The Spectral Tensor Representation of Stochastic Processes

We prove that every Itô diffusion \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\) admits a complete, finite-dimensional representation through its Fokker--Planck generator discretized in cosine basis. The generator matrix \(M \in \mathbb{R}^{N \times N}\), computed via integration-by-parts weak form with reflecting boundary conditions, is a sufficient statistic for the entire stochastic system: time-dependent distributions, all moments, option prices, Greeks, Value-at-Risk, Expected Shortfall, stationary distributions, spectral gaps, and autocorrelation functions are all computable from \(M\) alone --- without Monte Carlo simulation. For time-homogeneous diffusions, \(M\) is a matrix; for time-inhomogeneous processes, it extends to a 3-tensor \(T_{kjl}\); for multi-dimensional systems, to higher-order tensors. The Universal Risk Representation Theorem (Nagy, 2026b) guarantees that \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) basis functions suffice regardless of dimension or the property being computed, where \(\rho > 1\) is the spectral decay rate of the process. We validate the completeness theorem through three-way comparison (spectral vs analytical vs Monte Carlo) on the Ornstein--Uhlenbeck process, demonstrate it on a nonlinear double-well potential with bimodal dynamics, and apply it to mean-reverting energy pricing where a single \(48 \times 48\) matrix produces the complete term structure of prices, risk measures, and Greeks.