On the strict value of the non-linear optimal stopping problem

We address the non-linear strict value problem in the case of a general filtration and a completely irregular pay-off process (ξt). While the value process (Vt) of the non-linear problem is only right-uppersemicontinuous, we show that the strict value process (V+t) is necessarily right-continuous. Moreover, the strict value process (V+t) coincides with the process of right-limits (Vt+) of the value process. As an auxiliary result, we obtain that a strong non-linear f-supermartingale is right-continuous if and only if it is right-continuous along stopping times in conditional f-expectation.