A Second-Order Ordinary Differential Equation for Studying Price Dynamics in Market Equilibrium

This research studies Brent crude oil price dynamics using a second-order ordinary differential equation framework with a geopolitical shock represented by a conflict dummy. Monthly Brent price data from January 2003 to January 2026 are used to calibrate the model through two approaches: a continuous-time nonlinear optimization method and a discrete-time least-squares method derived from finite-difference approximations. The continuous approach proved numerically unstable, since the optimization converged to the imposed parameter bounds and is therefore interpreted only as a numerical diagnostic.
The discrete approach was more stable and produced a positive discriminant, formally corresponding to an overdamped adjustment regime. However, the calibrated roots indicate weak and slow convergence, and the results should be interpreted as illustrative rather than conclusive. The project contributes by and interpret commodity price adjustment.