## Abstract

In the paper, we prove probabilistic analogs of the universal approximation theorems and link continuous random variables of a certain type and monotonic feedforward artificial neural networks with one-dimensional input, output and one hidden layer. In particular, we show that any continuous infinitely divisible random variable can be successfully approximated with a mix of logistic distributions. Based on the theorems proved in the current paper, we develop a new approach for developing Monte Carlo methods combined with artificial neural networks for pricing options in Lévy models. In contrast to straightforward incorporation of neural networks into Monte Carlo methods, we approximate the cumulative distribution function, but not its inverse. Moreover, we give a clear probabilistic interpretation of the constructed approximator that helps us simulate the Lévy process by using only separate components of our neural network.

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This work is supported by Russian Science Foundation Grant No. 23-21-00474, https://rscf.ru/project/23-21-00474/

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Kudryavtsev, O., Danilova, N. APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS TO SIMULATING LÉVY PROCESSES.
*J Math Sci* **271**, 421–433 (2023). https://doi.org/10.1007/s10958-023-06580-1

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DOI: https://doi.org/10.1007/s10958-023-06580-1