Impulse neural networks application in solving the Wiener - Hopf factorization problem

Abstract

The paper is devoted to studying the possibilities of using artificial spike neural networks to solve the problem of approximate Wiener — Hopf factorization for Levy processes within an intelligent machine learning system. One of the applications of Wiener — Hopf factorization is pricing barrier options, and therefore the problem under consideration has an important applied aspect for computational finance in terms of designing hybrid numerical methods that combine modern technologies of third-generation neural networks and classical methods of computational mathematics. Within the framework of the paper, a spike neural network with an “integrate-and-fire” model with leaks is proposed for factorization of a trigonometric polynomial, the coefficients of which are a probability distribution. The sought-for factor polynomials have a similar probabilistic interpretation, while the first half of the coefficients of the first factor is zero, and the second half is zero for the second factor. The probabilistic interpretation of the problem allows one to do without encoding and decoding the input and output data into spikes and back. The network is trained for one set of polynomial coefficients in order to minimize the approximation error of this polynomial by the product of factors whose coefficients are predicted by the network, for which purpose its own loss function is implemented in software. In contrast to the traditional approach to fitting the model parameters to the training sample this paper proposes to minimize the approximation error of a specified characteristic function of the Levy process by the product of polynomial factors. In this case, the model does not use the actual values of the factor coefficients during training, but only the values of the polynomials calculated using the fast Fourier transform. In the framework of computational experiments, an example of factorization of a 255th-degree polynomial associated with the Gaussian Levy process using a spike neural network is presented. The software implementation of the approach to solving the factorization problem proposed in the article is written in the Python programming language using the pyTorch machine learning framework and the snnTorch library of pulse neural networks.