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

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 −lnP(u)/u as u tends to infinity, which depends essentially on the correlation ρ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.