**4. Conclusions and Discussions**

In the multi-dimensional risk theory, the so-called "ruin" can be defined in different manner. Motivated by diffusion approximation approach, in this paper we modelled the risk process via a multi-dimensional BM with drift. We analyzed the component-wise infinite-time ruin probability for dimension *d* = 2 by solving a two-layer optimization problem, which by the use of Theorem 1 from De¸bicki et al. (2010) led to the logarithmic asymptotics for *<sup>P</sup>*(*u*) as *u* → <sup>∞</sup>, given by explicit form of the adjustment coefficient *γ* = *g*(*<sup>t</sup>*0)/2 (see (8)). An important tool here is Lemma 5 on the quadratic programming, cited from Hashorva (2005). In this way we were also able to identify the dominating points by careful analysis of different regimes for *ρ* and specify three regimes with different formulas for *γ* (see Theorem 1). An open and difficult problem is the derivation of exact asymptotics for *<sup>P</sup>*(*u*) in (4), for which the problem of finding dominating points would be the first step. A refined double-sum method as in De¸bicki et al. (2018) might be suitable for this purpose. A detailed analysis of the case for dimensions *d* > 2 seems to be technically very complicated, even for getting the logarithmic asymptotics. We also note that a more natural problem of considering *Ri*(*t*) = *αiu* + *μit* − *Xi*(*t*), with general *αi* > 0, *i* = 1, 2, leads to much more difficult technicalities with the analysis of *γ*.

Define the ruin time of component *i*, 1 ≤ *i* ≤ *d*, by *Ti* = min{*t* : *Ri*(*t*) < 0} and let *<sup>T</sup>*(1) ≤ *<sup>T</sup>*(2) ≤ ... ≤ *<sup>T</sup>*(*d*) be the order statistics of ruin times. Then the component-wise infinite-time ruin probability is equivalent to P *<sup>T</sup>*(*d*) < ∞ while the ruin time of at least one business line is *T*min = *<sup>T</sup>*(1) = min*i Ti*. Other interesting problems like P *<sup>T</sup>*(*j*) < ∞ have not ye<sup>t</sup> been analysed. For instance, it would be interesting for *d* = 3 to study the case *<sup>T</sup>*(2). The general scheme on how to obtain logarithmic asymptotics for such problems was discussed in De¸bicki et al. (2010).

Random vector *X* **¯** = (sup*t*≥0(*<sup>X</sup>*1(*t*) − *p*1*<sup>t</sup>*), ... , sup*t*≥0(*Xd*(*t*) − *pd<sup>t</sup>*))' has exponential marginals and if it is not concentrated on a subspace of dimension less than *d*, it defines a multi-variate exponential distribution. In this paper for dimension *d* = 2, we derived some asymptotic properties of such distribution. Little is known about properties of this multi-variate distriution and more studies on it would be of interest. For example a correlation structure of *X* **¯** is unknown. In particular, in the context of findings presented

in this contribution, it would be interesting to find the correlation between sup*t*≥0(*<sup>X</sup>*1(*t*) − *μ*1*<sup>t</sup>*) and sup*t*≥0(*<sup>X</sup>*2(*t*) − *μ*2*<sup>t</sup>*).

**Author Contributions:** Investigation, K.D., L.J., T.R.; writing–original draft preparation, L.J.; writing–review and editing, K.D., T.R.

**Funding:** T.R. & K.D. acknowledge partial support by NCN gran<sup>t</sup> number 2015/17/B/ST1/01102 (2016-2019).

**Acknowledgments:** We are thankful to the referees for their carefully reading and constructive suggestions which significantly improved the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
