Optimal stopping and Dynkin games with non-linear $f$-expectations: beyond right-continuity
A talk in the Bielefeld Stochastic Afternoon - Math Finance Session series by
Miryana Grigorova from Bielefeld University
| Abstract: | We consider the optimal stopping problem with non-linear $f$-expectation (which
is a non-linear pricing operator induced by a Backward Stochastic Differential
Equation) and with payoff process $\xi$ on which we do not make any regularity assumptions. We characterize the value process of this optimal stopping problem as
the non-linear $\mathcal{E}^f$ -Snell envelope of $\xi$. We also establish an in infinitesimal characterization of the value process in terms of a Reflected BSDE with $\xi$ as the lower
obstacle. To do this, we prove some useful results on Reflected BSDEs with completely irregular obstacles, in particular, an existence and uniqueness result and a comparison theorem. In the second part of the talk, we formulate a non-linear $\mathcal{E}^f$-Dynkin game, that is, a game problem over stopping times with non-linear $f$-expectation, between two agents whose payoff processes $\xi$ and $\zeta$ are only right upper-semicontinuous (but not necessarily right-continuous). Under a technical assumption (a Mokobodzki- type condition), we show that our $\mathcal{E}^f$-Dynkin game has a value and that the value is characterized in terms of the solution of a Doubly Reflected BSDE where $\xi$ and $\zeta$ play the role of lower and upper obstacles. The case where the obstacles $\xi$ and $\zeta$ are completely irregular is more involved: we will see that in this case the solution of the doubly reflected BSDE is related to the value of "an extension" of the previous non-linear $\mathcal{E}^f$-game problem over a larger set of "stopping strategies" than the set of stopping times. Based on joint works with P. Imkeller, Y. Ouknine, and M.-C. Quenez. |