Random Processes and Their Applications in Financial Mathematics

Random processes are fundamental tools in modern financial mathematics, widely applied in fields such as financial derivatives pricing, risk management, and interest rate models. This paper systematically introduces the basic theory of random processes, including definitions and classifications, Brownian motion and geometric Brownian motion, and stochastic differential equations. Furthermore, the paper explores specific applications of random processes in financial mathematics, focusing on financial derivatives pricing, risk management and portfolio optimization, and applications in interest rate models and credit risk. Through theoretical and empirical research, this paper reveals the significant role and development prospects of random processes in financial mathematics, providing new ideas and methods for risk management and pricing models in financial markets.