Hyers-Ulam (HU) Stability and Mittag-Leffler-HU Stability of Additive Brownian Motion (Louis Bachelier's Model)

The concept of Hyers-Ulam stability provides a powerful framework for analysing the robustness of functional and differential equations (DE’s) under small perturbations. In this study, we investigate the Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam stability of Louis Bachelier’s stochastic DE’s, identifying conditions under which approximate solutions remain close to exact ones. By extending the classical stability concept, we establish sufficient criteria ensuring that if a function approximately satisfies the given stochastic equation, then a true solution exists in its neighbourhood. The analysis emphasizes the relationship between bounded deviations and the structural properties of the equation, offering insight into the qualitative behaviour of dynamic systems influenced by uncertainty. These results enrich stability theory and have applications in stochastic processes, control theory, numerical analysis, and mathematical modelling. Furthermore, we examine the relevance of the Bachelier’s model to carbon control modelling. The findings show that the system exhibits stability over finite time intervals, meaning small perturbations produce proportionally small deviations. This supports reliable short-term forecasting and policy decisions in carbon pricing. However, due to the unbounded variance of Brownian motion, stability does not persist over long horizons, limiting its effectiveness for long-term environmental planning.