Optimal stopping with f -expectations: the irregular case

Miryana Grigorova (1) , Peter Imkeller (2) , Youssef Ouknine (3) , Marie-Claire Quenez (4)

1 KVW - Centre for Risk and Insurance, Hanover
2 Institut für Mathematik [Berlin]
3 Faculté des Sciences Semlalia [Marrakech]
4 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. 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

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]; Optimisation et contrôle [math.OC]

Miryana Grigorova, Peter Imkeller, Youssef Ouknine, Marie-Claire Quenez. Optimal stopping with f -expectations: the irregular case. 2017. ⟨hal-01403616v2⟩