochastic continuity and no explosion of jumps, allowing the unique extension of the cylinder set pre-measure to a probability measure on the full path space $D(T,X)$ (Erreygers et al., 2023).

Key examples include the (classical) Poisson process and continuous-time Markov chains on countable spaces, where the Kolmogorov consistency and regularity conditions are easily checked, and stochastic continuity ensures càdlàg paths without any further modification of the probability space.

3. Stochastically Continuous Markov Processes and Graph Dynamics

Stochastic continuity is also central in the construction of continuous-time Markov jump processes on finite spaces, such as the space $\mathbb{G}$ of all graphs on a fixed finite vertex set (Butts, 2022).

A Markov process $\{G_t : t \geq 0\}$ is called stochastically continuous if for every $g \in \mathbb{G}$ and $t \geq 0$,

$$\lim_{\delta \to 0^+} \mathbb{P}(G_{t+\delta} \neq g \mid G_t = g) = 0.$$

Equivalently, this ensures that, with high probability, the process remains at $g$ in short time intervals, and the infinitesimal generator $Q$ characterizing the process leads to a well-behaved stochastic pathwise structure.

Continuous-time ERGM-generating processes, including separable temporal ERGM (STERGM) families, fulfill this property. Their construction via infinitesimal jump rates guarantees both the Markov and stochastic continuity properties, facilitating tractable simulation, equilibrium analysis, and parameter interpretation (Butts, 2022).

4. Modifications and Strong Extensions for Stochastic Differential Equations

For solutions of SDEs, the existence of stochastically continuous modifications is fundamental. Under a global one-sided Lipschitz (monotonicity) condition on the coefficients, it is possible to construct a jointly continuous semiflow $\phi_{s,t}(x)$ such that for almost all $\omega$, $(s, t, x) \mapsto \phi_{s,t}(x, \omega)$ is continuous and satisfies the semiflow property: $\phi_{t, u}(\phi_{s, t}(x)) = \phi_{s, u}(x)$ (Scheutzow et al., 2016).

This is achieved by:

• Obtaining moment bounds on increments,
• Applying exponential Gronwall inequalities,
• Using the Kolmogorov–Chentsov criterion for joint Hölder continuity in all variables.

Extensions are possible to strong $\Delta$-completeness: if $A \subset \mathbb{R}^d$ is a set of upper Minkowski dimension $\leq \Delta$, then on a full-measure event, the semiflow restricted to $A$ is continuous (Scheutzow et al., 2016).

5. Skeleton-Extension Methods for Measurable and Continuous Flows

In non-Euclidean state spaces (e.g., metric spaces, graphs), the strong measurable continuous extension is constructed via the skeleton-extension technique (Raimond et al., 2023):

• Build a random skeleton, i.e., a dense collection of trajectories with desired properties.
• Interpolate deterministically between skeleton paths via measurable selection.
• Use “bifurcation set” patching to enforce strong flow properties (i.e., deterministic composition rules).
• Ensure that, by continuity-in-probability and Feller conditions, the resulting modifications have jointly continuous trajectories and preserve the original finite-dimensional distributions.

This method applies to coalescing flows, including Walsh Brownian motion and Brownian motion on metric graphs (Raimond et al., 2023).

6. Continuous Weak and Strong Extensions in Simulation

For stochastic numerical schemes, stochastically continuous extensions manifest as dense-output, pathwise continuous families of random variables produced by continuous Runge–Kutta methods and general one-step approximations (Debrabant et al., 2013). The continuous extension (e.g., CRDI2WM, the continuous Platen scheme) ensures that for all $t$ and all functionals $f$ of polynomial growth,

$$\sup_{t\in [t_0, T]} \lvert \mathbb{E} f(Y(t)) - \mathbb{E} f(X(t)) \rvert \leq C h^p,$$

yielding uniform weak order of convergence to the law of the solution over the entire interval, not just at the mesh points. This avoids the need to resimulate increments for intermediate times and supports accurate weak approximation on path space (Debrabant et al., 2013).

7. Advanced Examples: Multistable Lévy Motions and Localisability

In the context of Lévy and multistable processes, stochastically continuous extensions ensure pathwise regularity in the following sense:

• Multistable Lévy motion (MsLM), with time-varying stability function $\beta:[0,T] \to (0,2]$, possesses independent increments and admits both Poissonian and series representations (Fan et al., 2015).
• Under mild uniform continuity of $\beta$, MsLM admits continuous approximations, is stochastically Hölder continuous with optimal exponent $<1/\beta_{\max}$, and is strongly localizable at each point, meaning its infinitesimal scaling limit is a stable Lévy motion with index $\beta(t_0)$.

This framework provides both theoretical interpolation between stable laws with varying index and concrete weighted-sum schemes for simulation, with limit theorems and tightness ensured by explicit tail and increment controls (Fan et al., 2015).

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In all these cases, stochastic continuity or regularity ensures the existence of extensions or modifications of probabilistic objects with desirable pathwise continuity. These constructions are fundamental for both theoretical development and applied modeling across Markovian, non-Markovian, and high-dimensional stochastic systems.