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Exit times for semimartingales under nonlinear expectation

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  • Liu, Guomin

Abstract

Let Eˆ be the upper expectation of a weakly compact but possibly non-dominated family P of probability measures. Assume that Y is a d-dimensional P-semimartingale under Eˆ. Given an open set Q⊂Rd, the exit time of Y from Q is defined by τQ≔inf{t≥0:Yt∈Qc}.The main objective of this paper is to study the quasi-continuity properties of τQ under the nonlinear expectation Eˆ. Under some additional assumptions on the growth and regularity of Y, we prove that τQ∧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 obtain the quasi-continuity of exit times for multi-dimensional G-martingales, which nontrivially generalizes the previous one-dimensional result of Song (2011).

Suggested Citation

  • Liu, Guomin, 2020. "Exit times for semimartingales under nonlinear expectation," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7338-7362.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:12:p:7338-7362
    DOI: 10.1016/j.spa.2020.07.017
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    References listed on IDEAS

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    1. Gao, Fuqing, 2009. "Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3356-3382, October.
    2. Soner, H. Mete & Touzi, Nizar & Zhang, Jianfeng, 2011. "Martingale representation theorem for the G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 265-287, February.
    3. Nutz, Marcel & van Handel, Ramon, 2013. "Constructing sublinear expectations on path space," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3100-3121.
    4. Hu, Mingshang & Wang, Falei & Zheng, Guoqiang, 2016. "Quasi-continuous random variables and processes under the G-expectation framework," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2367-2387.
    5. Hu, Mingshang & Ji, Shaolin & Peng, Shige & Song, Yongsheng, 2014. "Backward stochastic differential equations driven by G-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 759-784.
    6. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
    7. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    8. Song, Yongsheng, 2011. "Properties of hitting times for G-martingales and their applications," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1770-1784, August.
    9. Hu, Ying & Lin, Yiqing & Soumana Hima, Abdoulaye, 2018. "Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3724-3750.
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