# On ruin probabilities in the presence of risky investments and random switching

## TL;DR

This paper analyzes the asymptotic ruin probabilities in a model with claims arriving via renewal process and investments modeled as geometric Brownian motion with random, switching coefficients, using implicit renewal theory.

## Contribution

It introduces a model with random switching of investment parameters and derives power-function bounds for ruin probabilities, extending classical risk models.

## Key findings

- Power-function bounds for ruin probabilities established.
- Conditions for Lundberg's exponent existence identified.
- Asymptotic behavior characterized under switching investment parameters.

## Abstract

We study the asymptotic behavior of ruin probabilities, as the initial reserve goes to infinity, for a reserve process model where claims arrive according to a renewal process, while between the claim times the process has the dynamics of geometric Brownian motion-type It\^o processes with time-dependent random coefficients. These coefficients are ``reset" after each claim time, switching to new values independent of the past history of the process. We use the implicit renewal theory to obtain power-function bounds for the eventual ruin probability. In the special case when the random drift and diffusion coefficients of the investment returns process remain unchanged between consecutive claim arrivals, we obtain conditions for existence of Lundberg's exponent for our model ensuring the power function behaviour for the ruin probability.