Dynamical Consistency in Sequential Models

• Dynamical Consistency is defined as the property that sequential decision, inference, or optimization problems remain mutually compatible when re-optimized with new information.
• It leverages dynamic programming by requiring state variables to encapsulate all necessary data, ensuring that initially optimal solutions remain valid over time.
• DC is applied across stochastic control, risk management, constraint satisfaction, and optimization, highlighting its significance in maintaining coherent decision processes in diverse disciplines.

Dynamical Consistency (DC) is a foundational concept across stochastic control, mathematical finance, constraint satisfaction, temporal planning, statistical inference, optimization, and the physical sciences. At its core, DC formalizes the requirement that a sequence of decision, inference, or optimization problems—posed and solved at successive points in time or information—produce solutions that remain mutually compatible as new information arrives or as the process evolves. This article surveys the mathematical underpinnings, representative formalizations, algorithmic implications, and cross-disciplinary applications of dynamical consistency in both deterministic and stochastic settings.

1. Fundamental Definition and Motivation

Dynamical Consistency, often also termed time consistency, is the property that, when a multistage or sequential problem is formulated and solved at an initial stage (e.g., $t_0$), the optimal solution remains optimal when the problem is re-solved at a later stage $t_1$ after observing new information or experiencing additional system evolution. In stochastic optimal control, this is formalized by considering a family of decision rules $(\Phi_{t_0}, \Phi_{t_1}, ...,\Phi_T)$: the policy computed at $t_0$ prescribes controls for every future time, and at each time $t_j$, re-solving the control problem starting from the realized state $x_{t_j}$ should yield a continuation of the original policy.

In deterministic contexts, if the optimal control depends only on the current state and not on the initial condition, the plan is dynamically consistent. More generally, DC ensures that receding-horizon re-optimization at future stages does not require deviation from the plan previously adopted, provided the state variable captures all relevant information.

2. Dynamic Programming Principle and the Role of the State Variable

A central mechanistic underpinning of dynamical consistency in stochastic control is the Dynamic Programming (DP) principle. Under Markovian assumptions (Hypothesis 1 in (Carpentier et al., 2010)), DP recursion for value functions is given by: $$V_T(x) = K(x), \quad V_t(x) = \min_u \mathbb{E}[L_t(x,u,w_{t+1}) + V_{t+1}(f(x,u,w_{t+1}))]$$ where $x$ is the "state variable".

Dynamic consistency is guaranteed if the state variable at each time $t$ encapsulates the minimal sufficient information required for subsequent optimal decisions. This is formalized by the feedback structure: $$\Phi^*_{t_0, t}: \mathcal{X}_t \rightarrow \mathcal{U}_t$$ where policies are expressed as functions of state (not history) and their optimality is preserved under re-initialization at any time with the correct state.

If the state variable is chosen too "narrowly"—i.e., omits relevant distributional or historical information—future re-optimization may result in different, inconsistent policies, breaking DC. Conversely, if the state is appropriately "enlarged" (for instance, to include the full probability law $\mu_t$), consistency is restored.

3. DC in the Presence of Constraints and Risk Measures

When additional constraints—such as expectation constraints (e.g., $\mathbb{E}[g(x_T)] \leq a$) or risk measures—are imposed, the minimal sufficient state variable may no longer be just $x_t$ but must include distributional information about the system. For such problems, the DP principle must be generalized: $$V_T(\mu) = \begin{cases} \langle K, \mu \rangle & \text{if } \langle g, \mu \rangle \leq a, \ +\infty & \text{otherwise} \end{cases}$$

$$V_t(\mu) = \min_{\Phi_t} \left\{ \langle \Lambda_t^{(\Phi_t)}, \mu \rangle + V_{t+1}((A_t^{(\Phi_t)})^*\mu) \right\}$$

with $\mu_t$ the distribution of the state at time $t$, and $(A_t^{(\Phi_t)})^*$ the adjoint evolution operator. This distributed (measure-valued) formulation ensures DC when risk or distributional constraints "remember" more than the instantaneous state—a phenomenon often seen in risk-averse control and stochastic programming (Carpentier et al., 2010).

4. Connections to Risk, Acceptability, and Probabilistic Decision Theory

The concept of DC has deep analogues in mathematical finance and decision theory, especially in the literature on coherent risk measures and dynamic acceptability indices (Bielecki et al., 2010). Dynamic consistent risk measures admit robust dual representations as

$$\rho_t(D) = -\inf_{\mathbb{Q}\in\mathcal{Q}_t} \mathbb{E}_{\mathbb{Q}}\left[\sum_{s=t}^T D_s | \mathcal{F}_t\right]$$

where $\{\mathcal{Q}_t\}$ is a dynamically consistent sequence of sets of probability measures. Time consistency demands that

$$1_A \min_{\omega\in A} \left\{ \inf_{\mathbb{Q}\in\mathcal{Q}_{t+1}} \mathbb{E}_{\mathbb{Q}}[X|\mathcal{F}_{t+1}](\omega)\right\} \leq 1_A \inf_{\mathbb{Q}\in\mathcal{Q}_t} \mathbb{E}_{\mathbb{Q}}[X|\mathcal{F}_t] \leq 1_A \max_{\omega\in A} \left\{ \inf_{\mathbb{Q}\in\mathcal{Q}_{t+1}} \mathbb{E}_{\mathbb{Q}}[X|\mathcal{F}_{t+1}](\omega)\right\}$$

holds for all $A\in\mathcal{F}_t$, linking DC in risk to recursive admissibility of dynamic policies.

This framework generalizes also to set-valued risk measures for markets with transaction costs (Feinstein et al., 2012), where "multi-portfolio time consistency" ensures that vector-valued acceptance sets propagate recursively, and thus, that no-arbitrage risk assessments remain stable across time.

In decision analysis, the extension of DC to vacuous belief models (Giang, 2012) rigorously characterizes certainty equivalence operators and sequential (folding-back) consistency even without explicit probabilities, providing foundational insights for robust artificial intelligence agents.

5. DC in Constraint Satisfaction and Optimization

In constraint satisfaction problems (CSPs), dynamical consistency appears as "dual consistency" (DC) (Lecoutre et al., 2014), a second-order local consistency which ensures that prunings in one variable's domain (under some partial assignment) are consistently reflected in all other variables, and vice versa. The DC property can be enforced algorithmically by iterative singleton assignments and GAC propagation, striking a balance between pruning power and computational tractability. In the binary case, DC and path consistency coincide, but DC generalizes efficiently to non-binary CSPs.

In nonconvex optimization, including sparse and cone-constrained settings (Thi et al., 2014, Dolgopolik, 2021), DC also refers to the consistency of approximate solutions (e.g., replacing the $\ell_0$ "zero-norm" by continuous penalizations) with the original problem's minima. Exact penalty methods and careful design of difference-of-convex (DC) algorithms guarantee that, at suitable parameter thresholds, local or global optima of the approximation coincide with those of the original, establishing "dynamical consistency between models".

Tables:

Domain	DC Mechanism	Minimal Information Requirement
Stochastic Control	DP recursion, feedback policies	Markov state $x_t$, or $(x_t, \mu_t)$
Risk Measures/Finance	Recursion of coherent risk measures	Sequence of measure sets $\mathcal{Q}_t$
Constraint Satisfaction	Dual/path consistency, singleton GAC	Local pairwise consistency
Sparse Optimization	DC penalty approximations, DCA schemes	Thresholded penalty equivalence
Temporal Planning	Dynamic execution strategies	Observation history and state

6. DC in Temporal Planning, Inference, and Physical Sciences

Advanced planning and scheduling under uncertainty is now routinely encoded as Conditional Simple Temporal Networks (CSTN) (Comin et al., 2015, Cairo et al., 2016). Here, DC is the requirement that an agent's schedule remains feasible after any admissible sequence of observation events, with algorithmic solutions drawing on reductions to Mean Payoff Games and refined notions such as $\epsilon$-dynamic consistency (accounting for reaction time) and $\pi$-dynamic consistency (for instantaneous reaction scenarios).

In statistical inference for dynamical systems, DC is equivalent to consistency of parameter estimates as the data length increases. Here, information-theoretic arguments (e.g., via Shannon–McMillan–Breiman theorem, entropy rate comparisons) are used to show that under mixing, identifiability, and regularity conditions, posterior or MLE estimates converge to the true structural parameters, even in highly nonlinear and chaotic systems (McGoff et al., 2013, Su et al., 2021).

In physical and cosmological models, DC underpins the construction of robust dynamical models, such as dynamically consistent cosmological constants with stable linearized perturbations (Donato et al., 5 Mar 2025), or inflationary models constrained by nonlinear PDEs derived from field dynamics and symmetry principles (Anguelova et al., 2022).

7. Implications, Extensions, and Research Directions

Dynamical consistency has broad implications:

• The architecture of multistage decision systems must be designed with sufficient "state" enrichment to ensure DC, especially in the presence of path-dependent constraints or risk.
• In finance, the duality between acceptability indices and dynamically consistent risk measures ensures that risk/acceptability evaluations do not become arbitrageable through dynamic rebalancing.
• Algorithmic advances, such as efficient verification of DC in CSTNs or enforcing DC-inspired consistencies in CSP and optimization, have led to practical gains in robust planning, AI, and combinatorial optimization.
• Behavioral and economic decision models have been refined to account for cases in which dynamic consistency may purposely be relaxed (e.g., dynamic conservatism, confirmation bias models (Kovach, 2021)).
• In estimation and inverse problems, DC-inspired loss functions (e.g., distributional consistency loss) outperform classic pointwise fidelities by aligning candidate solutions with the empirical distribution of measurement noise, thus improving generalization and convergence characteristics (Webber et al., 15 Oct 2025).

Open questions concern computational complexity (e.g., for general set-valued risk in high dimensions), further connections between information theory and DC in continuous-time systems, the interaction of DC with learning and adaptivity, and the empirical consequences of deliberate violations of DC in behavioral models.

8. Conclusion

Dynamical consistency is a unifying principle expressing the demand for temporal and informational coherence in multistage decision, optimization, inference, and control problems. Across domains—from stochastic control and risk management to planning, optimization, and statistical learning—the rigorous specification of state, information, and updating rules is essential to guarantee DC. Many effective methodologies, both classical (dynamic programming, recursive risk evaluation) and modern (exact-penalty DCA, distributional loss functions), are deeply structured by the requirements of dynamic consistency. As complex, high-dimensional, or nonclassical systems become increasingly common in science and engineering, the precise articulation and enforcement of DC remains a central theme in theory and applications.