Optimal stopping with f -expectations: the irregular case
Résumé
We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.
Mots clés
optimal stopping
backward stochastic differential equation
f-expectation
nonlinear expectation
aggregation
American option
dynamic risk measure
strong $\mathcal{E}^f$ -supermartingale
Snell envelope
reflected backward stochastic differential equation
comparison theorem
Tanaka-type formula
General filtration
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