PhD position 04 – MSCA COFUND, AI4theSciences (PSL, France) - “Breaking the curse of high-dimensional PDE’S and applications to mathematical finance”

Artificial intelligence for the Sciences” (AI4theSciences) is an innovative, interdisciplinary and intersectoral PhD programme, led by Université Paris Sciences et Lettres (PSL) and co-funded by the European Commission. Supported by the European innovation and research programme Horizon 2020-Marie Sklodowska-Curie Actions, AI4theSciences is uniquely shaped to train a new generation of researchers at the highest academic level in their main discipline (Physics, Engineering, Biology, Human and Social Sciences) and master the latest technologies in Artificial Intelligence and Machine Learning which apply in their own field.

26 doctoral students will join the PSL university's doctoral schools in 2 academic cohorts to carry out work on subjects suggested and defined by PSL's scientific community. The 2020 call will offer up to 15 PhD positions on 24 PhD research projects. The candidates will be recruited through HR processes of high standard, based on transparency, equal opportunities and excellence.

Description of the PhD subject: “Breaking the curse of high-dimensional PDE’S and applications to mathematical finance”

Context – Motivation

Understanding the relative power of machine learning techniques for supervised and unsupervised learning, while a burgeoning area, is still in its infancy, see [E et al., 2020] for a comprehensive review. These successful results in supervised learning on very different datatypes, imaging, speech processing, natural language applications appear to break the curse of dimensionality in some ways. The curse of dimensionality also directly affects solving high-dimensional PDE’s, where reaching a given accuracy has a numerical complexity which is exponential in the domain dimension and its effect already appears in dimension greater to 5 or 6. The simplest example in mathematical finance is the Black-Scholes PDE for which recent works [Grohs et al., 2018] give an actual proof that the curse of dimensionality can be addressed through the use of artificial neural networks parametrization. Also Monte-Carlo methods and related developments of it, have been proposed with the same goal [E et al., 2020], as well as tensor train or network decompositions [Khoromskij, 2014].

Machine learning techniques themselves can benefit from the well-explored mathematical knowledge of ODE and PDE techniques such as adjoint equation used in [Chen et al., 2018] to alleviate the memory footprint. Other related developments can be found in [Chen et al., 2018, Zhang et al., 2020,Yildiz et al., 2019, Ruthotto and Haber, 2019]. The connection between deep learning and dynamical systems enables to leverage optimal control tools as proposed in [Vialard et al., 2020]. In this work, supervised learning is reformulated as an optimal control problem from which a new parametrization of the deep neural network is proposed through a collection of particles. This collection of particles solves a high-dimensional PDE resulting from optimal control, namely the EPDiff equation [Mumford and Michor, 2012] which is connected with advection dominated PDE equations and applications in imaging.

A last connection between machine learning and PDE’s can be found in model reduction methods, where the challenge is to approximate the solution of a PDE which is numerically costly for a given initial data, when one has access to a large computational budget offline. Many proposed approaches consisted in approximating the original and computationally expensive model with a simpler one, see [Benner et al., 2017] for a review. Prior information on the ”geometry” of the under-lying PDE has been also explored in this context of model reduction in [Mowlavi and Sapsis, 2018,Afkham and Hesthaven, 2017, Ehrlacher et al., 2019].

The use of PDE in finance is a long story, going back to the seminal work of Black and Scholes. However, thanks to the Feynman Kac representation formula, it is often possible to avoid solving the PDE and instead compute the solution via Monte-Carlo simulation. This is however possible only in a linear setting, i.e. when the corresponding Black-Scholes equation is linear. Many problems of modern finance are in fact non-linear, and high dimensional. To cite only one example, managing the credit risk of a large portfolio of options implies solving a very high (the number of underlyings) dimensional problem, since by essence, credit risk has to be managed at a global level, and not option by option. Many other applications such as: pricing with market im-pact [Loeper, 2018], [Bouchard et al., 2019], model calibration [Guo et al., 2019], [Guo and Loeper, ],[Guo et al., 2020], model free pricing [Tan and Touzi, 2013], involve solving fully non-linear PDEs,and are so far tractable only in low dimension.

Scientific objectives, methodology & expected results

The first goal of the PhD is to extend the knowledge in approximations of solutions of PDE’srelated to mathematical finance, in particular nonlinear parabolic equations and propose new nu-merical and algorithmic solutions in this particular setting. The second goal of the PhD is to state a theoretical result for the proposed methods on its ability to break the curse of dimensionality and propose some complexity measure associated to the class of PDE which could quantify the result. Finally, some applications to concrete problems of derivative pricing such as high dimensionalmodel calibration will be studied.

International mobility

Not offered

Thesis supervision

Jean-David Benamou and Grégoire Loeper

PSL

Created in 2012, Université PSL is aiming at developing interdisciplinary training programmes and science projects of excellence within its members. Its 140 laboratories and 2,900 researchers carry out high-level disciplinary research, both fundamental and applied, fostering a strong interdisciplinary approach. The scope of Université PSL covers all areas of knowledge and creation (Sciences, Humanities and Social Science, Engineering, the Arts). Its eleven component schools gather 17,000 students and have won more than 200 ERC. PSL has been ranked 36^th in the 2020 Shanghai ranking (ARWU).