Computational Linearization of Nonlinear Kolmogorov Equations via the Generalized Newton Method in LSV Models

This study develops a generalized Newton method to address the nonlinear Kolmogorov forward equation (KFE) under the local stochastic volatility (LSV) framework. Analytical convergence conditions are derived via the geometric series theorem, and empirical validation is conducted using 15 years of monthly crude oil spot price data (2011–2025), with the parameters set using maximum likelihood estimation. Sensitivity analyses confirm the stable convergence of the iterative scheme under realistic scenarios while also identifying parameter ranges that may lead to divergence. These findings demonstrate that the proposed methodology provides a tractable and accurate approach to probability density estimation in commodity markets, with a clear potential for extension to multi-dimensional settings and richer datasets.