Pricing-Sufficient Filtrations

• Pricing-sufficient filtrations are the minimal information flows that enable asset prices to be derived as conditional expectations under unique martingale measures.
• They are constructed via rigorous methods such as martingale representation and optional projection, ensuring minimality, uniqueness, and stability across discrete, continuous, and categorical markets.
• Applications include market microstructure models and reduced filtrations in jump-diffusion markets, highlighting their role in enforcing no-arbitrage pricing and dynamic completeness.

A pricing-sufficient filtration is a minimal information structure with respect to which all economically relevant (arbitrage-free, replication-consistent) prices for a prescribed class of assets can be derived as conditional expectations under (typically unique) martingale measures. The concept encodes the principle that market prices depend not on the entirety of background randomness, but only on the flow of information necessary to enforce the no-arbitrage pricing and dynamic completeness properties within a given market or submarket. Pricing-sufficiency has been rigorously formalized across discrete, continuous, and categorical market models, including cases with endogenous, reduced, and generalized filtrations.

1. Definition and Characterization of Pricing-Sufficient Filtrations

A filtration $\mathbb{G}$ (or, in categorical settings, a functor $F:T\to \mathbf{Prob}$) is called pricing-sufficient for a given set of asset price processes $S=(S^1,\dots,S^n)$ if, for some equivalent (local) martingale measure $\mathbb{Q} \sim \mathbb{P}$, $S^i$ is a (local) $\mathbb{Q}$-martingale with respect to $\mathbb{G}$ for all $i$, and $\mathbb{G}$ is minimal among all such filtrations, unique up to null sets (Dominguez, 18 Jan 2026, Adachi et al., 2020, Adachi et al., 2019). For discrete models, $G$ is pricing-sufficient with respect to a risk-neutral measure $Q$ if and only if prices conditional on $G$ coincide $Q$-a.s. with those conditional on the full filtration at each time and for every (bounded) contingent claim (Adachi et al., 2020).

Formally, for a price process $S$, the canonical pricing-sufficient filtration is

$$\mathcal{F}^S_t = \bigcap_{u>t} \sigma\{ S^i_s : 0 \leq s \leq u,\, i=1,\dots,n \} \vee \mathcal{N}$$

where $\mathcal{N}$ is the null sets of the reference filtration, and the right-continuous completed version is taken (Dominguez, 18 Jan 2026).

2. Minimality, Uniqueness, and Stability Properties

Pricing-sufficient filtrations exhibit the following structural properties (Dominguez, 18 Jan 2026, Adachi et al., 2020):

• Minimality: No strictly smaller filtration retains the martingale property for the designated assets under any equivalent measure.
• Uniqueness (up to null sets): If two filtrations both satisfy pricing sufficiency for the same collection of assets, they coincide up to null sets.
• Stability under restriction: If $B \subseteq A$ (subsets of assets), $\mathcal{F}^{S^B} \subseteq \mathcal{F}^{S^A}$.
• Stability under aggregation: If $S^A$ and $S^B$ can be priced under a common measure and filtration, so can $S^{A\cup B}$; their aggregated natural filtration remains pricing-sufficient.

These properties ensure that the pricing-sufficient filtration encodes only information essential for asset pricing, generalizing the concept of the natural filtration generated by price processes.

3. Construction in Endogenous and Reduced Filtration Settings

In models with endogenous information or filtration reduction, pricing-sufficient filtrations arise naturally:

• Market microstructure models: In the Glosten–Milgrom paradigm, trade flows reveal only partial information about the (unobservable) true asset value. The pricing-sufficient filtration is generated by the process of actual buys and sells—specifically, by the $\sigma$-algebra generated by executed trade times on each side (Kühn et al., 2012). The market maker’s filter for latent value is then updated via a pure jump SDE driven by executed trade counting processes.
• Jump-diffusion and incomplete markets: Pricing-sufficient filtrations can be constructed by filtration reduction: projecting price processes onto a reduced subfiltration containing only those sources of risk the trader chooses to monitor or hedge. The trader then prices claims using the unique equivalent martingale measure on the projected market, with risk premia assigned only for the observed risk factors. The original market measure is recovered via a consistent uplift, assigning zero price to neglected risks (Grigorian et al., 2023).

Setting	Filtration Construction	Pricing Property
Market microstructure	Generated by executed trade times	Prices as conditional expectations
Incomplete jump models	Reduced to hedged sources; projected dynamics	Unique EMM on reduced filtration
General semimartingale	Minimal filtration generated by prices	Arbitrage-free pricing local to assets

4. Categorical, Functional, and Generalized Perspectives

In categorical binomial pricing models, a generalized filtration is formulated as a functor from a time-index category to the category of probability spaces and null-preserving maps (Adachi et al., 2019, Adachi et al., 2020). Here, pricing sufficiency requires that for every claim, the conditional price computed under the subfiltration coincides (almost surely) with that computed under the full filtration. Characterization theorems state that this holds if and only if the subfiltration is, up to null sets, as fine as the original at each time, i.e., collapse of states that matter for pricing invalidates sufficiency. Classic and coarsened filtrations in the binomial model illustrate when this property holds or fails.

5. Two-Filtration and Pricing with Partial Information

In large or continuous-time markets, asset price observability may be separated from trading information. The two-filtration approach demonstrates that the smaller (observable) filtration suffices for arbitrage-free pricing and super-replication as long as trading strategies are based on the observed filtration and prices are projected accordingly via optional projection (Cuchiero et al., 2017). This is a strong and general form of pricing-sufficiency: all economically meaningful notions of arbitrage and replication can be formulated and solved in the smaller filtration, with prices calculated as conditional expectations under suitably constructed martingale measures.

6. Pricing-Sufficiency in Markets with Enlarged or Insider Filtrations

In information-asymmetric models, such as credit risk or insider options frameworks, pricing-sufficiency is realized as the smallest filtration necessary to render the value process Markovian or to accommodate additional information (e.g., default thresholds, global extrema) (Hillairet et al., 2010, Gapeev et al., 4 Jul 2025). In these settings:

• For defaultable assets, the manager’s filtration (enlarged with the default barrier) is sufficient for full-information pricing; investor filtrations are not sufficient unless the full barrier information is present.
• For lookback or American options with respect to global extrema, the progressively enlarged filtration with the corresponding "honest time" yields a Markov process in extended state and a tractable free-boundary valuation—demonstrating that this specific enlargement is both necessary and sufficient for exact pricing.

7. Limitations: Non-Existence and Compatibility Barriers

Despite their local sufficiency, pricing-sufficient filtrations do not always extend globally in complex markets. Notably, when three or more independent, unspanned drivers are present, it can be proven that there is, in general, no single admissible (pricing-sufficient) filtration supporting a global equivalent local martingale measure for all assets simultaneously (Dominguez, 18 Jan 2026). This failure is a sharp manifestation of incomplete dynamic completeness and highlights the fundamental limitation of extending local pricing sufficiency to a universal information structure across all submarkets. Numerical diagnostics via Doob–Meyer decompositions in discrete-time reinforce this incompatibility, showing predictable drift components where sufficiency fails.

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In summary, pricing-sufficient filtrations constitute the minimal, necessary, and often unique information flows for arbitrage-free, replication-consistent asset pricing. Their construction provides both a theoretical and practical framework for modeling information and pricing in continuous, discrete, endogenous, reduced, categorical, and asymmetric-information financial markets. Their sufficiency, uniqueness, and limitations are established through a combination of martingale representation, optional projection, measure change, and structural filtration analysis (Kühn et al., 2012, Grigorian et al., 2023, Dominguez, 18 Jan 2026, Adachi et al., 2019, Adachi et al., 2020, Cuchiero et al., 2017, Hillairet et al., 2010, Gapeev et al., 4 Jul 2025).