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Valuation Reveals Uncertainty
Valuation and uncertainty are two fundamental objects in economics. A substantial body of research has examined their relationship, and robust valuation is one of the most widely used frameworks in this literature. A robust valuation rule assigns each payoff its worst-case expected value over a set of plausible models. In the conventional approach, uncertainty is specified a priori, and the corresponding valuation rule is derived from it. This paper takes the converse perspective. In practice, valuation is often observable through market prices, whereas the underlying uncertainty remains latent. We investigate the relationship between these two objects and show that, under suitable conditions, uncertainty can be recovered from valuation.
This paper explores two types of continuous-time valuation rules. The first is a dynamic sublinear valuation rule, which is formulated based on axiomatic economic principles. We define it as a family of operators $ \mathcal T=\{\mathcal T_{t,T}\}_{0\le t\le T<\infty}$ that adhere to monotonicity, stability, and time consistency (see Definition (ref) for a precise formulation). These properties encapsulate the core aspects of meaningful economic valuation in dynamic settings, ensuring coherence and consistency across time and states. A significant feature of this approach is that it does not require an underlying probabilistic structure, such as probability measures or state processes, for its definition. Instead, the valuation rule is characterized solely by these economic axioms, offering a flexible framework that is not tied to any specific model and can adapt to a variety of uncertainty scenarios.
The second type is a dynamic robust valuation under uncertainty, a continuous-time valuation rule that assigns each payoff its worst-case discounted expected value over a family of plausible models. While much of the existing literature focuses on uncertainty in the dynamics of the state process, our framework also incorporates uncertainty in discounting. The robust valuation is formulated within a probabilistic framework as follows. Let \(X\) be an underlying state process with state space \(D\subset\mathbb R^d\). For each time \(t\ge0\) and state \(x\in D\), let \(\mathcal U_{t,x}\) be a family of pairs \((A,\mathbb Q)\), where \(A\) is a cumulative discounting process and \(\mathbb Q\) specifies a law of \(X\) starting from \(x\). The class \(\mathcal U_{t,x}\) captures the uncertainty at time \(t\) and state \(x\), with each pair \((A,\mathbb Q)\in\mathcal U_{t,x}\) specifying a particular model. Given this class of plausible models, the robust valuation of a payoff function \(f\) at time \(t\) and state \(x\) is defined by \[ \mathcal{T}_{t,T}^{\mathcal{U}} f(x) := \sup_{(A,\mathbb{Q}) \in \mathcal{U}_{t,x}} \mathbb{E}^{\mathbb{Q}}\!\left[e^{-A_T} f(X_T)\right]. \] We refer to the family \(\mathcal U=\{\mathcal U_{t,x}\}_{(t,x)\in[0,\infty)\times D}\) as an uncertainty structure and to the family \(\mathcal T^{\mathcal U}=\{\mathcal T_{t,T}^{\mathcal U}\}_{0\le t\le T<\infty}\) as the robust valuation rule under \(\mathcal U\).
This paper makes four contributions that illuminate the relationship between dynamic valuation rules and robust valuation under uncertainty. First, we show that every dynamic sublinear valuation rule admits a representation as a robust valuation under an uncertainty structure. More precisely, given a dynamic valuation rule $ \mathcal T$, we construct an uncertainty structure \(\mathcal U\) such that \[ \mathcal{T} = \mathcal{T}^{\mathcal{U}} \,. \] Moreover, we provide an explicit procedure for recovering \(\mathcal U\) from the given valuation rule. This is the most technically demanding part of the paper, as it requires constructing a probabilistic uncertainty structure from a valuation rule initially specified solely through economic axioms, without any probabilistic primitives. The construction is developed in detail in Section (ref) and summarized in Figure (ref).
Second, we develop the notion of time-consistent uncertainty structures. Time consistency is one of the central properties of valuation rules in continuous-time settings. A key challenge is to determine how the time consistency of a dynamic sublinear valuation rule $\mathcal{T}$ should be reflected in the underlying uncertainty structure $\mathcal{U}$. To address this question, we introduce dynamic uncertainty structures (DUSs), formally defined in Definition (ref). We show that robust valuations under DUSs form dynamic sublinear valuation rules and, conversely, that every dynamic sublinear valuation rule admits a robust representation under a suitable DUS. Thus, time consistency of a sublinear valuation rule and the DUS property of its underlying uncertainty structure can be viewed as equivalent valuation-side and model-side formulations of the same recursive principle.
Third, we characterize the class of DUSs that represent a given valuation rule. Although every dynamic sublinear valuation rule admits a robust representation under a DUS, this representation need not be unique, since distinct DUSs may yield the same valuation rule: \[ \mathcal U^1\neq\mathcal U^2, \qquad \mathcal T^{\mathcal U^1} = \mathcal T^{\mathcal U^2}. \] We therefore identify the essential properties shared by all representing DUSs. Our characterization provides economically meaningful necessary and sufficient conditions for a DUS to represent the given dynamic sublinear valuation rule. This result shows that the valuation rule itself contains sufficient information to identify not merely a single latent uncertainty structure, but the entire class of uncertainty structures that reproduce it.
Finally, we turn to the practical recovery of uncertainty from limited valuation data. In empirical applications, a valuation rule is typically observed only through a restricted set of data. Under partial observation, the valuation rule consistent with the available data need not be uniquely determined. We identify the most conservative valuation rule consistent with the observations and develop nonparametric estimators for both this valuation rule and its underlying uncertainty structure. Even with limited valuation data, our estimators can reveal the latent uncertainty encoded in the observed valuations.
Our results provide a new perspective on the role of valuation in economics and finance. Valuation is not merely an outcome of uncertainty but also a source of information about the uncertainty structures that govern it. In this sense, our findings overturn the traditional Knightian view that uncertainty is inherently non-measurable knight1921risk. Indeed, valuation contains sufficient information to identify and characterize the underlying uncertainty, reveal its economically relevant components, and permit its recovery from data. This perspective provides a new framework for studying and quantifying latent uncertainty and opens a broad range of directions for future theoretical and empirical research in economic systems.
A substantial body of work in economics and finance has studied uncertainty through several closely related formulations, including multiple-prior models, rectangular belief systems, variational or entropy penalization, and admissible classes of model distortions; see, for example, hansen2001robust, chen2002ambiguity, anderson2003quartet, epstein2003recursive, maenhout2004robust, cheridito2006dynamic, hansen2006robust, maccheroni2006ambiguity, maccheroni2006dynamic, hansen2007beliefs, peng2007g, follmer2011stochastic, and epstein2013ambiguous. A related mathematical literature develops nonlinear expectations, quasi-sure analysis, and robust valuation under nondominated families of probability measures; see denis2006functional, nutz2012quasi, nutzsoner2012superhedging, nutz2013random, and neufeld2017nonlinear. Abstract representations of sublinear or convex semigroups on path space in terms of probability measures are studied in criens2025representation and criens2025stochastic. The construction of sublinear expectations on path space, together with the analysis of the conditioning and concatenation properties of uncertainty structures, is studied in nutz2013constructing.
The remainder of this paper is organized as follows. Section (ref) introduces two economic objects: dynamic sublinear valuation rules and robust valuations under uncertainty structures. Section (ref) shows that every dynamic sublinear valuation rule admits a representation as a robust valuation under uncertainty and provides an explicit procedure for constructing the associated uncertainty structure. Section (ref) introduces the notion of a time-consistent uncertainty structure and establishes its equivalence with time consistency of the associated robust valuation rule. Section (ref) characterizes the class of dynamic uncertainty structures that represent a given dynamic sublinear valuation rule. Section (ref) studies the recovery of uncertainty structures from partial observations of the valuation rule. Section (ref) concludes the paper. The proofs of all main results are provided in the appendix.
The present paper studies two economic objects: dynamic sublinear valuation rules and robust valuations under uncertainty structures. In this section, we introduce these two objects within a mathematically rigorous framework.
\paragraph*{Notation}
We begin by fixing the state space and the space of contingent payoffs. Let $D\subset\mathbb{R}^d$ be a convex open domain, possibly unbounded, which can be exhausted by bounded convex subdomains $D_m$ with smooth boundary satisfying $\overline D_m\subset D_{m+1}$ for all $m\ge1$. We consider contingent payoffs given by bounded continuous functions on $D$, so that the contingent payoff space is $C_b(D)$. We equip $C_b(D)$ with the mixed topology,\footnote{That is, the Mackey topology associated with the dual pair $(C_b(D),\mathcal M(D))$, where $\mathcal M(D)$ denotes the space of finite signed countably additive measures on $D$. Equivalently, it is the strongest locally convex topology on $C_b(D)$ whose continuous dual is $\mathcal M(D)$. This is the natural choice for the probabilistic duality used throughout the paper; see Appendix (ref) for details.} and, unless stated otherwise, all limits in $C_b(D)$ are understood with respect to this topology.
Within this framework, we now formulate dynamic sublinear valuation rules axiomatically, guided by the economic principles of monotonicity, stability, and time consistency.
The above definition collects the basic economic and analytic requirements of a dynamic sublinear valuation rule. Condition (ref) encodes sublinearity and monotonicity, capturing coherence and the absence of arbitrage. Conditions (ref), (ref), and (ref) impose stability: (ref) reflects the non-negativity of discounting, (ref) ensures monotone order regularity with respect to contingent claims, and (ref) provides temporal continuity. Finally, condition (ref) imposes time consistency through the semigroup property. It ensures that valuation over $t+s$ is obtained recursively by valuing first over $s$ and then over the remaining horizon $t$.
In this section, we introduce the concepts of uncertainty structures and their associated robust valuation rules. We begin by describing the underlying mathematical framework, following pinsky1995positive. Let $\hat D:=D\cup\{\triangle\}$ denote the cemetery-augmented state space, given by the one-point compactification of $D$, equipped with the Riemannian metric $\rho_D$. We consider the canonical path space $\hat\Omega$, consisting of continuous paths in $\hat D$ that are absorbed at $\triangle$ once they reach it, together with its Borel $\sigma$-field $\hat{\mathcal F}$ and canonical filtration $(\hat{\mathcal F}_t)_{t\ge0}$. The space $\hat\Omega$ is Polish under its natural topology, and its Borel $\sigma$-field is generated by the canonical filtration. We denote by $X$ the canonical process on $\hat\Omega$. The exit times are defined by \[ \tau_n(\omega):=\inf\{t>0:\omega(t)\notin D_n\}, \qquad \tau_{\mathrm{exp}}(\omega):=\lim_{n\to\infty}\tau_n(\omega). \] The cemetery state $\triangle$ represents explosion of the state process. Explosion means that the state process enters the absorbing terminal state, corresponding to irreversible exit from the feasible domain. In particular, once the process reaches $\triangle$, it remains there permanently and no further evolution takes place. From an economic perspective, this framework encompasses phenomena such as default, market exit, and structural regime change. Accordingly, uncertainty is characterized by a family of state-process laws that may admit explosion in finite time.
We consider two sources of uncertainty: uncertainty about discounting and uncertainty about the law of the underlying state process. Accordingly, a model in our framework is represented by a pair $(A,\mathbb Q)$, where \(A\) is a cumulative discounting process and \(\mathbb Q\) specifies a law of the canonical process \(X\). We introduce the corresponding pair space \(\mathfrak U\) below; its topology and measurable structure are provided in Appendix (ref).
For each $(t,x)\in[0,\infty)\times\hat D$, let $\mathcal U_{t,x}$ be a class of models, that is, $\mathcal U_{t,x}\subseteq\mathfrak U$. The class $\mathcal U_{t,x}$ represents the uncertainty at time $t$ when the state is $x$. A family of model classes $\mathcal U=\{\mathcal U_{t,x}\}_{(t,x)\in[0,\infty)\times\hat D}$ is called an uncertainty structure. We say that an uncertainty structure $\mathcal U=\{\mathcal U_{t,x}\}_{(t,x)\in[0,\infty)\times\hat D}$ is time-homogeneous if \[ \mathcal U_{t,x} = \mathcal U_{0,x}\circ\theta_t^{-1}, \qquad (t,x)\in[0,\infty)\times\hat D, \] where \(\theta_t:\hat\Omega\to\hat\Omega\) denotes the time-\(t\) shift operator defined by \[ (\theta_t\omega)(s):=\omega((s-t)\vee0), \qquad s\ge0. \] In the time-homogeneous case, we write \[ \mathcal U_x:=\mathcal U_{0,x}, \qquad x\in\hat D, \] and simply refer to \(\{\mathcal U_x\}_{x\in\hat D}\) as the uncertainty structure. The entire family \(\{\mathcal U_{t,x}\}_{(t,x)\in[0,\infty)\times\hat D}\) is then determined by \(\{\mathcal U_x\}_{x\in\hat D}\) through the time-shift operator.
In this section, we show that any dynamic sublinear valuation rule admits a representation as a robust valuation under uncertainty. More precisely, for any given dynamic valuation rule $\{\mathcal T_{t,T}\}_{0\le t\le T<\infty}$, we construct an uncertainty structure $\mathcal{U}$ such that $$\mathcal T_{t,T}=\mathcal T_{t,T}^\mathcal{U}\;\textnormal{ for all }\;0\le t\le T<\infty\,.$$ We emphasize that a dynamic sublinear valuation rule is defined purely axiomatically, with no reference to an underlying probability space or stochastic model.
Throughout the remainder of the paper, we restrict attention to the time-homogeneous case $\{\mathcal T_t\}_{t\ge0}$ unless stated otherwise. This entails no loss of generality, since any time-inhomogeneous setting can be reduced to a time-homogeneous one by enlarging the state space to incorporate time itself, namely, \[ \tilde X_t=(t,X_t), \qquad t\ge0. \] Accordingly, the time-homogeneous framework considered here also covers the time-inhomogeneous case.
The recovery of uncertainty from valuation proceeds in four steps. Figure (ref) illustrates the procedure. First, we extract the infinitesimal generator associated with the valuation rule $\{\mathcal T_t\}_{t\ge0}$ and describe its local behavior at each state $x\in D$ through a generating function $G$. Second, motivated by convex duality theory, we construct the support sets $\{A(x)\}_{x\in D}$ corresponding to the sublinear function $G(x,\cdot)$. Third, we construct an uncertainty structure $\mathcal U(G)$ from these support sets. Finally, we show that the robust valuation associated with the uncertainty structure $\mathcal U(G)$ coincides with the original valuation rule. The following subsections implement these steps in detail.
The first step in recovering uncertainty from valuation is to extract the infinitesimal generator, motivated by classical semigroup theory. This infinitesimal generator characterizes the local behavior of the valuation rule.
We restrict our attention to valuation rules whose infinitesimal dynamics are local. Economically, the following assumption means that prices are driven by local market information: the instantaneous change at state $x$ depends only on nearby variations in fundamentals and payoffs. Thus, the generator $\mathcal G$ is restricted to the continuous-path, diffusion-type regime and excludes genuinely nonlocal effects such as jumps, crashes, or discrete policy interventions. This is a limitation of the present analysis, not of the valuation-based framework. Treating nonlocal generators would require a corresponding inverse theory for jump-type dynamics and is left for future work.
Under Assumption (ref), the infinitesimal generator admits a local pointwise representation: for each \(x\in D\), the value \(\mathcal G[f](x)\) depends only on \(x\), \(f(x)\), \(\nabla f(x)\), and \(\nabla^2 f(x)\). The following theorem makes this statement precise and introduces the associated generating function. The proof is deferred to Appendix (ref).
This function $G$ plays a central role throughout the paper. It is a spatially local object determined by the valuation rule in a neighborhood of each point $x$, while the valuation rule itself is a global object determined by its behavior on the entire domain $D$.
We now introduce the parabolic comparison principle for generating functions. We say that a generating function $G$ satisfies the parabolic comparison principle if, for every $T>0$, whenever $v^+$ is a bounded viscosity supersolution and $v^-$ is a bounded viscosity subsolution of (ref), one has \[ v^+\ge v^- \qquad\text{on } [0,T)\times D. \] In recovering uncertainty from valuation, a central point is that the local valuation mechanism $G$ should uniquely determine the valuation rule, which is a global object. This uniqueness is ensured by the parabolic comparison principle.
The precise relationship between the valuation rule and the associated nonlinear PDE is given in the proposition below, with the proof postponed to Appendix (ref). We emphasize that this result is fully model-free: it relies solely on the economic axioms imposed on the valuation rule and does not require any probabilistic assumptions. When a particular model is specified, the PDE (ref) specializes to a Feynman--Kac-type equation. In particular, under the Black--Scholes specification, (ref) reduces precisely to the classical Black--Scholes pricing PDE.
The generating function $G$ yields an analytic description of the valuation rule through a nonlinear parabolic equation. For each payoff $f\in C_b(D)$, the valuation function $v(t,x):=\mathcal T_t f(x)$ satisfies (ref). While recovering uncertainty from valuation, this PDE representation is useful because it makes explicit how the local valuation mechanism $G$ determines the global evolution of the valuation function. In other words, it provides the analytic bridge from the infinitesimal object recovered from the valuation rule to the full dynamic valuation itself. Because smooth solutions need not exist in degenerate cases, and because within our axiomatic framework it is not known a priori whether the recovered generating function $G$ is degenerate or nondegenerate, we work entirely within the viscosity-solution framework.\footnote{See crandall1992user or crandall2000lp for the formal definition of viscosity solutions.}
We next introduce the support sets associated with a generating function $G$. For $V=(C,B,\Sigma)\in \mathbb R\times\mathbb R^d\times\mathbb S(d)$, let \[ L^V(r,p,X) := \frac12\operatorname{tr}(\Sigma X) +B\cdot p +Cr, \qquad (r,p,X)\in\mathbb R\times\mathbb R^d\times\mathbb S(d). \] For each $x\in D$, the support set of $G(x,\cdot)$ is defined as
Since the map $U\mapsto G(x,U)$ is sublinear, the classical dual representation theorem for sublinear functionals (see, e.g., rockafellar2015convex) implies that $A(x)$ is nonempty, compact, and convex. Moreover, $G$ admits the representation \[ G(x,U)=\sup_{V\in A(x)}L^V(U)\,. \]
We now pass from the support sets to a probabilistic uncertainty structure. A progressively measurable process \[ \beta=(C,B,\Sigma):[0,\infty)\times\hat{\Omega} \to (-\infty,0]\times\mathbb{R}^d\times\mathbb{S}^+(d) \] is called a coefficient field. A coefficient field $\beta$ is admissible if
Equivalently, admissibility is characterized by
where $L^\beta(t,\omega,\cdot): \mathbb{R}\times\mathbb{R}^d\times\mathbb{S}(d) \to \mathbb{R}$ denotes the linear functional associated with the coefficient field $\beta$, defined by \[ L^\beta(t,\omega,U) := \frac12\operatorname{tr}\bigl(\Sigma(t,\omega)X\bigr) + B(t,\omega)\cdot p + C(t,\omega)r, \qquad U=(r,p,X). \] Thus, admissibility means that the linear functional associated with $\beta$ is pointwise dominated by the generating function $G$. We write $\mathcal B_{\mathrm{ad}}(G)$ for the collection of all admissible coefficient fields. For a coefficient field $\beta=(C,B,\Sigma)$, the value $\beta(t,\omega)\in (-\infty,0]\times\mathbb{R}^d\times\mathbb{S}^+(d)$ at the time-path pair $(t,\omega)$ is called the local characteristic of $\beta$ at $(t,\omega)$. The support set $A(x)$ therefore represents the collection of all possible local characteristics of admissible coefficient fields at time-path pairs satisfying $\omega(t)=x$.
For any $x\in D$ and any admissible coefficient field $\beta=(C,B,\Sigma)$, we construct a cumulative discounting process and a class of laws for the underlying state process. The cumulative discounting process is determined by the $C$-component of $\beta$. Let $k:=-C$ and define \[ A_t^k:=\int_0^t k_s\,ds, \qquad t\ge0.\footnote{The integral is defined pathwise and therefore does not depend on any underlying probability measure.} \] Next, letting $\gamma:=(B,\Sigma)$, we define $\mathcal P_x(L^\gamma)$ as the collection of solutions to the generalized $L^\gamma$-martingale problem starting from $x$ (Remark (ref)), where
Each element \(\mathbb Q\in\mathcal P_x(L)\) represents a possible law of the underlying state process.
This construction leads to the definition of uncertainty structures. The family $\mathcal U_x(G)$ introduced below consists of pairs of a cumulative discounting process and a law for the underlying state process associated with admissible coefficient fields. Note that $\delta_{\triangle}$ denotes the Dirac measure concentrated on the constant path identically equal to the cemetery state $\triangle$.
We recall the definition of a solution to a generalized martingale problem. Let $L=L(t,\omega,p,X):[0,\infty)\times \hat\Omega\times\mathbb{R}^d\times\mathbb{S}(d)\to\mathbb{R}$ be a measurable function that is linear in $(p,X)$. A probability measure $\mathbb{Q}$ on the extended canonical path space $(\hat{\Omega},\hat{\mathcal{F}},(\hat{\mathcal F}_t)_{t\ge0})$ is called a solution to the generalized $L$-martingale problem starting from $x\in D$ if
A solution to the generalized $L$-martingale problem may fail to exist or may not be unique. We denote by $\mathcal P_x(L)$ the collection of all solutions starting from $x$. Refer to pinsky1995positive for further details.
For a given dynamic valuation rule $\{\mathcal T_t\}_{t\ge0}$, we have constructed the uncertainty structure $\mathcal U(G)$. It remains to show that the robust valuation rule under this uncertainty structure coincides with the original dynamic valuation rule. Establishing this equivalence completes the cycle
illustrated in Figure (ref).
The following Lyapunov condition provides a convenient sufficient criterion for completing this cycle. To ensure that the robust valuation rule under the uncertainty structure $\mathcal U(G)$ satisfies the stability axioms (ref) and (ref) of Definition (ref), we require each model class $\mathcal U_x(G)$ to be weakly compact. At a conceptual level, weak compactness provides control over the tail behavior of the corresponding state-process laws. We therefore impose a Lyapunov-type condition that guarantees this compactness property for the family $\mathcal U(G)$; see Proposition (ref). Such conditions are standard in the martingale-problem literature, broad enough for the economic applications considered here, and typically straightforward to verify.
The next theorem shows that, under the comparison principle and the Lyapunov condition above, the uncertainty structure $\mathcal U(G)$ generates a dynamic sublinear valuation rule whose infinitesimal generator is precisely the original generating function $G$. The proof is given in Appendix (ref). Recall that the robust valuation rule associated with the uncertainty structure $\mathcal U(G)$ is given by
for $(t,x)\in[0,\infty)\times D$ and $f\in C_b(D)$.
The next corollary provides a stochastic representation of dynamic sublinear valuation rules and constitutes one of the main results of this paper. It completes the cycle in (ref) by showing that the robust valuation rule under the uncertainty structure $\mathcal U(G)$ coincides with the original valuation rule. The proof is an immediate consequence of Proposition (ref) and Theorem (ref).
Consequently, this completes the first step of our uncertainty identification theory: under suitable conditions, every dynamic sublinear valuation rule admits a representation as a robust valuation under an uncertainty structure. Our construction identifies the latent models $(A,\mathbb Q)$ underlying the valuation rule by specifying the probabilistic laws governing both the discounting process and the underlying state process. In this way, the valuation rule itself reveals the latent uncertainty structure under which payoffs are evaluated.
A key insight of our uncertainty identification theory is that a global uncertainty structure can be recovered from local information encoded in a valuation rule. The generating function and its support set are local objects: their values at a state \(x\in D\) are determined by information in a neighborhood of \(x\). By contrast, uncertainty structures and robust valuation rules are global objects, since their values depend on the evolution of the state process over the entire domain \(D\). Corollary (ref) shows that piecing together these local objects extracted from a dynamic valuation rule recovers the latent uncertainty structure \(\mathcal U(G)\).
Time consistency ((ref) in Definition (ref)) is one of the fundamental properties of dynamic valuation rules. For a general uncertainty structure $\mathcal U$, however, the associated robust valuation rule $\mathcal T^{\mathcal U}$ need not be time-consistent. A natural question is therefore how time consistency of a valuation rule is reflected in the underlying uncertainty structure. We introduce the notion of a time-consistent uncertainty structure and show that it is equivalent to time consistency of the associated robust valuation rule.
To formulate this notion rigorously, we define the operations of conditioning and concatenation for models in $\mathfrak U$. Given a model $(A,\mathbb Q)$ and a stopping time $\tau$, the conditioned model $(A,\mathbb Q)^{\tau,\omega}$ represents the continuation model obtained after observing the history $\omega$ up to time $\tau(\omega)$: the state-law component is conditioned in the usual regular-conditional-probability sense, while the cumulative discounting process is reset after the conditioning time. Conversely, if $\nu:\hat\Omega\to\mathfrak U$ is a continuation kernel, the concatenated model $(A,\mathbb Q)\otimes_\tau \nu$ is obtained by following $(A,\mathbb Q)$ up to $\tau$ and then pasting the continuation model $\nu(\omega)$ after $\tau(\omega)$; the state-law component is pasted probabilistically, and the cumulative discounting component is pasted additively. The precise definitions are given in Appendix (ref).
We now introduce the notion of a dynamic uncertainty structure. A dynamic uncertainty structure \( \mathcal U=\{\mathcal U_{t,x}\}_{(t,x)\in[0,\infty)\times\hat D} \) possesses stability and recursive properties at the level of models, expressed through compactness, conditioning, and concatenation. The conditions in Definition (ref) are natural model-side counterparts of the axioms imposed on dynamic sublinear valuation rules. Condition (ref) imposes weak compactness and upper hemicontinuity of the model classes, mirroring the stability requirements underlying order continuity (ref) and temporal continuity (ref). Conditions (ref) and (ref) encode the recursive structure of uncertainty through conditioning and concatenation, thereby corresponding to the time-consistency axiom (ref). Thus, dynamic uncertainty structures provide a model-side formulation of the stability and time-consistency properties of dynamic sublinear valuation rules.
Proposition (ref) shows that the uncertainty structure $\mathcal U(G)$ is a dynamic uncertainty structure. Conditions (ref) and (ref) play a role analogous to the rectangularity and stability-under-conditioning-and-pasting conditions that appear in the literature on recursive multiple priors, dynamic risk measures, and sublinear expectations on path space; see, for example, epstein2003recursive, cheridito2006dynamic, and nutz2013constructing. Such conditions are known to provide the model-side mechanism for the dynamic programming principle, or tower property of nonlinear expectations. The next proposition shows that, in the present setting, this mechanism yields a dynamic sublinear valuation rule. The proof is deferred to Appendix (ref).
Combined with Corollary (ref), the following proposition implies that every dynamic sublinear valuation rule can be represented as a robust valuation under a DUS. The proof is given in Appendix (ref) and Appendix (ref).
Propositions (ref) and (ref), together with Corollary (ref), show that dynamic uncertainty structures provide the model-side counterpart of time consistency for robust valuation rules. On the one hand, under mild regularity conditions, a DUS induces a dynamic sublinear valuation rule and hence a time-consistent valuation rule. On the other hand, every dynamic sublinear valuation rule admits a robust valuation representation under a DUS, namely the uncertainty structure \(\mathcal U(G)\) recovered from its generating function $G$. Thus, DUSs are not merely a sufficient class of uncertainty structures for generating time-consistent robust valuations, but the natural model-side formulation of time consistency itself.
In this section, we characterize the class of DUSs that represent a given dynamic sublinear valuation rule. The preceding sections established that every dynamic sublinear valuation rule admits a representation as a robust valuation under a DUS. Such a representation, however, need not be unique. Indeed, distinct DUSs may induce the same valuation rule: \[ \mathcal U^1\neq\mathcal U^2, \qquad \mathcal T_t^{\mathcal U^1} = \mathcal T_t^{\mathcal U^2}. \] A natural question is therefore which DUSs represent a given valuation rule. We answer this question by providing an economically meaningful characterization of the class of all such DUSs.
The uncertainty structure \(\mathcal U(G)\) constructed in Section (ref) plays a central role in this characterization. The following theorem shows that \(\mathcal U(G)\) is maximal among all DUSs representing the given dynamic sublinear valuation rule. Consequently, \(\mathcal U(G)\) serves as an upper envelope for the class of all representing DUSs. For this reason, \(\mathcal U(G)\) may be interpreted as the largest, or most robust, DUS representing the given valuation rule. The proof of this theorem is given in Appendix (ref).
We now introduce subgradient sets and effective coefficient fields. For \(x\in D\) and \(U\in\mathbb R\times\mathbb R^d\times\mathbb S(d)\), we denote by \(\nabla G(x,U)\) the subgradient set of the sublinear map \(G(x,\cdot)\) at \(U\), that is, \[ \nabla G(x,U) := \Bigl\{ V\in A(x): L^V(U)=G(x,U) \Bigr\}. \] While the support set \(A(x)\) collects all local characteristics of admissible coefficient fields at time-path pairs \((t,\omega)\) with \(\omega(t)=x\), the subgradient set \(\nabla G(x,U)\) selects those characteristics for which the domination is binding at the jet \(U\). For $\varphi\in C_b^\infty(D)$, a coefficient field $\beta$ is said to be $\varphi$-effective if
We denote by $\mathcal B_{\mathrm{eff}}(G;\varphi)$ the collection of all $\varphi$-effective coefficient fields. In particular, every \(\beta\in\mathcal B_{\mathrm{eff}}(G;\varphi)\) satisfies the admissibility condition (ref) and the pointwise binding condition \[ L^\beta\bigl(t,\omega,\varphi(\omega(t)),\nabla\varphi(\omega(t)),\nabla^2\varphi(\omega(t))\bigr) = G\bigl(\omega(t),\varphi(\omega(t)),\nabla\varphi(\omega(t)),\nabla^2\varphi(\omega(t))\bigr) \] for all $(t,\omega)$ with $t<\tau_{\mathrm{exp}}(\omega)$. Economically, \(\varphi\) serves as a local test payoff, and the \(\varphi\)-effective coefficient fields are precisely those admissible coefficient fields that attain the generating function \(G\) along the jet of \(\varphi\).
Parallel to the construction in Section (ref), each $\beta=(-k,\gamma)\in\mathcal B_{\mathrm{eff}}(G;\varphi)$ determines a cumulative discounting process $A^k$ and a class $\mathcal P_x(L^\gamma)$ of laws for the underlying state process. This leads to the following definition.
We are now ready to state the main result of this section, which gives an if-and-only-if characterization of the DUSs representing a given dynamic sublinear valuation rule. The criterion consists of two conditions, (ref) and (ref) in Theorem (ref). Condition (ref) is an outer admissibility requirement inherited from Theorem (ref): every representing model must belong to the maximal uncertainty structure \(\mathcal U(G)\). Condition (ref) is an inner effectiveness requirement: for each smooth test payoff, the representing class must contain at least one model that is locally binding for that test. The proof is deferred to Appendix (ref).
Consequently, the generating function \(G\) identifies not only the dynamic valuation rule, but also the class of dynamic uncertainty structures that represent it. Recall from Proposition (ref) that \(G\) uniquely determines the global valuation rule through the associated parabolic equation (ref). Theorem (ref) goes further by showing that, under suitable additional conditions, \(G\) also determines which dynamic uncertainty structures represent the same valuation rule. More precisely, it provides an explicit characterization in terms of discounting--state-process-law pairs, which encode the local characteristics of uncertainty. Thus, \(G\) does not merely describe the local valuation mechanism. Its geometry reveals the local characteristics of uncertainty and characterizes the global dynamic uncertainty structures representing the valuation rule. In this sense, dynamic uncertainty structures can be viewed as the probabilistic shadow cast by the geometry of the generating function $G$.
This section studies the recovery of uncertainty structures from partial observations of the valuation rule. The preceding sections showed how to recover the uncertainty structure under full knowledge of the values \(\mathcal T_t f(x)\) for all payoffs \(f\in C_b(D)\), states \(x\in D\), and times \(t\ge0\). In practice, however, valuation data are available only for a restricted set of observable payoffs, states, and times. The central question is therefore whether the underlying uncertainty structure can still be recovered from such limited valuation information.
Throughout this section, fix \(T>0\), and let \(\mathcal K\subset C_b(D)\) denote the observable payoff set. For each \(f\in\mathcal K\), the valuation function $v^f:[0,T]\times D\to\mathbb R$ is defined by \[ v^f(t,x):=\mathcal T_t f(x)\,. \] We impose the following assumption on \(\mathcal K\).
Assumption (ref) is economically and structurally natural. Closedness of \(\mathcal K\) ensures stability of the observable payoff set under mixed-topology limits. Positive homogeneity reflects a basic implication of sublinearity: if a payoff is rescaled by \(c>0\), then its value is rescaled by the same factor. Thus, even when \(cf\) is not directly observed, it can be included in the observable class without loss of generality whenever \(f\) is observed.
Section (ref) considers the case in which the values \(v^f(t,x)\) are known for all observable payoffs \(f\in\mathcal K\) and all \((t,x)\in[0,T]\times D\). Section (ref) then turns to the finite-data setting, where the values \(v^f(t,x)\) are observed for all \(f\in\mathcal K\), but only at finitely many points \((t,x)\in[0,T]\times D\).
We study the recovery of uncertainty when the valuation functions \(\{v^f\}_{f\in\mathcal K}\) are known. Since the uncertainty structure is fully encoded in the generating function \(G\), our main objective is to determine how much of \(G\) can be recovered from this partial valuation information. To this end, we characterize the class of generating functions that are consistent with the observable valuation functions \(\{v^f\}_{f\in\mathcal K}\). We then identify pointwise lower and upper bounds for this class. Among all consistent generating functions, we single out a canonical choice, namely the pointwise largest one. This generating function corresponds to the most conservative valuation rule consistent with the observable valuation data.
We begin by introducing two envelopes, \(\overline G\) and \(\underline G\), motivated by the viscosity inequalities. Let \(v:[0,T]\times D\to\mathbb R\) be a continuous function. The parabolic second-order subjet of \(v\) at \((t,x)\in(0,T]\times D\) is defined by
In the definition above, local minima are taken with respect to the backward parabolic topology, that is, \(v-\varphi\) attains its minimum in a neighborhood of \((t,x)\) of the form \[ \mathcal C_r^-(t,x):=(t-r,t]\times B_r(x). \] The parabolic second-order superjet is defined by \[ \mathcal J^{2,+}v(t,x):=-\mathcal J^{2,-}(-v)(t,x). \]
For \(x\in D\) and \(U=(r,p,X)\in\mathbb R\times\mathbb R^d\times\mathbb S(d)\), define the upper and lower jet-derivative sets by
where
The set \(\mathcal Z(U)\) is included to enforce the structural properties (ref) and (ref), which are necessary requirements for a function to be the generating function of a dynamic sublinear valuation rule.\footnote{Conditions (ref) and (ref) are equivalent to \(G(x,1,0,0)\le0\) and \(G(x,0,0,X)\le0\) for \(X\le0\), respectively.} We define the envelopes \(\overline G\) and \(\underline G\) by
with the conventions \(\inf\varnothing=+\infty\) and \(\sup\varnothing=-\infty\). These envelopes are determined solely by the observable valuation functions.
The envelopes \(\overline G\) and \(\underline G\) characterize the pointwise upper and lower bounds of all generating functions consistent with the observable valuation functions. Indeed, if \(v^f\) is a viscosity supersolution of (ref), then every \(q\in\mathcal D^-_{\mathcal K}(x,U)\) must satisfy \(q\ge G(x,U)\). Similarly, if \(v^f\) is a viscosity subsolution, then every \(q\in\mathcal D^+_{\mathcal K}(x,U)\) must satisfy \(q\le G(x,U)\). Thus, by Theorems (ref) and (ref), any generating function consistent with the observable valuation functions must satisfy
The next theorem shows that these inequalities are the tightest possible pointwise bounds imposed by the observable valuation functions. It is worth noting that the envelopes \(\overline G(x,\cdot)\) and \(\underline G(x,\cdot)\) themselves need not be sublinear. The proof is provided in Appendix (ref).
Although Theorem (ref) is stated in terms of generating functions, its implications go beyond the identification of \(G\). As discussed in Section (ref), a generating function completely characterizes the class of dynamic uncertainty structures representing the corresponding dynamic sublinear valuation rule. Thus, the theorem does not merely describe the set of generating functions consistent with the observable valuation data. It also characterizes the dynamic uncertainty structures that remain consistent with those observations.
We now construct the largest generating function consistent with the observable valuation functions. The construction relies on the dual representation of sublinear functions and depends only on the upper envelope \(\overline G\). The proof is provided in Appendix (ref).
The next corollary is one of the main results of this section. The function \(G_{\max}\) and its support set \(A_{\max}\) generate a dynamic uncertainty structure and a dynamic valuation rule through the procedure \[ G_{\max} \;\longrightarrow\; A_{\max}(\cdot) \;\longrightarrow\; \mathcal U(G_{\max}) \;\longrightarrow\; \{\mathcal T_t^{\max}\}_{t\ge0}, \] through the recovery chain presented in (ref). The corollary shows that \(\{\mathcal T_t^{\max}\}_{t\ge0}\) is the largest dynamic sublinear valuation rule consistent with the observable valuation functions. It also shows that \(\mathcal U(G_{\max})\) is maximal among all dynamic uncertainty structures representing this valuation rule.
In the previous section, we identified the largest generating function \(G_{\max}\) and the corresponding support sets \(A_{\max}(\cdot)\), which generate the largest valuation rule \(\{\mathcal T_t^{\max}\}_{t\ge0}\) consistent with the observable valuation functions \(\{v^f\}_{f\in\mathcal K}\) on \([0,T]\times D\). In practice, however, the observable valuation functions \(v^f\) are not available on the entire domain; rather, their values are sampled only at finitely many points in \([0,T]\times D\). The objective of this section is to construct finite-sample approximations of \(G_{\max}\) and \(A_{\max}(\cdot)\), derive the corresponding valuation rule, and estimate the discrepancy between this approximate valuation rule and the original largest valuation rule \(\{\mathcal T_t^{\max}\}_{t\ge0}\).
Recall that $D\subset\mathbb R^d$ is a convex open domain, possibly unbounded, which can be exhausted by bounded convex subdomains $D_m$ with smooth boundary satisfying $\overline D_m\subset D_{m+1}$ for all $m\ge1$. On each truncated domain \([0,T]\times D_m\), sample data are available only on a rectangular grid \[ \mathcal I_{m,n}:=\mathbb T_n\times\Gamma_{m,n}\,,\quad n\ge1, \] where \[ \mathbb T_n=\{t_1^n,\dots,t_{N_n}^n\}\subset(0,T]\,,\quad \Gamma_{m,n}\subset D_m \] are finite temporal and spatial grids, respectively. For fixed $m$, the stage-$n$ sample data are given by $\{v^f(t,x)\}_{f\in\mathcal K,\,(t,x)\in\mathcal I_{m,n}}$ for the observable payoff set $\mathcal{K}$. Let $ P_{m,n}:=\operatorname{conv}(\Gamma_{m,n})$ be the convex hull of the spatial grid \(\Gamma_{m,n}\). A triangulation of \(P_{m,n}\) with vertices in \(\Gamma_{m,n}\) is a family \(\mathfrak S\) of \(d\)-simplices of the form \[ S=\operatorname{conv}\{x_0,\ldots,x_d\}\,,\;\quad x_0,\ldots,x_d\in\Gamma_{m,n}, \] such that \[ P_{m,n}=\bigcup_{S\in\mathfrak S}S \] and the intersection of any two simplices in \(\mathfrak S\) is either empty or a common face of both. We denote by $\operatorname{Tri}(P_{m,n};\Gamma_{m,n})$ the collection of all such triangulations. Define \[ \Delta_{m,n} := \inf_{\mathfrak S\in \operatorname{Tri}(P_{m,n};\Gamma_{m,n})} \max_{S\in\mathfrak S}\operatorname{diam}(S), \] with the convention \(\inf\varnothing:=\infty\).
We measure the mesh size of \(\mathcal I_{m,n}\) on \([0,T]\times\overline D_m\) by \[ \|\mathcal I_{m,n}\|_{T,m} := \max\left\{ d_H(\mathbb T_n,[0,T]), d_H(P_{m,n},\overline D_m), \Delta_{m,n} \right\}, \] where \(d_H\) denotes the Hausdorff distance. Throughout this section, we work in the regime \[ \|\mathcal I_{m,n}\|_{T,m}\to0 \qquad \text{as } n\to\infty \] for each fixed \(m\ge1\). This condition means that the temporal grid, the spatial convex hull, and the spatial triangulation become increasingly fine. In particular, \(\bigcup_{n\ge1}\mathcal I_{m,n}\) is dense in \([0,T]\times\overline D_m\); intuitively, the rectangular grids asymptotically fill the truncated domain.
We now impose a regularity condition on the family of valuation functions $\{v^f\}_{f\in\mathcal K,\,\|f\|_\infty=1}$. The following assumption requires only equi-H\"older continuity, which is significantly weaker than smoothness. It plays a key role in the proofs of Theorems (ref) and (ref).
We now approximate the lower jet-derivative set $\mathcal D^-_{\mathcal K}(x,U)$ using the sample data $\{v^f(t,x)\}_{f\in\mathcal K,\,(t,x)\in\mathcal I_{m,n}}$. Suppose that $G_{\max}$ is finite and continuous. For $R>0$, define
Since $\overline D_m$ is compact and $G_{\max}(x,\cdot\,)$ is sublinear, there exists $N_m>0$ such that $A_{\max}(x)\subseteq\mathbb B_{N_m}'$ for all $x\in\overline D_m$. Fix $\beta_m\in(0,\alpha_m/2)$ and $\delta_m\in(0,\min\{\alpha_m-2\beta_m,\alpha_m\beta_m\})$, and define \[ \varepsilon_{m,n}:=\|\mathcal I_{m,n}\|_{T,m}^{\beta_m}\,, \quad R_{m,n}:=\|\mathcal I_{m,n}\|_{T,m}^{-\delta_m}. \] For each $m\ge1$, let $(\eta_{m,n})_{n\ge1}$ be a sequence of positive numbers such that $\eta_{m,n}\downarrow0$ as $n\to\infty$ and
For instance, one may take $\eta_{m,n}=\|\mathcal I_{m,n}\|_{T,m}^{\theta_m}$ with $2\beta_m<\theta_m<\alpha_m-\delta_m$. For $x\in\Gamma_{m,n}$, $U=(r,p,X)\in \mathbb R\times\mathbb R^d\times\mathbb S(d)$, and $\ell>0$, define \[ \mathcal D_{\mathcal K,m,n}^{-,\ell}(x,U) := \left\{ a\in\mathbb R:
\right\} \cup \mathcal Z(U), \] where $$P^{(a,p,X)}(s,z;t,y):=a(s-t) +p\cdot(z-y)+\frac12(z-y)^\top X(z-y)\,.$$
This set is a data-driven approximation of the lower jet-derivative set $\mathcal D^-_{\mathcal K}(x,U)$ based on the sample data $\{v^f(t,x)\}_{f\in\mathcal K,\,(t,x)\in\mathcal I_{m,n}}$. The point $y\in\Gamma_{m,n}\cap B_\ell(x)$ serves as a grid-based proxy for the state $x$, allowing a spatial tolerance of radius $\ell$. The restriction $\|f\|_\infty\le R_{m,n}$ limits attention to bounded observable payoffs. Since $R_{m,n}\to\infty$ as $n\to\infty$ for each fixed $m\ge1$, this restriction becomes asymptotically negligible and eventually recovers the full observable class $\mathcal K$. In the definition of $\mathcal D^-_{\mathcal K}(x,U)$, the condition that $v^f-\varphi\ge0$ in a neighborhood of $(t,x)$, for a test function $\varphi\in C_b^\infty((0,T)\times D)$, is approximated here by the discrete inequality \[ v^f(s,z)\ge v^f(t,y)+P^{(a,p,X)}(s,z;t,y)-\eta_{m,n} \quad \text{for all }(s,z)\in\mathcal I_{m,n}\cap\mathcal C_{\varepsilon_{m,n}}^-(t,y)\,. \] The quadratic polynomial $P^{(a,p,X)}(\cdot,\cdot;t,y)$ replaces the smooth test function $\varphi$, making the construction tractable in practice. The parameter $\eta_{m,n}$ serves as a tolerance level, allowing errors of size $\eta_{m,n}$ both in the discrete inequality and in the anchoring condition: the exact identity $v^f(t,x)=r$ is relaxed to $|v^f(t,y)-r|\le \eta_{m,n}$.
We now construct estimators for the support set $A_{\max}$ and the generating function $G_{\max}$ using the approximation $\mathcal D_{\mathcal K,m,n}^{-,\ell}(x,U)$ of the lower jet-derivative set. Choose \((\lambda_{m,\ell})_{\ell>0}\) such that \[ \lambda_{m,\ell}\downarrow0, \qquad \omega_m(\ell)=o(\lambda_{m,\ell}) \quad\text{as }\ell\downarrow0, \] where $\omega_m:[0,\infty)\to[0,\infty)$ denotes the modulus of continuity of $G_{\max}$ on \(\overline D_m\), defined by \[ \omega_m(\ell) := \sup_{\substack{x,y\in\overline D_m,\ |x-y|\le \ell\\ U\in\mathbb B_1}} \left| G_{\max}(x,U)-G_{\max}(y,U) \right|. \] For $x\in \Gamma_{m,n}$, define the support-set estimator by
and the generating-function estimator by
We next extend these estimators from $\Gamma_{m,n}$ to $\overline D_m$. Fix a triangulation $\mathfrak S_{m,n}\in \operatorname{Tri}(P_{m,n};\Gamma_{m,n})$ such that $\max_{S\in\mathfrak S_{m,n}}\operatorname{diam}(S)\le 2\|\mathcal I_{m,n}\|_{T,m}$. For $x\in\overline D_m$, let $y$ be the projection of $x$ onto the convex set $P_{m,n}$, and choose a simplex $S\in\mathfrak S_{m,n}$ containing $y$. Let \((\mu_0,\ldots,\mu_d)\) be the barycentric coordinates of $y$ relative to $S$.\footnote{For \(y\in S:=\operatorname{conv}\{x_0,\ldots,x_d\}\), the barycentric coordinates of \(y\) relative to $S$ are a tuple \((\mu_0,\ldots,\mu_d)\in \mathbb{R}^{d+1}\) such that \[ y=\sum_{i=0}^d \mu_i x_i,\qquad \mu_i\ge 0,\quad \sum_{i=0}^d \mu_i=1. \] } We define \[ G_{\max,m,n}^{\ell}(x,U):=\sum_{i=0}^d \mu_i\,G_{\max,m,n}^{\ell}(x_i,U) \quad\mbox{and}\quad A_{\max,m,n}^{\ell}(x):=\sum_{i=0}^d \mu_i\,A_{\max,m,n}^{\ell}(x_i), \] where the latter denotes the Minkowski convex combination. These extensions agree with the original estimators on $\Gamma_{m,n}$ and preserve the dual relation \[ G_{\max,m,n}^{\ell}(x,U) = \sup_{V\in A_{\max,m,n}^{\ell}(x)}L^V(U), \qquad (x,U)\in\overline D_m\times\mathbb R\times\mathbb R^d\times\mathbb S(d). \]
The next theorem establishes the convergence of these estimators. While the largest generating function $G_{\max}$ and its support set $A_{\max}$ are constructed from valuation functions defined on the entire domain $[0,T]\times D$, the estimators $G_{\max,m,n}^{\ell}$ and $A_{\max,m,n}^{\ell}$ are constructed only from the sample data observed on $\mathcal I_{m,n}$. The theorem shows that, as $n\to\infty$ and $\ell\downarrow0$, these estimators converge to $G_{\max}$ and $A_{\max}$, respectively. The detailed proof is deferred to Appendix (ref).
We emphasize that the method is genuinely nonparametric. Starting from discrete observations of valuation functions, it recovers the maximal support sets $A_{\max}(x)$ and the associated maximal generating function $G_{\max}$ without imposing any parametric specification on the valuation mechanism or on the underlying uncertainty structure.
We now develop a procedure for recovering the uncertainty structure and its associated valuation rule from sample data. We first construct an approximate valuation rule and then establish its convergence to the largest valuation rule \(\{\mathcal T_t^{\max}\}_{t\ge0}\). Let \[ \tau_m := \inf\{t \ge 0 : X_t \notin D_m\} \] denote the first exit time from $D_m$. Since \(G_{\max,m,n}^{\ell}\) and \(A_{\max,m,n}^{\ell}\) are defined on \(\overline{D}_m\), we consider coefficient fields stopped at \(\tau_m\). More precisely, let \(\mathcal B_{\mathrm{ad}}^{m}(G_{\max,m,n}^{\ell})\) be the collection of progressively measurable coefficient fields \[ \beta = (-k,\gamma) : [0,\infty)\times\hat\Omega \to (-\infty,0]\times\mathbb R^d\times\mathbb S^+(d) \] satisfying \[
\] For \(x \in D_m\), define \[ \mathcal U_x^{m}(G_{\max,m,n}^{\ell}) := \left\{ (A^k,\mathbb Q)\in\mathfrak U : (-k,\gamma)\in \mathcal B_{\mathrm{ad}}^{m}(G_{\max,m,n}^{\ell}) \ \text{and}\ \mathbb Q \in \mathcal P_x(L^\gamma) \right\}, \] where \(A_t^k := \int_0^t k_s\,ds\) and \(\mathcal P_x(L^\gamma)\) denotes the class of laws solving the generalized \(L^\gamma\)-martingale problem. For each \(n \ge 1\) and \(\ell > 0\), define the $D_m$-truncated robust valuation rule generated by \(G_{\max,m,n}^{\ell}\) by
for \((t,x)\in [0,\infty)\times D_m\).
The next theorem shows that the $D_m$-truncated robust valuation rule \(\{\mathcal T_t^{\max,m,n,\ell}\}_{t\ge0}\) converges to \(\{\mathcal T_t^{\max}\}_{t\ge0}\) as \(n\to\infty\), \(\ell\downarrow0\), and \(m\to\infty\). While Theorem (ref) establishes the convergence of the estimators of the largest generating function and its support set, Theorem (ref) extends this convergence result to the corresponding largest dynamic valuation rule. The proof is deferred to Appendix (ref). We recall the Lyapunov pair \((C,\phi)\) from Assumption (ref).
Recall from Theorem (ref) that the largest generating function and its support set can be fully recovered on \(D_m\) from sample data collected on \(D_m\) as \(n\to\infty\) and \(\ell\downarrow0\). This recovery result, however, does not extend directly to the largest valuation rule. The generating function and its support set are local objects: their values at a state \(x\) are determined by information in a neighborhood of \(x\), so data restricted to \(D_m\) suffice to recover them on \(D_m\). By contrast, a valuation rule is a global object: the value it assigns at a state \(x\) depends on the evolution of the state process over the entire domain \(D\). Consequently, data from a fixed subdomain \(D_m\) are insufficient to fully recover the largest valuation rule on \(D\).
Nevertheless, Theorem (ref) shows that the error between the largest valuation rule and its approximation based on sample data from \(D_m\) is controlled and vanishes as \(n\to\infty\), \(\ell\downarrow0\), and \(m\to\infty\). The theorem also provides an explicit error bound in terms of the Lyapunov pair \((C,\phi)\). Thus, sufficiently rich data on \(D\) allow the largest valuation rule to be approximated accurately. In this sense, our method provides a nonparametric procedure for recovering continuous-time dynamic valuation from finite-sample data. The resulting estimators recover not only the largest generating function and its support set, but also the associated largest valuation rule.
This paper develops a unified framework connecting dynamic sublinear valuation rules with robust valuation under uncertainty and makes four main contributions. First, we show that every dynamic sublinear valuation rule admits a representation as a robust valuation under uncertainty and provide an explicit procedure for identifying the underlying uncertainty structure from the valuation rule. Second, we introduce the notion of a dynamic uncertainty structure (DUS) as the model-side counterpart of time consistency in valuation. Third, we characterize the entire class of DUSs that represent a given valuation rule. Finally, we develop nonparametric estimators for recovering uncertainty from limited valuation data and establish their convergence. Taken together, these results show that valuation contains sufficient information to identify, characterize, and statistically recover the uncertainty structures underlying it.
Several directions remain for future research. One natural extension is to move beyond the Markovian and sublinear settings by considering path-dependent models and more general convex valuation rules. It would also be valuable to examine whether uncertainty structures can be recovered when valuation data are noisy, incomplete, or available only over restricted time intervals. Another direction is to adapt and apply the present framework to empirical asset pricing and dynamic decision problems. Observable valuations may reveal economically meaningful information about latent beliefs, market frictions, and ambiguity. We hope that the framework developed in this paper provides a useful foundation for these theoretical extensions and empirical applications.