Monte Carlo method in stock trading research based on accelerated diffusion theory with jumps

This study proposes an optimized diffusion model based on accelerated Monte Carlo methods and stochastic differential equations (SDEs) with jumps for simulating stock prices in financial markets and generating trading strategies. The model accelerates the traditional diffusion process by introducing a skew-symmetric matrix B, while incorporating a jump term dJ to capture sudden market fluctuations. The core equation is expressed as: dSt=(μE+βB)Stdt+σStdW(t)+δStdJ(t) Here, βB serves as an irreversible perturbation term that accelerates convergence by expanding the spectral gap of the generator. The jump process is modeled using a Poisson distribution to simulate abrupt price changes. The Monte Carlo method is optimized through stratified time discretization and adaptive sampling, significantly reducing computational complexity for high-dimensional data processing. Experimental results demonstrate that, compared to traditional jump SDE models, this model achieves a 20-60x improvement in computational efficiency (e.g., a 64x speedup on the SSE50 dataset) while maintaining prediction errors below 0.001. Through rolling time window validation and maximum drawdown analysis, the model exhibits stable generalization capabilities and superior risk-return performance across five datasets, including Apple Inc. (AAPL), C3.ai Inc. (AI), HDFC Bank Limited (HDFC), NIFTY 50 Index (NIFTY50), and SSE 50 Index (SSE50). Machine-driven trading strategies based on this model achieve annualized returns of 59%–211% in backtesting, outperforming passive holding strategies by 62%–165%. This research provides an efficient technical pathway for real-time decision-making and high-dimensional data processing in financial markets, laying a theoretical foundation for derivative applications such as option pricing and risk management.