Exit times for semimartingales under nonlinear expectation

Abstract

Let $\hat{\mathbb{E}}$ be the upper expectation of a weakly compact but non-dominated family $\mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $\mathcal{P}$-semimartingale under $\hat{\mathbb{E}}$. Given an open set $Q\subset {\mathbb{R}}^d$, the exit time of $Y$ from $Q$ is defined by \[ τ_{Q}:=\inf\{t\geq0:Y_{t}\in Q^{c}\}. \] The main objective of this paper is to study the quasi-continuity properties of ${\tau }_Q$ under the nonlinear expectation $\hat{\mathbb{E}}$. Under some additional assumptions on the growth and regularity of $Y$, we prove that ${\tau }_Q\wedge t$ is quasi-continuous if $Q$ satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we get the quasi-continuity of exit times for multi-dimensional $G$-martingales, which nontrivially generalizes the previous one-dimensional result of Song.