# Chaotic Hedging with Iterated Integrals and Neural Networks

@article{Neufeld2022ChaoticHW,
title={Chaotic Hedging with Iterated Integrals and Neural Networks},
author={Ariel Neufeld and Philipp Schmocker},
journal={ArXiv},
year={2022},
volume={abs/2209.10166}
}
• Published 21 September 2022
• Mathematics
• ArXiv
A BSTRACT . In this paper, we extend the Wiener-Ito chaos decomposition to the class of diffusion processes, whose drift and diffusion coefﬁcient are of linear growth. By omitting the orthogonality in the chaos expansion, we are able to show that every p -integrable functional, for p ∈ [1 , ∞ ) , can be represented as sum of iterated integrals of the underlying process. Using a truncated sum of this expansion and (possibly random) neural networks for the integrands, whose parameters are learned…
4 Citations

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## References

SHOWING 1-10 OF 65 REFERENCES

• J. Lelong
• Computer Science, Mathematics
SIAM J. Financial Math.
• 2018
In this work, we propose an algorithm to price American options by directly solving thedual minimization problem introduced by Rogers. Our approach relies on approximating the set of uniformly square
This paper extends a recent martingale representation result of [N-S] for a L´evy process to filtrations generated by a rather large class of semimartingales. As in [N-S], we assume the underlying
If F is a Frechet differentiable functional on is a Brownian motion, and clark's formula states that where is the measure defining the Frechet derivative of F at b.In this paper we extend Clark's
• Mathematics
• 2008
The Continuous Case: Brownian Motion.- The Wiener-Ito Chaos Expansion.- The Skorohod Integral.- Malliavin Derivative via Chaos Expansion.- Integral Representations and the Clark-Ocone formula.- White
• Computer Science
ArXiv
• 2020
It is proved that, as long as the unknown function, functional, or dynamical system is sufficiently regular, it is possible to draw the internal weights of the random neural network from a generic distribution and quantify the error in terms of the number of neurons and the hyper parameters.
• Mathematics
SIAM J. Sci. Comput.
• 2002
This work represents the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error.
• Mathematics
Finance Stochastics
• 2016
This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest
Abstract The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L 2(P), then from this
• G. Cybenko
• Computer Science
Math. Control. Signals Syst.
• 1989
In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real