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

This job offer has expired

    Université PSL
    MathematicsApplied mathematics
    First Stage Researcher (R1)
    26/02/2021 23:00 - Europe/Brussels
    France › Paris
    H2020 / Marie Skłodowska-Curie Actions COFUND


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



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 36th in the 2020 Shanghai ranking (ARWU).

More Information


  • Opportunity to conduct academic research in a top 100 university in the world.
  • High-quality doctoral training rewarded by a PhD degree, prepared within Université Paris Dauphine - PSL and delivered by PSL.
  • Access to cutting-edge infrastructures for research & innovation.
  • Appointment for a period of 36 months (job contract delivered by the involved component school of PSL) based on a salary of 3100 € gross employer (including employer tax) per month or approximately a 2228 € gross salary per month.
  • Job contract under the French labour legislation in force, respecting health and safety, and social security: 35 hours per week contract, 25 days of annual leave per year (“congés annuels”). Eventual complementary activities may be accepted or proposed by the co-supervisors (maximum of 64h/year for teaching, 32 day/year for specific missions).
  • Short stay(s) or secondment in France or abroad are expected.
  • An international environment supported by the adherence to the European Charter & Code.
  • Access to AI training package, with a strong interdisciplinary focus, together with a Career development Plan.

Eligibility criteria

  • Applicants must have a Master’s degree (or be in the process of obtaining one) or have a University degree equivalent to a European Master’s (5-year duration) to be eligible at the time of the deadline of the relative call.
  • There is no nationality or age criteria, but applicants must not have resided or carried out their main activity (work, studies, etc.) in France for more than 12 months in the 3 years immediately before the deadline of the call (MSCA Mobility rule).
  • Applicants must declare to be available to start the programme on schedule.

For submitting your online application, go to: https://www.psl.eu/recherche/grands-projets-de-recherche/projets-europee...


The online application should contain the following documents:

  • English translated transcripts from the Master’s degree (or equivalent 5-year degree). A copy of the Master’s degree or a certificate of achievement will be required later on for the final registration.
  • International curriculum vitae and a cover letter explaining the reasons that lead him/her to prepare a PhD, why he/she applies to this offer and his/her professional project (guidelines will be given to the applicants in order to help him/her in the writing of his/her letter).
  • Two academic reference letters.
  • A statement duly signed on the mobility rules, availability, and conflicts of interest.


The applicants can only apply to one PhD project among the available ones. Multiple applications of one candidate will automatically make all his/her applications ineligible.

Selection process

The applications will be analysed by the Management Team for eligibility and completeness. Afterwards, the applications will be reviewed by the Selection Committee. In the pre-selection round (March-April 2021), applicants will be rated using a scoring system based on 3 criteria (academic excellence, experience, motivation, and qualities). A shortlist of qualified applicants will be interviewed during the selection round (June 2021) to further assess their qualifications and skills according to the predefined selection criteria.

All information regarding the applications (criteria, composition of the Selection Committee, requirements) can be found on the website of the programme, in greater detail.


The selection and recruitment processes of the PhD student will be in accordance with the European Charter for Researchers and Code of Conduct of the Recruitment of Researchers. The recruitment process will be open, transparent, impartial, equitable, and merit based. There will be no discrimination based on race, gender, sexual orientation, religion of belief, disability, or age.

Additional comments

As a constituent institution of the Université PSL, Dauphine - PSL contributes actively to its development, alongside 10 other institutions. The 3 concerned fields are: — research, through collaborations such as the PR[AI]RIE project with other PSL institutions ; — education, with PSL programs at every level from a Bachelor’s to a PhD; — the development of Shared Support Services (SMS): born of cooperative, mutual logic, these PSL services are operated by the constituent institutions for the entire community. Dauphine supports 3 of them — the Internal School, sports and housing, and actively contributes to others such as the EU Support Service for Calls for Proposals, the documentation and dissemination of knowledge.


Web site for additional job details

Required Research Experiences

    1 - 4

Offer Requirements

    Mathematics: Master Degree or equivalent
    ENGLISH: Excellent


  • Master’s degree in Mathematics.
  • Strong mathematical and/or computational background.
  • Good basis in probability / optimization / calculus of variation.
  • An interest in machine-learning techniques and computational methods.

Work location(s)
1 position(s) available at
Rue Simone Iff

EURAXESS offer ID: 584105


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