Comparative Analysis of Classical and Deep Learning-based Numerical Methods for Solving Differential Equations in Financial Modeling

Abstract

Talking specifically about mathematical modelling for financial markets, one can mention ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs) as among the most respective for their dynamics. This paper compares the numerical methods more common in practice and recent deep learning strategies for finding solutions to differential equations used in financial mathematics, such as the Black-Scholes formula and stochastic interest rate models. This paper aimed at comparing some of the classical methods in numerically solving SDEs, which include FDM, Runge-Kutta schemes, Euler-Maruyama, and Milstein methods, in terms of stability, accuracy and computation. These were compared with neurally trained PINNs and Deep BSDE solvers for high-dimensional and irregular domain problems. The experimental results show that for many problems in a more organised and lower-dimensional field, using traditional approaches still has great advantages and effectiveness, but in larger and highly-dimensional finance fields, deep learning solvers have better adaptability and extendibility. The designed work proves the significance of the mixed approach to carry out the numerical analysis of finance based on the formal concepts as well as the modern computational intelligence.