# Logarithmic asymptotics for probability of component-wise ruin in a two-dimensional Brownian model

## TL;DR

This paper analyzes the probability of simultaneous ruin in a two-dimensional correlated Brownian motion model, deriving asymptotic formulas for large initial capital and exploring the impact of correlation on ruin probabilities.

## Contribution

It provides a novel asymptotic analysis of the component-wise ruin probability in a two-dimensional Brownian model, including solving a complex two-layer optimization problem.

## Key findings

- The adjustment coefficient depends critically on the correlation between the two processes.
- Asymptotic ruin probabilities decay exponentially with initial capital.
- The paper offers explicit formulas for the ruin probability's logarithmic asymptotics.

## Abstract

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function $P(u)$ for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where $u$ is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of $-\ln P(u)/u$ as $u$ tends to infinity, which depends essentially on the correlation $\rho$ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.