Stochastic Integration on Stochastic Sets of Interval Type and Applications to Mathematical Finance

Abstract

In the existing works, stochastic sets $\mathbb{B}$ of interval type, along with $\mathbb{B}$-stochastic processes, were introduced within the framework of stochastic analysis. In this paper, we undertake the construction of $\mathbb{B}$-stochastic integration by exploring three novel types of $\mathbb{B}$-stochastic integrals: Stieltjes integrals of $\mathbb{B}$-predictable processes with respect to $\mathbb{B}$-adapted processes with finite variation, stochastic integrals of $\mathbb{B}$-predictable processes with respect to $\mathbb{B}$-inner local martingales, and stochastic integrals of $\mathbb{B}$-predictable processes with respect to $\mathbb{B}$-inner semimartingales. These $\mathbb{B}$-stochastic integrals are exclusively defined on subsets $\mathbb{B}$, with values outside the scope of $\mathbb{B}$ being deemed irrelevant. Additionally, we present several notable consequences, including the relationship between $\mathbb{B}$-stochastic integrals and existing stochastic integrals, as well as It\^{o}'s formula for $\mathbb{B}$-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 $\mathbb{B}$-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.