Adaptive Stochastic Gradient Flow: A Variational Framework for Optimal Continuous-Time Approximation of Stochastic Optimization

This paper introduces a novel high-order numerical methodology for solving the generalized Black-Scholes equation with variable coefficients. The proposed approach combines hp-adaptive spectral element discretization in space with implicit-explicit (IMEX) time integration schemes, addressing key challenges in financial derivative pricing. The spatial discretization employs an adaptive strategy that dynamically refines both mesh size (h) and polynomial degree (p) based on a posteriori error estimation, effectively resolving the singularity originating from the non-differentiable payoff function of European options. The temporal discretization utilizes third-order L-stable IMEX additive Runge-Kutta (IMEX-ARK) methods, which implicitly handle the stiff diffusion term while explicitly treating advection and reaction components. We establish comprehensive theoretical foundations including well-posedness analysis, stability proofs, and convergence estimates under appropriate regularity conditions. Numerical experiments demonstrate the method's efficacy for various financial scenarios including constant coefficient models, local volatility surfaces, and time-dependent parameters. The framework is extended to exotic options, particularly barrier options and American options via a penalty formulation. Comparative studies against established numerical techniques confirm the superior accuracy and computational efficiency of the proposed methodology across all test cases. MSC Classification: 62L20 , 65C30 , 68T05 , 90C15