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Exit times for semimartingales under nonlinear expectation
Stochastic Processes and their Applications ( IF 1.4 ) Pub Date : 2020-12-01 , DOI: 10.1016/j.spa.2020.07.017
Guomin Liu

Let $\mathbb{\hat{E}}$ 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 $\mathbb{\hat{E}}$. Given an open set $Q\subset\mathbb{R}^{d}$, the exit time of $Y$ from $Q$ is defined by \[ {\tau}_{Q}:=\inf\{t\geq0:Y_{t}\in Q^{c}\}. \] The main objective of this paper is to study the quasi-continuity properties of ${\tau}_{Q}$ under the nonlinear expectation $\mathbb{\hat{E}}$. 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.

中文翻译:

非线性期望下半鞅的退出时间

令 $\mathbb{\hat{E}}$ 是概率测度的弱紧凑但非支配族 $\mathcal{P}$ 的上期望。假设$Y$是$\mathbb{\hat{E}}$下的$d$维$\mathcal{P}$-semimartingale。给定一个开集 $Q\subset\mathbb{R}^{d}$,$Y$ 从 $Q$ 的退出时间定义为 \[ {\tau}_{Q}:=\inf\{t \geq0:Y_{t}\in Q^{c}\}。\] 本文的主要目的是研究${\tau}_{Q}$在非线性期望$\mathbb{\hat{E}}$下的拟连续性。在对$Y$ 的增长和规律性的一些额外假设下,我们证明如果$Q$ 满足外部球条件,则${\tau}_{Q}\wedge t$ 是准连续的。我们还给出了准连续过程的表征和停止过程的相关属性。特别是,
更新日期:2020-12-01
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