A second-order ordinary differential equation for studying price dynamics in market equilibrium

Resumo

This paper 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 is theoretically appealing but proved numerically unstable: the optimization converged to the imposed parameter bounds, so its parameters are not identified and are reported only as a numerical diagnostic. The discrete approach is computationally stable and yields a positive discriminant, which formally corresponds to an overdamped regime. The calibrated system, however, lies close to the boundary of stability. The slow characteristic root is approximately , implying a weak restoring force and a half-life of adjustment of roughly ten months, while the fast root falls outside the range in which the monthly finite-difference approximation is reliable and is therefore not interpreted. The paper concludes that the framework provides a useful structured description of price adjustment, but that its quantitative results should be read as illustrative rather than conclusive.