*Article* **Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model**

### **Krzysztof De¸bicki 1,†, Lanpeng Ji 2,\* and Tomasz Rolski 1,†**


Received: 14 June 2019; Accepted: 29 July 2019; Published: 1 August 2019

**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 *<sup>P</sup>*(*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 *<sup>P</sup>*(*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.

**Keywords:** adjustment coefficient; logarithmic asymptotics; quadratic programming problem; ruin probability; two-dimensional Brownian motion
