Joint identification of physical and pricing laws

FI-01 |  Asset pricing and identification

Research agenda formalization

Definitions and assumptions

Consider a one-period market on a standard Borel state space $\Omega$. The physical law is $P$, and a pricing law $Q$ is equivalent to $P$. Normalize the initial bond price to one and its terminal payoff to a known deterministic $R_f>0$. For bounded traded payoffs $H_j:\Omega\to\R$, observed initial prices $\pi_j$ satisfy \[ \E_Q[H_j]=R_f\pi_j,\qquad j\in J. \] Historical observations identify the population law $P\circ O^{-1}$ of an observable state map $O:\Omega\to\mathcal X$. This is a population identification formulation; a finite sample does not itself reveal that law exactly. Let $\mathcal C$ specify admissible model pairs, including structural and support restrictions. For data $D=(F_O,(\pi_j)_{j\in J})$, define \[ \mathcal I(D)=\{(P,Q)\in\mathcal C:Q\sim P,\ P\circ O^{-1}=F_O, \ \E_QH_j=R_f\pi_j\ \forall j\in J\}. \] Two pairs are observationally equivalent if they generate the same $D$.

Formal question

Characterize conditions on the observable map, payoff family and model class for $\mathcal I(D)$ to be a singleton, or for specified functionals of $(P,Q)$ to be constant on it. When both laws are identified, determine when the likelihood ratio $dQ/dP$ is identified up to $P$-null sets. When identification fails, characterize the sharp identified set and economically useful risk-premium functionals that remain identified. Develop estimation guarantees under an explicitly stated sampling process and completeness or regularity assumptions.

Scope and status

Without restrictions, neither unobserved physical-state probabilities nor pricing probabilities orthogonal to the traded payoff information can be recovered. Conversely, observing the full physical state law and a measure-determining family of correctly priced claims can identify both laws. The issue is to state informative intermediate conditions, rather than to claim an unconditional identification theorem. This entry formalizes a research frontier in the source, not a named conjecture.

References for the source of the problem

Zhang Chen and Chen Kay, Historical Developments in Probability Measures for Asset Pricing: From State Prices to Modern Pricing Kernels, arXiv:2605.27658 (2026), Section 20, "Open Problems and Directions Toward 2026," pp. 23--24; definitions of physical and pricing measures are developed earlier in the review. https://arxiv.org/pdf/2605.27658.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.