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arXiv:2601.14355

Mathematical Foundations of Quantum Pricing Theory

Tian Xin, Liang Aoqin

Published Jan 20, 2026• 0 citations
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Abstract

Let $M$ be a von Neumann algebra and let $(N_t)_{t\in[0,T]}$ be an increasing family of abelian von Neumann subalgebras encoding a (classical) information flow. Fix a faithful normal state $φ_ρ$ and a filtration of normal $φ_ρ$-preserving conditional expectations $E_t:M\to N_t$ satisfying the tower property. Using bounded functional-calculus cutoffs $f_n$, we introduce a truncation-stable notion of localized $(N_t,E_t)$-martingales for affiliated self-adjoint observables, and formulate a \emph{Local Informational Efficiency Principle} requiring symmetrically discounted traded prices to be martingales in this sense. Assuming a pricing state $φ^\star$ and a compatible family of normal $φ^\star$-preserving conditional expectations $(E_t^\star)$, we define for bounded terminal payoffs $X\in M_T$ the dynamic pricing operator \[ Π_t(X):=B_t^{1/2}\,E_t^\star\!\bigl(B_T^{-1/2}XB_T^{-1/2}\bigr)\,B_t^{1/2}, \] where $(B_t)$ is a strictly positive numéraire adapted to $(N_t)$. We prove that $(Π_t)$ is normal, completely positive, $N_t$-bimodular, and time-consistent, and satisfies $Π_t(\mathbf 1)=B_t$ (equivalently, $\widetildeΠ_t(X):=B_t^{-1/2}Π_t(X)B_t^{-1/2}$ is unital). In the commutative reduction it agrees with risk-neutral valuation by conditional expectation. Finally, we develop an $L^2(M,φ_ρ)$ prediction theory and introduce an operator-valued Fisher information relative to $(N_t)$, obtaining a noncommutative Cramér--Rao lower bound for conditional mean-square prediction error; we compute the bound for compound Poisson lattice-jump models under $\sum_αγ_α(e^{αΔx}-1)=r$.

DOI
10.48550/arxiv.2601.14355

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Given a normalized state-vector ψψψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψψψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A){\mathcal{C}(A)}C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A){\mathcal{C}(A)}C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” A‾{\underline A}A​ of A, which is a commutative Rickart C*-algebra in the topos of functors from CA{\mathcal{C}A}CA to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A‾{\underline A}A​ for A = B(H).
In quantum mechanics, many observables for a quantum system are represented as self-adjoint linear operators (possibly unbounded) on a complex Hilbert space H\mathcal{H}H. In this paper, we introduce a binary relation ⊥\perp⊥ on S(H)S(\mathcal{H})S(H) of all self-adjoint linear operators (including unbounded) on H\mathcal{H}H. We obtain that A⊥BA\perp BA⊥B implies that A,BA,BA,B are affiliated with some abelian von Neumann algebra at the same time. Depending on ⊥\perp⊥,S(H), S(\mathcal{H}),S(H) can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order ⪯\preceq⪯ and the traditional order ≤\leq≤ on S(H)S(\mathcal{H})S(H) and give some properties about these two orders. We prove that for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H), A⪯BA\preceq BA⪯B if and only if AAA has a value in Δ\DeltaΔ implies that BBB has a value in Δ\DeltaΔ for every Borel set Δ\DeltaΔ not containing 000. Moreover, we study the existence of the infimum A∧BA\wedge BA∧B and supremum A∨BA\vee BA∨B for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H) with respect to ⪯\preceq⪯. Finally, we prove that Q∧P=0Q\wedge P=0Q∧P=0 where QQQ and PPP are the significant quantum observables corresponding to the position and momentum.}
The relationship between expectation and price is commonly established with two principles: no-arbitrage, which asserts that both maps are positive; and equivalence, which asserts that the maps share the same null events. Constructed from the Arrow-Debreu securities, classical and quantum models of economics are then distinguished by their respective use of classical and quantum logic, following the program of von Neumann. In this essay, the operations and axioms of quantum groups are discovered in the minimal preconditions of stochastic and functional calculus, making this the natural domain for the axiomatic development of mathematical finance. Quantum economics emerges from the twin pillars of the Gelfand-Naimark-Segal construction, implementing the principle of no-arbitrage, and the Radon-Nikodym theorem, implementing the principle of equivalence. Exploiting quantum group duality, a holographic principle that exchanges the roles of state and observable creates two distinct economic models from the same set of elementary valuations. Advocating on the grounds that this contains and extends classical economics, noncommutativity is presented as a modelling resource, with novel applications in the pricing of options and other derivative securities.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed