Research on the Application of Neural Networks in Continuous-Time Mathematical Finance

This is a master's thesis research from the Department of Mathematics at ETH Zurich, focusing on the application of neural networks in continuous-time mathematical finance. The core lies in using deep learning techniques to solve complex problems in traditional financial mathematics, such as option pricing and risk hedging, breaking through limitations like the curse of dimensionality faced by traditional methods. The study combines technologies like neural differential equations and physics-informed neural networks to provide new solutions for high-dimensional and complex financial problems.

Continuous-time financial models use stochastic differential equations, Ito calculus, martingale theory, partial differential equations (PDEs) as core tools, but traditional methods (such as numerical PDE solving and Monte Carlo simulation) have obvious limitations: high-dimensional problems face the curse of dimensionality, path-dependent derivative pricing is complex, model calibration has high computational costs, and real-time performance is insufficient in trading scenarios. The universal function approximation ability of neural networks provides a new perspective for solving these problems.

The application framework of neural networks in financial mathematics includes: acting as a universal function approximator to learn price functions, strategy functions, or probability distributions; neural differential equations (Neural ODE/SDE) naturally fit continuous-time models. Key technical methods are: Physics-Informed Neural Networks (PINNs) embed PDE constraints into the loss function to solve the Black-Scholes equation; Deep Galerkin Method (DGM) handles high-dimensional parabolic PDEs; neural control variate method reduces the variance of Monte Carlo simulations; reinforcement learning (DDPG, A2C/A3C) learns hedging strategies.

Empirical cases include: 1. Pricing of 50-asset basket options (traditional PDEs cannot handle this; neural networks solve it by learning price functions and using asset correlations to reduce effective dimensionality); 2. Calibration of the Heston stochastic volatility model (neural networks reduce calibration time from minutes to milliseconds); 3. Early exercise boundary of American options (neural networks learn boundary parameterization and combine with Monte Carlo simulation for pricing).

The study provides theoretical support: neural networks can achieve arbitrary precision approximation with polynomial complexity for financial PDE solutions that meet regularity conditions; there exists an upper bound on generalization error (depending on network complexity, training sample size, etc.); by designing network architectures and training strategies, numerical stability under extreme market conditions (such as volatility approaching zero or infinity) is ensured.

Current challenges: insufficient interpretability (the black-box nature conflicts with financial regulatory transparency requirements), weak extrapolation ability, high training computational costs, and lack of systematic methods for model validation. Future directions: neural operators (learning operators to achieve cross-problem generalization), Bayesian neural networks for quantifying uncertainty, causal inference to avoid spurious correlations, and federated learning for distributed training under privacy protection.

The combination of neural networks and classical financial mathematics represents a cutting-edge direction in computational finance, breaking through the bottlenecks of traditional methods and opening up new paths for high-dimensional and complex financial problems. Although facing challenges such as interpretability and stability, with the maturity of algorithms and improvement of computing power, neural networks are expected to play a more important role in financial practice and promote quantitative finance to enter a new stage.