제목: Computational Methods in Financial Mathematics: Mirror-Padded Fourier Neural Operators and Penalized Backward Stochastic Differential Equations

This study investigates computational methods in financial mathematics from two directions: learning-based approximation of stochastic processes and numerical approximation of backward stochastic differential equations arising in option pricing. The first part develops the mirror-padded Fourier neural operator (MFNO), an operator-learning architecture for non-periodic stochastic time series, and establishes approximation results for path-dependent stochastic differential equations and fractional Brownian motion. The second part studies a penalty-based two-grid approximation scheme for doubly reflected backward stochastic differential equations with double obstacles, and derives explicit error bounds under suitable assumptions. Together, these results provide rigorous computational tools for stochastic process approximation and BSDE-based computation in finance.