Random walks with sticky barriers

Bohun, Vladyslav; Marynych, Alexander

doi:10.15559/22-vmsta202

Cited by 1 publication

(1 citation statement)

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“…The model ( 2) can be further extended, cf. E. Spare Andersen's model [2] or models considered in [6,7].…”

Section: Introductionmentioning

confidence: 99%

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Gervė,

Grigutis

2024

Modern Stochastics: Theory and Applications

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In this paper, the distribution function \[ \varphi (u)=\mathbb{P}\Bigg(\underset{n\geqslant 1}{\sup }{\sum \limits_{i=1}^{n}}({X_{i}}-\kappa )\lt u\Bigg),\] and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.

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“…The model ( 2) can be further extended, cf. E. Spare Andersen's model [2] or models considered in [6,7].…”

Section: Introductionmentioning

confidence: 99%

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Gervė,

Grigutis

2024

Modern Stochastics: Theory and Applications

View full text Add to dashboard Cite

show abstract

Random walks with sticky barriers

Abstract: A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.

Cited by 1 publication

References 9 publications

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

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