Spectral Time: Optimal Stopping, First Passage, and Subordination via the Fokker-Planck Generator

We develop a unified spectral framework for three fundamental problems in stochastic analysis: optimal stopping, first passage times, and time-changed (subordinated) processes. The Fokker--Planck generator \(M \in \mathbb{R}^{N \times N}\) of a diffusion process, discretized in cosine basis, serves as the single computational object from which all three are solved algebraically. Optimal stopping (American options) becomes backward iteration of the matrix exponential \(e^{M\Delta t}\) interleaved with pointwise maximization --- replacing Longstaff--Schwartz regression. First passage times reduce to one matrix inverse: \(\mathbb{E}[\tau_b] = -\mathbf{1}^\top M_{\text{killed}}^{-1} A(0)\), where \(M_{\text{killed}}\) is the generator with absorbing boundary. Subordination (Variance Gamma, NIG, CGMY processes) becomes a matrix function \(\psi(M)\) that remaps eigenvalues while preserving eigenvectors --- the spatial modes are unchanged; only the temporal decay rates are transformed. Numerically, the first passage time computation achieves \(>10{,}000\times\) speedup over Monte Carlo (0.003s vs 37s), the subordinated generator produces Variance Gamma distributions with correct excess kurtosis and fat tails from a single eigendecomposition, and the optimal stopping backward induction runs 38\(\times\) faster than Longstaff--Schwartz. The unifying insight: the eigenvalue spectrum of \(M\) IS the intrinsic time structure of the process.