La professoressa
Marie-Amélie Morlais
del Laboratoire Manceau de Mathématiques (LMM), Institut du Risque et de l’Assurance IRA, Le Mans Université,
terrà un seminario dal titolo
Max-min and min-max systems of variational inequalities: connection with multidimensional BSDEs with bilateral obstacles and switching game
il giorno
lunedì 19 marzo 2018 alle ore 14:00
in sala rappresentanza del Dipartimento di Matematica dell'Università di Milano (via Saldini 50, 20133 Milano).
Tutti gli interessati sono cordialmente invitati. La prof.ssa Morlais resterà in visita fino a giovedì 22 marzo.
Cordiali saluti, Marco Fuhrman
Abstract.
This talk is based on two joint works (published in 2015 and 2017) with Prof. S. Hamadène (LMM Le Mans), Prof. B. Djehiche (KTH Stockholm Sweden) and X. Zhao (former PhD student of S. Hamadène) . These two papers focus on the connection between systems of interconnected PDEs, related multidimensional BSDEs and a general switching game (second paper).
As an introduction, I recall the (rich) literature which links the optimal switching problem with explicit multidimensional BSDEs and related systems of variational inequalities. The relationship between those objects shall be given as well. I will finally present the so called min-max and max-min systems of PDEs whose viscosity solutions are heuristically identified (via the verification result) as the lower and upper value of the switching game.
In the second part (first paper), I will explain the main steps allowing to construct solutions to both the min max and max min systems of variational inequalities. I stress the fact that, due to the non-linearities of those systems and the presence of interconnected obstacles, those solutions are a priori irregular: thus, to construct a solution, the use of viscosity theory is useful.
In a third part (main result of the second paper) the connection with a multidimensional BSDE with bilateral obstacles is established (existence and uniqueness for the BSDE solution are then established in a Markovian framework of randomness).
Finally and as an application, we provide an explicit relationship with the switching game: this consists in providing (sufficient) conditions ensuring the existence of both the switching game value and an optimal mixed strategy (as well as the relationship with the BSDE solution).
Some open questions remaining unsolved shall be presented as a conclusion.