2025/06/18 by Jia Yue, Yue, Jia, Ming-hui Wang +3
Decision Sciences · Mathematics · #60H05 #91G10 #FOS: Mathematics #Fuzzy Systems and Optimization #Probability (math.PR) #Risk and Portfolio Optimization
paper · pdf · doi:10.48550/arxiv.2506.15044
openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
In the existing works, stochastic sets \mathbbB of interval type, along with \mathbbB-stochastic processes, were introduced within the framework of stochastic analysis. In this paper, we undertake the construction of \mathbbB-stochastic integration by exploring three novel types of \mathbbB-stochastic integrals: Stieltjes integrals of \mathbbB-predictable processes with respect to \mathbbB-adapted processes with finite variation, stochastic integrals of \mathbbB-predictable processes with respect to \mathbbB-inner local martingales, and stochastic integrals of \mathbbB-predictable processes with respect to \mathbbB-inner semimartingales. These \mathbbB-stochastic integrals are exclusively defined on subsets \mathbbB, with values outside the scope of \mathbbB being deemed irrelevant. Additionally, we present several notable consequences, including the relationship between \mathbbB-stochastic integrals and existing stochastic integrals, as well as Itô's formula for \mathbbB-inner semimartingales. In the context of models pertaining to uncertain time-horizons in mathematical finance, we establish essentials of mathematical finance for general markets characterized by sudden-stop horizons. This is achieved by defining self-financing strategies, admissible strategies, and no-arbitrary conditions. In such financial markets, the exclusivity characteristic inherent in \mathbbB-stochastic integrals offers investors a viable alternative approach. This approach enables them to effectively filter out unnecessary information pertaining to asset price dynamics and portfolio strategies that extend beyond the predefined time-horizons.