Non-linear mixed optimal control/stopping (game) problems and applications to American options in incomplete markets with imperfections


We study the superhedging problem for American options with irregular payoffs in a non-linear and incomplete financial market. The market consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. We study the case when there are imperfections or constraints which are encoded in the non-linearity of the portfolio dynamics. We prove a nonlinear pricing-hedging duality formula for the seller of the American option in terms of the value of a non-linear mixed control/stopping problem, involving a suitable set of equivalent probability measures, which we call f-martingale probability measures.  We also characterize the seller’s price process as the minimal supersolution of a reflected Backward Stochastic Differential Equation with constraints. We then prove a duality result for the buyer’s price in terms of the value of a non-linear control/stopping game problem.

(based on joint works with Miryana Grigorova, University of Leeds and Marie-Claire Quenez, LPSM, Université Paris Denis Diderot)


M. Grigorova,  M.C. Quenez,  A. Sulem : European  options in a non-linear incomplete market with default, SIAM J. Fin.  Math., 11(3), (2020)
M. Grigorova,  M.C. Quenez,  A. Sulem :  American options in a non-linear incomplete market model with default, Stoch. Proc.  App., to appear.


Agnès Sulem is a senior researcher (Directeur de Recherche Classe exceptionnelle) at the Paris Research Center of INRIA, the French National Institute for Computer Science and Applied Mathematics. She has created and is leading the “MATHRISK” research group on stochastic analysis and financial mathematics (https://www.inria.fr/en/teams/mathrisk). She is also the director of the Premia consortium, which provides a numerical platform for quantitative finance. (http://www.premia.fr).

Agnès Sulem is teaching in the doctoral program at the University of Luxemburg.

Her fields of research are stochastic control, numerical and stochastic analysis, and mathematical finance. She is the author of 2 books (in 2019 has appeared the 3rd edition of her book “Applied Stochastic Control of Jump diffusions” with Bernt Øksendal, Springer, Universitext), and more than 100 research articles. She has been supervising fifteen PhD theses.

She is a member of the Editorial Boards of SIAM Journal on Financial Mathematics, the Journal of Mathematical Analysis and Applications, Mathematics, and is a reviewer for Mathematical Reviews.

Agnès Sulem studied at “Pierre and Marie Curie” Paris University and obtained her PhD in Mathematics under the supervision of Prof. Alain Bensoussan and her «Habilitation pour Diriger des Recherches» from the University Paris-Dauphine.

Agnès Sulem has been awarded the title of “Chevalier de l’Ordre de la Légion d’Honneur” for her scientific career. Besides mathematics, Agnès Sulem enjoys playing the violin.


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