Stochastic modeling of AAPL stock using classical stochastic differential equation models

Abstract. This study investigates the ability of classical stochastic differential equations models to replicate the real behavior of AAPL stock prices over the ۲۰۲۰–۲۰۲۴ period. Several models from stochastic calculus—including standard Brownian motion, geometric Brownian motion, Brownian motion with drift, Brownian bridge, and the Ornstein–Uhlenbeck process—are implemented and analyzed, along with their corresponding simulation algorithms. Using actual adjusted price data, each model is simulated and compared with the observed price trajectory, with particular attention to key features such as hitting times, exit times, long-term trends, and mean-reversion behavior. The results indicate how these models can be applied to capture the structural and random components of financial time series. The analysis suggests that while classical models provide useful insights into certain statistical aspects of asset prices, they also exhibit limitations in capturing complex market behavior. To ensure full reproducibility and methodological transparency, all simulations were performed in R, and the appendix includes the corresponding code, visualizations, and explicit algorithms for all models. This comparative framework may support future efforts in selecting or combining stochastic models for more robust financial forecasting.

کلیدواژه ها:

Geometric Brownian motion ، Ornstein–Uhlenbeck process ، Brownian bridge ، hitting time ، exit time