**1. Introduction**

In classical risk theory, the surplus process of an insurance company is modelled by the compound Poisson risk model. For both applied and theoretical investigations, calculation of ruin probabilities for such model is of particular interest. In order to avoid technical calculations, *diffusion approximation* is often considered e.g., (Asmussen and Albrecher 2010; Grandell 1991; Iglehart 1969; Klugman et al. 2012), which results in tractable approximations for the interested finite-time or infinite-time ruin probabilities. The basic premise for the approximation is to let the number of claims grow in a unit time interval and to make the claim sizes smaller in such a way that the risk process converges to a Brownian motion with drift. Precisely, the Brownian motion risk process is defined by

$$\mathcal{R}(t) = \mathbf{x} + p\mathbf{t} - \sigma \mathcal{B}(\mathbf{t}), \quad \mathbf{t} \ge \mathbf{0}\_r$$

where *x* > 0 is the *initial capital*, *p* > 0 is the *net profit rate* and *σ<sup>B</sup>*(*t*) models the net loss process with *σ* > 0 the volatility coefficient. Roughly speaking, *σ<sup>B</sup>*(*t*) is an approximation of the total claim amount process by time *t* minus its expectation, the latter is usually called the *pure premium* amount and calculated to cover the average payments of claims. The net profit, also called *safety loading*, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. Ruin related problems for Brownian models have been well studied; see, for example, Asmussen and Albrecher (2010); Gerber and Shiu (2004).

In recent years, multi-dimensional risk models have been introduced to model the surplus of multiple business lines of an insurance company or the suplus of collaborating companies (e.g., insurance and reinsurance). We refer to Asmussen and Albrecher (2010) [Chapter XIII 9] and Avram and Loke (2018); Avram and Minca (2017); Avram et al. (2008a, 2008b); Albrecher et al. (2017); Azcue and Muler (2018); Azcue et al. (2019); Foss et al. (2017); Ji and Robert (2018) for relevant recent discussions. It is concluded in the literature that in comparison with the well-understood 1-dimensional risk models, study of multi-dimensional risk models is much more challenging. It was shown recently in Delsing et al. (2019) that multi-dimensional Brownian model can serve as approximation of a multi-dimensional classical risk model in a Markovian environment. Therefore, obtained results for multi-dimensional Brownian model can serve as approximations of the multi-dimensional classical risk models in a Markovian environment; ruin probability approximation has been used in the aforementioned paper. Actually, multi-dimensional Brownian models have drawn a lot of attention due to its tractability and practical relevancy.

A *d*-dimensional Brownian model can be defined in a matrix form as

$$\mathcal{R}(t) = \mathbf{x} + \mathbf{p}t - \mathbf{X}(t), \ t \ge 0, \text{ with } \mathbf{X}(t) = AB(t),$$

where *x* = (*<sup>x</sup>*1, ··· , *xd*)', *p* = (*p*1, ··· , *pd*)' ∈ (0, ∞)*<sup>d</sup>* are, respectively, (column) vectors representing the initial capital and net profit rate, *A* ∈ R*d*×*<sup>d</sup>* is a non-singular matrix modelling dependence between different business lines and *<sup>B</sup>*(*t*)=(*<sup>B</sup>*1(*t*), ... , *Bd*(*t*))', *t* ≥ 0 is a standard *d*-dimensional Brownian motion (BM) with independent coordinates. Here ' is the transpose sign. In what follows, vectors are understood as column vectors written in bold letters.

Different types of ruin can be considered in multi-dimensional models, which are relevant to the probability that the surplus of one or more of the business lines drops below zero in a certain time interval [0, *T*] with *T* either a finite constant or infinity. One of the commonly studied is the so-called *simultaneous ruin probability* defined as

$$Q\_T(\mathfrak{x}) := \mathbb{P}\left\{ \exists\_{t \in [0,T]} \bigcap\_{i=1}^d \left\{ \mathcal{R}\_i(t) < 0 \right\} \right\},$$

which is the probability that at a certain time *t* ∈ [0, *T*] all the surpluses become negative. Here for *T* < <sup>∞</sup>, *QT*(*x*) is called finite-time simultaneous ruin probability, and *Q* ∞(*x*) is called infinite-time simultaneous ruin probability. Simultaneous ruin probability, which is essentially the hitting probability of *R*(*t*) to the orthant {*y* ∈ R*<sup>d</sup>* : *yi* < 0, *i* = 1, . . . , *d*}, has been discussed for multi-dimensional Brownian models in different contexts; see De¸bicki et al. (2018); Garbit and Raschel (2014). In Garbit and Raschel (2014), for fixed *x* the asymptotic behaviour of *QT*(*x*) as *T* → ∞ has been discussed. Whereas, in De¸bicki et al. (2018), the asymptotic behaviour, as *u* → <sup>∞</sup>, of the infinite-time ruin probability *Q* <sup>∞</sup>(*x*), with *x* = *αu* = (*<sup>α</sup>*1*u*, *α*2*u*, ... , *<sup>α</sup>d<sup>u</sup>*)', *αi* > 0, 1 ≤ *i* ≤ *d* has been obtained. Note that it is common in risk theory to derive the later type of asymptotic results for ruin probabilities; see, for example, Avram et al. (2008a); Embrechts et al. (1997); Mikosch (2008).

Another type of ruin probability is the *component-wise (or joint) ruin probability* defined as

$$P\_T(\mathbf{x}) := \mathbb{P}\left\{\bigcap\_{i=1}^d \left\{\exists\_{t \in [0,T]} \mathcal{R}\_i(t) < 0\right\}\right\} = \mathbb{P}\left\{\bigcap\_{i=1}^d \left\{\sup\_{t\_i \in [0,T]} (X\_i(t\_i) - p\_i t\_i) > \mathbf{x}\_i\right\}\right\},\tag{1}$$

which is the probability that all surpluses ge<sup>t</sup> below zero but possibly at different times. It is this possibility that makes the study of *PT*(*x*) more difficult.

The study of joint distribution of the extrema of multi-dimensional BM over finite-time interval has been proved to be important in quantitative finance; see, for example, He et al. (1998); Kou and Zhong (2016).

We refer to Delsing et al. (2019) for a comprehensive summary of related results. Due to the complexity of the problem, two-dimensional case has been the focus in the literature and for this case some explicit formulas can be obtained by using a PDE approach. Of particular relevance to the ruin probability *PT*(*x*) is a result derived in He et al. (1998) which shows that

$$\begin{aligned} &\mathbb{P}\left\{\sup\_{t\in[0,T]}(X\_1(t) - p\_1t) \le x\_1, \sup\_{s\in[0,T]}(X\_2(s) - p\_2s) \le x\_2\right\}, \\ &y = e^{a\_1x\_1 + a\_2x\_2 + bT}f(x\_1, x\_2, T), \end{aligned}$$

where *a*1, *a*2, *b* are known constants and *f* is a function defined in terms of infinite-series, double-integral and Bessel function. Using the above formula one can derive an expression for *PT*(*x*) in two-dimensional case as follows

$$\begin{split} \mathbb{P}\_{\mathbb{P}}(\mathbf{x}) &= 1 - \mathbb{P}\left\{ \sup\_{t \in [0,T]} (X\_1(t) - p\_1 t) \le x\_1 \right\} - \mathbb{P}\left\{ \sup\_{s \in [0,T]} (X\_2(s) - p\_2 s) \le x\_2 \right\} \\ &+ \mathbb{P}\left\{ \sup\_{t \in [0,T]} (X\_1(t) - p\_1 t) \le x\_1, \sup\_{s \in [0,T]} (X\_2(s) - p\_2 s) \le x\_2 \right\}, \end{split} \tag{2}$$

where the expression for the distribution of single supremum is also known; see He et al. (1998). Note that even though we have obtained explicit expression of *PT*(*x*) in (2) for the two-dimensional case, it seems difficult to derive the explicit form of the corresponding infinite-time ruin probability *<sup>P</sup>*∞(*x*) by simply putting *T* → ∞ in (2).

By assuming *x* = *αu* = (*<sup>α</sup>*1*u*, *α*2*u*, ... , *<sup>α</sup>d<sup>u</sup>*)', *αi* > 0, 1 ≤ *i* ≤ *d*, we aim to analyse the asymptotic behaviour of the infinite-time ruin probability *<sup>P</sup>*∞(*x*) as *u* → ∞. Applying Theorem 1 in De¸bicki et al. (2010) we arrive at the following logarithmic asymptotics

$$-\frac{1}{\mu}\ln P\_{\boldsymbol{\Theta}}(\boldsymbol{x}) \quad \sim \quad \frac{1}{2}\inf\_{t>0} \inf\_{\boldsymbol{v}\geq\boldsymbol{a}+pt} \boldsymbol{v}^{\top}\boldsymbol{\Sigma}\_{\boldsymbol{t}}^{-1}\boldsymbol{v}, \quad \text{as } \boldsymbol{u} \to \infty \tag{3}$$

provided Σ*t* is non-singular, where *pt* := (*p*1*t*1, ··· , *pdtd*)', inequality of vectors are meant component-wise, and Σ−<sup>1</sup> *t* is the inverse matrix of the covariance function Σ*t* of (*<sup>X</sup>*1(*<sup>t</sup>*1), ··· , *Xd*(*td*)), with *t* = (*<sup>t</sup>*1, ··· , *td*)' and **0** = (0, ··· , 0)' ∈ R*d*. Let us recall that conventionally for two given positive functions *f*(·) and *h*(·), we write *f*(*x*) ∼ *h*(*x*) if lim*x*→∞ *f*(*x*)/*h*(*x*) = 1.

For more precise analysis on *<sup>P</sup>*∞(*x*), it seems crucial to first solve the two-layer optimization problem in (3) and find the optimization points *t*0. As it can be recognized in the following, when dealing with *d*-dimensional case with *d* > 2 the calculations become highly nontrivial and complicated. Therefore, in this contribution we only discuss a tractable two-dimensional model and aim for an explicit logarithmic asymptotics by solving the minimization problem in (3).

In the classical ruin theory when analysing the compound Poisson model or Sparre Andersen model, the so-called *adjustment coefficient* is used as a measure of goodness; see, for example, Asmussen and Albrecher (2010) or Rolski et al. (2009). It is of interest to obtain the solution of the minimization problem in (3) from a practical point of view, as it can be seen as an analogue of the adjustment coefficient and thus we could ge<sup>t</sup> some insights about the risk that the company is facing. As discussed in Asmussen and Albrecher (2010) and Li et al. (2007) it is also of interest to know how the dependence between different risks influences the joint ruin probability, which can be easily analysed through the obtained logarithmic asymptotics; see Remark 2.

The rest of this paper is organised as follows. In Section 2, we formulate the two-dimensional Brownian model and give the main results of this paper. The main lines of proof with auxiliary lemmas are displayed in Section 3. In Section 4 we conclude the paper. All technical proofs of the lemmas in Section 3 are presented in Appendix A.

### **2. Model Formulation and Main Results**

Due to the fact that component-wise ruin probability *<sup>P</sup>*∞(*x*) does not change under scaling, we can simply assume that the volatility coefficient for all business lines is equal to 1. Furthermore, noting that the timelines for different business lines should be distinguished as shown in (1) and (3), we introduce a two-parameter extension of correlated two-dimensional BM defined as

$$(X\_1(t), X\_2(s)) = \left(B\_1(t), \,\rho B\_1(s) + \sqrt{1 - \rho^2} B\_2(s)\right), \,\, t, s \ge 0,$$

with *ρ* ∈ (−1, 1) and mutually independent Brownian motions *B*1, *B*2. We shall consider the following two dependent insurance risk processes

$$R\_i(t) = u + \mu\_i t - X\_i(t), \ t \ge 0, \quad i = 1, 2, 3$$

where *μ*1, *μ*2 > 0 are net profit rates, *u* is the initial capital (which is assumed to be the same for both business lines, as otherwise, the calculations become rather complicated). We shall assume without loss of generality that *μ*1 ≤ *μ*2. Here, *μi* is different from *pi* (see (1)) in the sense that it corresponds to the (scaled) model with volatility coefficient standardized to be 1.

In this contribution, we shall focus on the logarithmic asymptotics of

$$\begin{split} P(u) := P\_{\mathbb{R}}(u(1,1)^{\top}) &= \mathbb{P}\left\{ \left\{ \exists\_{t \ge 0} R\_1(t) < 0 \right\} \cap \left\{ \exists\_{s \ge 0} R\_2(s) < 0 \right\} \right\} \\ &= \mathbb{P}\left\{ \sup\_{t \ge 0} (X\_1(t) - \mu\_1 t) > u, \sup\_{s \ge 0} (X\_2(s) - \mu\_2 s) > u \right\}, \text{ as } u \to \infty. \end{split} \tag{4}$$

Define

$$\hat{\rho}\_1 = \frac{\mu\_1 + \mu\_2 - \sqrt{(\mu\_1 + \mu\_2)^2 - 4\mu\_1(\mu\_2 - \mu\_1)}}{4\mu\_1} \in [0, \frac{1}{2}), \quad \hat{\rho}\_2 = \frac{\mu\_1 + \mu\_2}{2\mu\_2} \tag{5}$$

and let

$$t^\* = t^\*(\rho) = s^\* = s^\*(\rho) := \sqrt{\frac{2(1-\rho)}{\mu\_1^2 + \mu\_2^2 - 2\rho\mu\_1\mu\_2}}.\tag{6}$$

The following theorem constitutes the main result of this contribution.

**Theorem 1.** *For the joint infinite-time ruin probability* (4) *we have, as u* → <sup>∞</sup>*,* 

$$-\frac{\log(P(u))}{u} \sim \begin{cases} \frac{2(\mu\_2 + (1 - 2\rho)\mu\_1)}{\frac{\mu\_1 + \mu\_2 + 2/t^\*}{1 + \rho}}, & \text{if } \rho\_1 < \rho < \rho\_2 \\\ 2\mu\_{2\prime} & \text{if } \rho\_2 \le \rho < 1. \end{cases}$$

**Remark 2.** *(a) Following the classical one-dimensional risk theory we can call quantities on the right hand side in Theorem 1 as adjustment coefficients. They serve sometimes as a measure of goodness for a risk business.*

*(b) One can easily check that adjustment coefficient as a function of ρ is continuous, strictly decreasing on* (−1, *ρ*ˆ2] *and it is constant, equal to* 2*μ*2 *on* [*ρ*ˆ2, 1)*. This means that as the two lines of business becomes more positively correlated the risk of ruin becomes larger, which is consistent with the intuition.*

Define

$$\log(t,s) := \inf\_{\substack{x \geq 1 + \mu\_1 t \\ y \geq 1 + \mu\_2 s}} \left( \mathbf{x}, y \right) \Sigma\_{ts}^{-1} \left( \mathbf{x}, y \right)^{\top}, \quad t, s > 0,\tag{7}$$

where Σ−<sup>1</sup> *ts* is the inverse matrix of Σ*ts* = *t ρ t* ∧ *s ρ t* ∧ *s s* , with *t* ∧ *s* = min(*<sup>t</sup>*,*<sup>s</sup>*) and *ρ* ∈ (−1, <sup>1</sup>).

The proof of Theorem 1 follows from (3) which implies that the logarithmic asymptotics for *<sup>P</sup>*(*u*) is of the form

$$-\frac{1}{u}\ln P(u) \quad \sim \quad \frac{g(\mathbf{t}\_0)}{2}, \quad u \to \infty,\tag{8}$$

where

$$\mathbf{g}(\mathbf{t}\mathbf{o}) = \inf\_{(t,\mathbf{s})\in(0,\infty)^2} \mathbf{g}(t,\mathbf{s}),\tag{9}$$

and Proposition 3 below, wherein we list dominating points *t*0 that optimize the function *g* over (0, ∞)<sup>2</sup> and the corresponding optimal values *g*(*<sup>t</sup>*0).

In order to solve the two-layer minimization problem in (9) (see also (7)) we define for *t*,*<sup>s</sup>* > 0 the following functions:

$$\begin{aligned} \mathbf{g}\_1(t) &= \frac{(1+\mu\_1 t)^2}{t}, & \mathbf{g}\_2(s) &= \frac{(1+\mu\_2 s)^2}{s}, \\ \mathbf{g}\_3(t,s) &= (1+\mu\_1 t, 1+\mu\_2 s) \,\,\Sigma\_{ts}^{-1} \left(1+\mu\_1 t, 1+\mu\_2 s\right)^\top. \end{aligned}$$

Since *t* ∧ *s* appears in the above formula, we shall consider a partition of the quadrant (0, ∞)2, namely

$$(0, \infty)^2 = A \cup L \cup B, \quad A = \{\mathbf{s} < t\}, \; L = \{\mathbf{s} = t\}, \; B = \{\mathbf{s} > t\}. \tag{10}$$

For convenience we denote *A* = {*s* ≤ *t*} = *A* ∪ *L* and *B* = {*s* ≥ *t*} = *B* ∪ *L*. Hereafter, all sets are defined on (0, ∞)2, so (*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, ∞)<sup>2</sup> will be omitted.

Note that *g*3(*<sup>t</sup>*,*<sup>s</sup>*) can be represented in the following form:

$$\begin{array}{rcl} \mathcal{g}\_{3}(t,s) & = & \begin{cases} \end{cases} \mathcal{g}\_{A}(t,s) := \frac{(1+\mu\_{2}s)^{2}}{s} + \frac{((1+\mu\_{1}t)-\rho(1+\mu\_{2}s))^{2}}{t-\rho^{2}s}, & \text{if } (t,s) \in \overline{A} \\\ \mathcal{g}\_{\overline{B}}(t,s) := \frac{(1+\mu\_{1}t)^{2}}{t} + \frac{((1+\mu\_{2}s)-\rho(1+\mu\_{1}t))^{2}}{s-\rho^{2}t}, & \text{if } (t,s) \in \overline{B}. \end{array} \tag{11}$$

Denote further

$$\mathcal{g}\_L(s) := \mathcal{g}\_A(s,s) = \mathcal{g}\_B(s,s) = \frac{(1+\mu\_1 s)^2 + (1+\mu\_2 s)^2 - 2\rho(1+\mu\_1 s)(1+\mu\_2 s)}{(1-\rho^2)s}, \quad s > 0. \tag{12}$$

In the next proposition we identify the so-called dominating points, that is, points *t*0 for which function defined in (7) achieves its minimum. This identification might also be useful for deriving a more subtle asymptotics for *<sup>P</sup>*(*u*).

**Notation:** *In the following, in order to keep the notation consistent, ρ* ≤ *μ*1/*μ*2 *is understood as ρ* < 1 *if μ*1 = *μ*2.
