Weak Form Recovery of Heston Type Stochastic Dynamics

We study whether a spatial weak-form LASSO pipeline can recover the Heston stochastic-volatility model. Gaussian test functions convert local increment moments into weak targets for drift, diffusion, and return--variance covariance. Using 50 Gaussian kernels, the linear library ${1,v}$, column normalization, five-fold cross-validated LASSO, and contemporaneous indexing, 30 independent 100-year simulations recover $\xi$, $\rho$, and $\rho\xi$ with median relative errors of 0.90%, 0.53%, and 1.80%, while drift recovery is less accurate (12.29% for $\kappa$, 6.61% for $\theta$). Under noisy variance observations, leverage estimates degrade gradually without a sharp phase transition. Applied to S&P 500 data (2007--2010), the method estimates $\kappa=2.361$, $\theta=0.0360$, $\xi=0.530$, and $\rho=-0.329$. A quadratic-drift falsification selects spurious terms in 93% of null simulations, indicating the method is reliable for coefficient estimation within a specified Heston library but not unrestricted model discovery.