# On the Compound Beta-Binomial Risk Model with Delayed Claims and Randomized Dividends

## TL;DR

This paper introduces a novel discrete-time compound Beta-Binomial risk model incorporating delayed claims and randomized dividends, providing recursive formulas for ruin probabilities and related risk measures.

## Contribution

It develops a new risk model with random claim and dividend probabilities following Beta distributions, and derives recursive formulas for key risk measures.

## Key findings

- Recursive formulas for ruin probability and deficit at ruin.
- Explicit expressions for probability of claim causing ruin.
- Analysis of the Gerber-Shiu function for different dividend thresholds.

## Abstract

In this paper, we propose the discrete time Compound Beta-Binomial Risk Model with by-claims, delayed by-claims and randomized dividends. We then analyze the Gerber-Shiu function for the cases where the dividend threshold $d=0$ and $d>0$ under the assumption that the constant discount rate $\nu \in (0,1)$. More specifically, we study the discrete time compound binomial risk model subject to the assumption that the probabilities with which the claims, by-claims occur and the dividends are issued are not fixed(constant), instead the probabilities are random and follow a Beta distribution with parameters $a_{i}$ and $b_{i}$, $i = 1, 2, 3$. Recursive expressions for the Gerber-Shiu function corresponding to the proposed model are obtained. The recursive relations are further utilized to obtain significant ruin related quantities of interest. Recursive relations for probability of ruin, the probability of the deficit at ruin, the generating function of the deficit at ruin and the probability of surplus at ruin and for the probability of the claim causing ruin are obtained.