arXiv subjects

Assylliya K. Zhunussova

Publications and source records attributed to Assylliya K. Zhunussova.

2 recordsLinked to original sources

arXiv2026

Properties of a Special Type of Filtration and its Martingale Criteria

This article investigates the structural properties of stochastic processes relative to a generalized single jump filtration, extending the framework introduced by A.A. Gushchin (2020) to the case of a non-trivial initial $σ$-algebra $\mathscr{H}$. By leveraging the general theory of processes and optional projection techniques, we establish fundamental measurability criteria for random variables and a complete characterization of stopping times and adapted processes. Furthermore, we derive comprehensive martingale and local martingale criteria, providing necessary and sufficient conditions for the preservation of the martingale property in this extended setting.

math.PR↗

arXiv2026

Explicit Predictable Compensators for Single Jump Processes with Initial Information

We study the predictable compensators of stochastic processes in a single jump filtration augmented with initial information represented by a sub-$σ$-algebra $\mathcal{H}$. We consider adapted càdlàg processes of finite variation and give an explicit construction of their predictable compensators. The main difficulty arises when the jump size has a heavy tail and lacks integrability, so that the classical Doob-Meyer decomposition does not apply. To overcome this, we use the theory of $σ$-martingales. We establish necessary and sufficient conditions for a process to be a $σ$-martingale and explicitly compute the compensator of a suitably weighted process. This yields an explicit relation between the continuous drift and the expected jump component of the original process.

math.PR↗