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We consider a two-dimensional ruin problem where the surplus process ofbusiness lines is modelled by a two-dimensional correlated Brownian motion withdrift. We study the ruin function $P(u)$ for the component-wise ruin (that isboth business lines are ruined in an infinite-time horizon), where $u$ is thesame initial capital for each line. We measure the goodness of the business byanalysing 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 thetwo surplus processes. In order to work out the adjustment coefficient we solvea two-layer optimization problem.

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