Chaotic Hedging with Iterated Integrals and Neural Networks

2022/09/21 by Ariel Neufeld, Philipp Schmocker, Neufeld, Ariel +1 · 1 citation
Computer Science · Physics and Astronomy · #Computational Finance (q-fin.CP) #FOS: Computer and information sciences #FOS: Economics and business #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Mathematical Finance (q-fin.MF) #Model Reduction and Neural Networks #Probability (math.PR) #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2209.10166

openalex publication_date 2022/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we derive an L^p-chaos expansion based on iterated Stratonovich integrals with respect to a given exponentially integrable continuous semimartingale. By omitting the orthogonality of the expansion, we show that every p-integrable functional, p ∈ [1,∞), can be approximated by a finite sum of iterated Stratonovich integrals. Using (possibly random) neural networks as integrands, we therefere obtain universal approximation results for p-integrable financial derivatives in the L^p-sense. Moreover, we can approximately solve the L^p-hedging problem (coinciding for p = 2 with the quadratic hedging problem), where the approximating hedging strategy can be computed in closed form within short runtime.

Cited by

• Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs

Related

• NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes
• Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion
• The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations
• Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to Mathematical Finance
• A Distributed Procedure for Computing Stochastic Expansions with\n Mathematica