An innovative extension of the geometric Brownian motion model is developed by incorporating a weighting factor and a stochastic function modeled as a mixture of power and trigonometric functions. Simulations based on this modified Brownian motion model, with optimal weighting factors selected by goodness-of-fit tests, substantially outperform the basic geometric Brownian motion model in terms of fitting the returns distribution of historic data price indices. Furthermore, we attempt to provide an interpretation of the additional stochastic term in relation to irrational behavior in financial markets and outline the importance of this novel model.