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

Abstract

We address the non-linear strict value problem in the case of a general filtration and a completely irregular pay-off process $(\xi _{t})$. While the value process $(V_{t})$ 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 $(V_{t+})$ 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.