Mathematical Foundations of Quantum Pricing Theory

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 $\varphi_\rho$ and a filtration of normal $\varphi_\rho$-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 $\varphi^\star$ and a compatible family of normal $\varphi^\star$-preserving conditional expectations $(E_t^\star)$, we define for bounded terminal payoffs $X\in M_T$ the dynamic pricing operator $$\Pi_t(X):=B_t^{1/2}\,E_t^\star\!\bigl(B_T^{-1/2}XB_T^{-1/2}\bigr)\,B_t^{1/2},$$ where $(B_t)$ is a strictly positive num\'eraire adapted to $(N_t)$. We prove that $(\Pi_t)$ is normal, completely positive, $N_t$-bimodular, and time-consistent, and satisfies $\Pi_t(\mathbf 1)=B_t$ (equivalently, $\widetilde{\Pi}_t(X):=B_t^{-1/2}\Pi_t(X)B_t^{-1/2}$ is unital). In the commutative reduction it agrees with risk-neutral valuation by conditional expectation. Finally, we develop an $L^2(M,\varphi_\rho)$ prediction theory and introduce an operator-valued Fisher information relative to $(N_t)$, obtaining a noncommutative Cram\'er--Rao lower bound for conditional mean-square prediction error; we compute the bound for compound Poisson lattice-jump models under $\sum_{\alpha}\gamma_\alpha(e^{\alpha\Delta x}-1)=r$.