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SIAM Activity Group on FME Virtual Talk Series
Join us for a series of online talks on topics related to mathematical finance and engineering and running every two weeks until further notice. The series is organized by the SIAM Activity Group on Financial Mathematics and Engineering.

Date: Thursday, April 29, 2021, 1PM-2PM

Speaker: Huyen Pham, Université de Paris

Title: DeepSet and their derivative networks for solving symmetric problems

Abstract: Machine learning methods represent a breakthrough for solving nonlinear partial differential equations (PDEs) and control problems in very high dimension, and have been the subject of intense research over the last five years. In this talk, we consider a widespread class of problems that are invariant to permutations of their inputs (state variables or model parameters). This occurs for example in multi- asset models for option pricing with exchangeable payoff, or for optimal trading portfolio with respect to the market price of covariance risk. Our main application comes actually from mean-field control problems and the corresponding PDEs in the Wasserstein space of probability measures. Their particle approximations, for which we provide a rate of convergence, lead to symmetric PDEs that are solved by deep learning algorithms based on certain types of neural networks, named DeepSet. We illustrate the performance and accuracy of the DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, and mean-variance problem. Finally, we show how the combination of DeepSet and DeepOnet, a network architecture recently proposed for learning operators, provides an efficient approximation for a family of optimal trading strategies in terms of market price of covariance risk coefficients.

Moderator: Agostino Capponi, Columbia University
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