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On dual risk models with proportional gains and dependencies

Ioannis Dimitriou

TL;DR

The work tackles ruin probabilities and the time-to-ruin in a dual risk model with proportional gains under various dependence structures between gain interarrival times and sizes, including causal dependence and (generalized) FGM copulas. It develops a Laplace-transform framework and contraction-based iterative methods to obtain explicit expressions for $R(x)$ and the distribution of $\tau_x$, covering exponential, Erlang, and mixed Erlang gain sizes, as well as scenarios with uniformly distributed proportional gains and two-sided jumps. The results specialize to independence as a limiting case and extend to several generalized dependence settings, providing actionable analytical tools and iterative schemes for computation. The findings have practical implications for insurance and startup finance where gains depend on inflow timing and magnitude, enabling more nuanced risk assessment under dependence.

Abstract

In this work, we consider extensions of the dual risk model with proportional gains by introducing a dependence structure between gain sizes and gain interrarrival times. Among others, we further consider the case where the proportional parameter is randomly chosen, the case where it is a uniformly random variable, as well as the case where we may have upwards as well as downwards jumps. Moreover, we consider the case with causal dependence structure, as well as the case where the dependence is based on the generalized Farlie-Gumbel-Morgenstern copula. The ruin probability and the distribution of the time to ruin are investigated.

On dual risk models with proportional gains and dependencies

TL;DR

The work tackles ruin probabilities and the time-to-ruin in a dual risk model with proportional gains under various dependence structures between gain interarrival times and sizes, including causal dependence and (generalized) FGM copulas. It develops a Laplace-transform framework and contraction-based iterative methods to obtain explicit expressions for and the distribution of , covering exponential, Erlang, and mixed Erlang gain sizes, as well as scenarios with uniformly distributed proportional gains and two-sided jumps. The results specialize to independence as a limiting case and extend to several generalized dependence settings, providing actionable analytical tools and iterative schemes for computation. The findings have practical implications for insurance and startup finance where gains depend on inflow timing and magnitude, enabling more nuanced risk assessment under dependence.

Abstract

In this work, we consider extensions of the dual risk model with proportional gains by introducing a dependence structure between gain sizes and gain interrarrival times. Among others, we further consider the case where the proportional parameter is randomly chosen, the case where it is a uniformly random variable, as well as the case where we may have upwards as well as downwards jumps. Moreover, we consider the case with causal dependence structure, as well as the case where the dependence is based on the generalized Farlie-Gumbel-Morgenstern copula. The ruin probability and the distribution of the time to ruin are investigated.
Paper Structure (11 sections, 6 theorems, 134 equations)

This paper contains 11 sections, 6 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. A dual risk model with causal dependence structure and proportional gains
  3. The ruin probability
  4. The time to ruin
  5. The dual risk model with additive/proportional gains and dependencies based on FGM copula
  6. The time to ruin
  7. An extension to the generalized FGM copula
  8. The dual risk model with proportional gains and a linear dependence among gain interarrival times and surplus level
  9. A dual risk model with upward and downward jumps and randomly proportional gains
  10. The dual risk model with uniformly proportional gains
  11. Conclusion & future work

Key Result

Theorem 5

The Laplace transform of the ruin probability $\rho(s)$ is given in ghy, where the terms $\rho^{(l)}(\mu)$, $l=0,1,\ldots,n-1$, $\rho^{(l)}(2\mu)$, $l=0,1,\ldots,2n-2$, are obtained as the solution of a system of $3n-1$ equations that is derived by differentiating $l$ times ghy with respect to $s$,

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Proposition 8
  • Remark 9
  • Proposition 10
  • ...and 7 more