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

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation and Main Results

**Theorem**

**1.**

**Remark**

**2.**

**Notation:**In the following, in order to keep the notation consistent, $\rho \le {\mu}_{1}/{\mu}_{2}$ is understood as $\rho <1$ if ${\mu}_{1}={\mu}_{2}.$

**Proposition**

**3.**

- (i)
- Suppose that $-1<\rho <0$.For ${\mu}_{1}<{\mu}_{2}$ we have$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{A}({t}_{A},{s}_{A})=4({\mu}_{2}+(1-2\rho ){\mu}_{1}),\end{array}$$For ${\mu}_{1}={\mu}_{2}=:\mu $ we have$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{A}({t}_{A},{s}_{A})={g}_{B}({t}_{B},{s}_{B})=8(1-\rho )\mu ,\end{array}$$
- (ii)
- Suppose that $0\le \rho <{\widehat{\rho}}_{1}$. We have$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{A}({t}_{A},{s}_{A})=4({\mu}_{2}+(1-2\rho ){\mu}_{1}),\end{array}$$
- (iii)
- Suppose that $\rho ={\widehat{\rho}}_{1}$. We have$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{A}({t}_{A},{s}_{A})=4({\mu}_{2}+(1-2\rho ){\mu}_{1}),\end{array}$$
- (iv)
- Suppose that ${\widehat{\rho}}_{1}<\rho <{\widehat{\rho}}_{2}$. We have$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{A}({t}^{*},{s}^{*})={g}_{L}\left({t}^{*}\right)=\frac{2}{1+\rho}({\mu}_{1}+{\mu}_{2}+2/{t}^{*}),\end{array}$$
- (v)
- Suppose that $\rho ={\widehat{\rho}}_{2}$. We have ${t}^{*}\left({\widehat{\rho}}_{2}\right)={s}^{*}\left({\widehat{\rho}}_{2}\right)=1/{\mu}_{2}$ and$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{A}(1/{\mu}_{2},1/{\mu}_{2})={g}_{L}(1/{\mu}_{2})={g}_{2}(1/{\mu}_{2})=4{\mu}_{2},\end{array}$$
- (vi)
- Suppose that ${\widehat{\rho}}_{2}<\rho <1$. We have$$\begin{array}{c}\hfill g\left({\mathit{t}}_{0}\right)={g}_{2}(1/{\mu}_{2})=4{\mu}_{2},\end{array}$$

**Remark**

**4.**

## 3. Proofs of Main Results

**Lemma**

**5.**

**Lemma**

**6.**

- (i)
- If $-1<\rho \le {\mu}_{1}/{\mu}_{2},$ then$$g(t,s)={g}_{3}(t,s),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(t,s)\in {(0,\infty )}^{2}.$$
- (ii)
- If $1>\rho >{\mu}_{1}/{\mu}_{2},$ then$$g(t,s)=\left\{\begin{array}{cc}{g}_{3}(t,s),& \mathit{if}\phantom{\rule{4.pt}{0ex}}(t,s)\in {D}_{1}\\ {g}_{2}\left(s\right),& \mathit{if}\phantom{\rule{4.pt}{0ex}}(t,s)\in {D}_{2}.\end{array}\right.$$

#### 3.1. Proof of Proposition 3

#### 3.1.1. Case $-1<\rho <0$

#### 3.1.2. Case $0\le \rho <1$

**Lemma**

**7.**

- (a)
- The function ${t}^{*}\left(\rho \right)$ is a decreasing function on $[0,1]$ and both ${t}_{B}\left(\rho \right)$ and ${s}_{1}^{*}\left(\rho \right)$ are decreasing functions on $({\mu}_{1}/{\mu}_{2},1)$.
- (b)
- The function ${t}^{**}\left(\rho \right)$ decreases from $1/{\mu}_{2}$ at $\rho =0$ to some positive value and then increases to $1/{\mu}_{2}$ at ${\widehat{\rho}}_{2}$ (defined in (5)) and then increases to $+\infty $ at the root $\widehat{\rho}\in (0,1]$ of the equation ${\mu}_{2}+\rho {\mu}_{1}-2{\mu}_{2}{\rho}^{2}=0.$
- (c)
- For $0\le \rho \le {\mu}_{1}/{\mu}_{2}$, we have$$\begin{array}{c}\hfill {t}_{B}\left(\rho \right)\ge {t}^{**}\left(\rho \right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{t}^{*}\left(\rho \right)\ge {t}^{**}\left(\rho \right),\end{array}$$
- (d)
- It holds that$$\begin{array}{c}\hfill {t}^{*}\left({\widehat{\rho}}_{2}\right)={t}_{B}\left({\widehat{\rho}}_{2}\right)={s}_{1}^{*}\left({\widehat{\rho}}_{2}\right)={t}^{**}\left({\widehat{\rho}}_{2}\right)=\frac{1}{{\mu}_{2}}.\end{array}$$

- (i)
- ${t}^{*}\left(\rho \right)<{s}_{1}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\mu}_{1}/{\mu}_{2},{\widehat{\rho}}_{2}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{t}^{*}\left(\rho \right)>{s}_{1}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{2},1).$
- (ii)
- ${t}_{B}\left(\rho \right)<{s}_{1}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\mu}_{1}/{\mu}_{2},{\widehat{\rho}}_{2}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{t}_{B}\left(\rho \right)>{s}_{1}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{2},1).$
- (iii)
- ${t}^{**}\left(\rho \right)<{s}_{1}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\mu}_{1}/{\mu}_{2},{\widehat{\rho}}_{2}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{t}^{**}\left(\rho \right)>{s}_{1}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{2},\widehat{\rho}).$
- (iv)
- ${t}^{**}\left(\rho \right)<{t}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\mu}_{1}/{\mu}_{2},{\widehat{\rho}}_{2}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{t}^{**}\left(\rho \right)>{t}^{*}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{2},\widehat{\rho}).$
- (v)
- ${t}^{**}\left(\rho \right)<{t}_{B}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\mu}_{1}/{\mu}_{2},{\widehat{\rho}}_{2}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{t}^{**}\left(\rho \right)>{t}_{B}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{2},\widehat{\rho}).$

**Lemma**

**8.**

- (i)
- If $0\le \rho <{\widehat{\rho}}_{2}$, then$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{B}}{inf}g(t,s)={g}_{L}\left({t}^{*}\right)=\frac{2}{1+\rho}({\mu}_{1}+{\mu}_{2}+2/{t}^{*}),\end{array}$$
- (ii)
- If $\rho ={\widehat{\rho}}_{2}$, then ${t}^{*}\left({\widehat{\rho}}_{2}\right)={s}^{*}\left({\widehat{\rho}}_{2}\right)=1/{\mu}_{2}$ and$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{B}}{inf}g(t,s)={g}_{L}(1/{\mu}_{2})={g}_{2}(1/{\mu}_{2})=4{\mu}_{2},\end{array}$$
- (iii)
- If ${\widehat{\rho}}_{2}<\rho <1$, then$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{B}}{inf}g(t,s)=\underset{(t,s)\in {D}_{2}}{inf}{g}_{2}\left(s\right)={g}_{2}(1/{\mu}_{2})=4{\mu}_{2},\end{array}$$

**Lemma**

**9.**

- (a)
- Both ${s}^{*}\left(\rho \right)$ and ${s}^{**}\left(\rho \right)$ are decreasing functions on $[0,1]$.
- (b)
- That ${\widehat{\rho}}_{1}$ is the unique point on $[0,1)$ such that$$\begin{array}{c}\hfill {s}_{A}\left({\widehat{\rho}}_{1}\right)={s}^{**}\left({\widehat{\rho}}_{1}\right)={s}^{*}\left({\widehat{\rho}}_{1}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\end{array}$$
- (i)
- ${s}_{A}\left(\rho \right)<{s}^{**}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in [0,{\widehat{\rho}}_{1}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{s}_{A}\left(\rho \right)>{s}^{**}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{1},1)$,
- (ii)
- ${s}^{*}\left(\rho \right)<{s}^{**}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in [0,{\widehat{\rho}}_{1}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{s}^{*}\left(\rho \right)>{s}^{**}\left(\rho \right)\phantom{\rule{4pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{all}\phantom{\rule{4pt}{0ex}}\rho \in ({\widehat{\rho}}_{1},1).$

- (c)
- For all ${\mu}_{1}/{\mu}_{2}<\rho <1$, it holds that ${s}^{**}\left(\rho \right)<{s}_{1}^{*}\left(\rho \right)$.

**Lemma**

**10.**

- (i)
- If $0\le \rho <{\widehat{\rho}}_{1}$, then$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{A}}{inf}g(t,s)={g}_{A}({t}_{A},{s}_{A})=4({\mu}_{2}+(1-2\rho ){\mu}_{1}),\end{array}$$
- (ii)
- If $\rho ={\widehat{\rho}}_{1}$, then$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{A}}{inf}g(t,s)={g}_{A}({t}_{A},{s}_{A})=4({\mu}_{2}+(1-2\rho ){\mu}_{1}),\end{array}$$
- (iii)
- If ${\widehat{\rho}}_{1}<\rho <{\widehat{\rho}}_{2}$, then$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{A}}{inf}g(t,s)={g}_{L}\left({s}^{*}\right)=\frac{2}{1+\rho}({\mu}_{1}+{\mu}_{2}+2/{s}^{*}),\end{array}$$
- (iv)
- If $\rho ={\widehat{\rho}}_{2}$, then ${t}^{*}\left({\widehat{\rho}}_{2}\right)={s}^{*}\left({\widehat{\rho}}_{2}\right)=1/{\mu}_{2}$ and$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{A}}{inf}g(t,s)={g}_{L}\left({s}^{*}\right)={g}_{2}(1/{\mu}_{2})=4{\mu}_{2},\end{array}$$
- (v)
- If ${\widehat{\rho}}_{2}<\rho <1$, then$$\begin{array}{c}\hfill \underset{(t,s)\in \overline{A}}{inf}g(t,s)={g}_{2}(1/{\mu}_{2})=4{\mu}_{2},\end{array}$$

## 4. Conclusions and Discussions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BM | Brownian motion |

## Appendix A

**Proof**

**of**

**Lemma**

**6.**

- (S1)
- On the set ${E}_{1}=\{(t,s):\rho \phantom{\rule{4pt}{0ex}}t\wedge s\phantom{\rule{4pt}{0ex}}{s}^{-1}(1+{\mu}_{2}s)\ge (1+{\mu}_{1}t)\}$, $g(t,s)={g}_{2}\left(s\right)$
- (S2)
- On the set ${E}_{2}=\{(t,s):\rho \phantom{\rule{4pt}{0ex}}t\wedge s\phantom{\rule{4pt}{0ex}}{t}^{-1}(1+{\mu}_{1}t)\ge (1+{\mu}_{2}s)\}$, $g(t,s)={g}_{1}\left(t\right)$
- (S3)
- On the set ${E}_{3}={(0,\infty )}^{2}\backslash ({E}_{1}\cup {E}_{2})$, $g(t,s)={g}_{3}(t,s)$.

**Proof**

**of**

**Lemma**

**7.**

- (a)
- The claim for ${t}^{*}\left(\rho \right)$ follows by noting its following representation:$$\begin{array}{c}\hfill {t}^{*}\left(\rho \right)={s}^{*}\left(\rho \right)=\sqrt{\frac{2(1-\rho )}{{\mu}_{1}^{2}+{\mu}_{2}^{2}-2{\mu}_{1}{\mu}_{2}+2{\mu}_{1}{\mu}_{2}-2\rho {\mu}_{1}{\mu}_{2}}}=\sqrt{\frac{2}{\frac{{({\mu}_{1}-{\mu}_{2})}^{2}}{1-\rho}+2{\mu}_{1}{\mu}_{2}}}.\end{array}$$The claims for ${t}_{B}\left(\rho \right)$ and ${s}_{1}^{*}\left(\rho \right)$ follow directly from their definition.
- (b)
- First note that$$\begin{array}{c}\hfill {t}^{**}\left(0\right)={t}^{**}\left({\widehat{\rho}}_{2}\right)=\frac{1}{{\mu}_{2}}.\end{array}$$Next it is calculated that$$\begin{array}{c}\hfill \frac{\partial {t}^{**}\left(\rho \right)}{\partial \rho}=\frac{-2{\mu}_{2}{\rho}^{2}+4{\mu}_{2}\rho -{\mu}_{1}-{\mu}_{2}}{{({\mu}_{2}+\rho {\mu}_{1}-2{\mu}_{2}{\rho}^{2})}^{2}}.\end{array}$$Thus, the claim of (b) follows by analysing the sign of $\frac{\partial {t}^{**}\left(\rho \right)}{\partial \rho}$ over $(0,\widehat{\rho})$.
- (c)
- For any $0\le \rho \le {\mu}_{1}/{\mu}_{2}$ we have $|{\mu}_{1}-2\rho {\mu}_{2}|\le {\mu}_{1}$ and thus$$\begin{array}{c}\hfill {t}_{B}\left(\rho \right)\ge \frac{1}{{u}_{1}}\ge \frac{1}{{u}_{2}}\ge \frac{1-\rho}{{u}_{2}(1-{\rho}^{2})}\ge \frac{1-\rho}{\rho ({\mu}_{1}-\rho {\mu}_{2})+{\mu}_{2}(1-{\rho}^{2})}={t}^{**}\left(\rho \right).\end{array}$$Further, since$$\begin{array}{c}\hfill {\mu}_{1}^{2}+{\mu}_{2}^{2}-2\rho {\mu}_{1}{\mu}_{2}={\mu}_{1}({\mu}_{1}-\rho {\mu}_{2})+{\mu}_{2}({\mu}_{2}-\rho {\mu}_{1})\le {\mu}_{2}({\mu}_{1}-\rho {\mu}_{2})+{\mu}_{2}({\mu}_{2}-\rho {\mu}_{1})\le 2{\mu}_{2}^{2}(1-\rho ),\end{array}$$$$\begin{array}{c}\hfill {t}^{*}\left(\rho \right)\ge \frac{1}{{\mu}_{2}}\ge {t}^{**}\left(\rho \right).\end{array}$$
- (d)
- It is easy to check that (26) holds. For (i) we have$$\begin{array}{c}\hfill {t}^{*}\left(\rho \right)-{s}_{1}^{*}\left(\rho \right)=(1-\rho )\left(\frac{1}{{f}_{1}\left(\rho \right)}-\frac{1}{{f}_{2}\left(\rho \right)}\right),\end{array}$$$$\begin{array}{ccc}\hfill {f}_{1}\left(\rho \right)& =& \sqrt{\frac{(1-\rho )({\mu}_{1}^{2}+{\mu}_{2}^{2}-2\rho {\mu}_{1}{\mu}_{2})}{2}}=\sqrt{{\mu}_{1}{\mu}_{2}{\rho}^{2}-\frac{{({\mu}_{1}+{\mu}_{2})}^{2}}{2}\rho +\frac{{\mu}_{1}^{2}+{\mu}_{2}^{2}}{2}}\hfill \\ \hfill {f}_{2}\left(\rho \right)& =& \rho {\mu}_{2}-{\mu}_{1}.\hfill \end{array}$$

**Proof**

**of**

**Lemma**

**8.**

**Proof**

**of**

**Lemma**

**9.**

- (a)
- The claim for ${s}^{*}\left(\rho \right)$ has been shown in the proof of (a) in Lemma 7. Next, we show the claim for ${s}^{**}\left(\rho \right)$, for which it is sufficient to show that $\frac{\partial {s}^{**}\left(\rho \right)}{\partial \rho}<0$ for all $\rho \in [0,1]$. In fact, we have$$\begin{array}{c}\hfill \frac{\partial {s}^{**}\left(\rho \right)}{\partial \rho}=\frac{-2{\mu}_{1}{\rho}^{2}+4{\mu}_{1}\rho -{\mu}_{1}-{\mu}_{2}}{{({\mu}_{1}+\rho {\mu}_{2}-2{\mu}_{1}{\rho}^{2})}^{2}}<0.\end{array}$$
- (b)
- In order to prove (i), the following two scenarios will be discussed separately:$$\begin{array}{c}\hfill \left(S1\right).\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{2}<2{\mu}_{1};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(S2\right).\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{2}\ge 2{\mu}_{1}.\end{array}$$

**Proof**

**of**

**Lemma**

**10.**

## References

- Albrecher, Hansjörg, Pablo Azcue, and Nora Muler. 2017. Optimal dividend strategies for two collaborating insurance companies. Advances in Applied Probability 49: 515–48. [Google Scholar] [CrossRef]
- Asmussen, Søren, and Hansjörg Albrecher. 2010. Ruin Probabilities, 2nd ed. Advanced Series on Statistical Science & Applied Probability, 14; Hackensack: World Scientific Publishing Co. Pte. Ltd. [Google Scholar]
- Avram, Florin, and Sooie-Hoe Loke. 2018. On central branch/reinsurance risk networks: Exact results and heuristics. Risks 6: 35. [Google Scholar] [CrossRef]
- Avram, Florin, and Andreea Minca. 2017. On the central management of risk networks. Advances in Applied Probability 49: 221–37. [Google Scholar] [CrossRef]
- Avram, Florin, Zbigniew Palmowski, and Martijn R. Pistorius. 2008a. Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results. Annals of Applied Probability 19: 2421–49. [Google Scholar] [CrossRef]
- Avram, Florin, Zbigniew Palmowski, and Martijn R. Pistorius. 2008b. A two-dimensional ruin problem on the positive quadrant. Insurance: Mathematics and Economics 42: 227–34. [Google Scholar] [CrossRef]
- Azcue, Pablo, and Nora Muler. 2018. A multidimensional problem of optimal dividends with irreversible switching: A convergent numerical scheme. arXiv. [Google Scholar]
- Azcue, Pablo, Nora Muler, and Zbigniew Palmowski. 2019. Optimal dividend payments for a two-dimensional insurance risk process. European Actuarial Journal 9: 241–72. [Google Scholar] [CrossRef]
- Dȩbicki, Krzysztof, Enkelejd Hashorva, Lanpeng Ji, and Tomasz Rolski. 2018. Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stochastic Processes and their Applications 12: 4171–206. [Google Scholar] [CrossRef]
- Dȩbicki, Krzysztof, Kamil MarcinKosiński, Michel Mandjes, and Tomasz Rolski. 2010. Extremes of multidimensional Gaussian processes. Stochastic Processes and their Applications 120: 2289–301. [Google Scholar] [CrossRef]
- Delsing, Guusje, Michel Mandjes, Peter Spreij, and Erik Winands. 2018. Asymptotics and approximations of ruin probabilities for multivariate risk processes in a Markovian environment. arXiv. [Google Scholar]
- Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch. 1997. Modelling Extremal Events of Applications of Mathematics (New York). Berlin: Springer, vol. 33. [Google Scholar]
- Foss, Sergey, Dmitry Korshunov, Zbigniew Palmowski, and Tomasz Rolski. 2017. Two-dimensional ruin probability for subexponential claim size. Probability and Mathematical Statistics 2: 319–35. [Google Scholar]
- Garbit, Rodolphe, and Kilian Raschel. 2014. On the exit time from a cone for Brownian motion with drift. Electronic Journal of Probability 19: 1–27. [Google Scholar] [CrossRef]
- Gerber, Hans U., and Elias SW Shiu. 2004. Optimal Dvidends: Analysis with Brownian Motion. North American Actuarial Journal 8: 1–20. [Google Scholar] [CrossRef]
- Grandell, Jan. 1991. Aspects of Risk Theory. New York: Springer. [Google Scholar]
- Hashorva, Enkelejd. 2005. Asymptotics and bounds for multivariate Gaussian tails. Journal of Theoretical Probability 18: 79–97. [Google Scholar] [CrossRef]
- Hashorva, Enkelejd, and Jürg Hüsler. 2002. On asymptotics of multivariate integrals with applications to records. Stochastic Models 18: 41–69. [Google Scholar] [CrossRef]
- He, Hua, William P. Keirstead, and Joachim Rebholz. 1998. Double lookbacks. Mathematical Finance 8: 201–28. [Google Scholar] [CrossRef]
- Iglehart, L. Donald. 1969. Diffusion approximations in collective risk theory. Journal of Applied Probability 6: 285–92. [Google Scholar] [CrossRef]
- Ji, Lanpeng, and Stephan Robert. 2018. Ruin problem of a two-dimensional fractional Brownian motion risk process. Stochastic Models 34: 73–97. [Google Scholar] [CrossRef]
- Klugman, Stuart A., Harry H. Panjer, and Gordon E. Willmot. 2012. Loss Models: From Data to Decisions. Hoboken: John Wiley and Sons. [Google Scholar]
- Kou, Steven, and Haowen Zhong. 2016. First-passage times of two-dimensional Brownian motion. Advances in Applied Probability 48: 1045–60. [Google Scholar] [CrossRef]
- Li, Junhai, Zaiming Liu, and Qihe Tang. 2007. On the ruin probabilities of a bidimensional perturbed risk model. Insurance: Mathematics and Economics 41: 185–95. [Google Scholar] [CrossRef]
- Mikosch, Thomas. 2008. Non-life Insurance Mathematics. An Introduction with Stochastic Processes. Berlin: Springer. [Google Scholar]
- Rolski, Tomasz, Hanspeter Schmidli, Volker Schmidt, and Jozef Teugels. 2009. Stochastic Processes for Insurance and Finance. Hoboken: John Wiley & Sons, vol. 505. [Google Scholar]

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**MDPI and ACS Style**

Dȩbicki, K.; Ji, L.; Rolski, T.
Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model. *Risks* **2019**, *7*, 83.
https://doi.org/10.3390/risks7030083

**AMA Style**

Dȩbicki K, Ji L, Rolski T.
Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model. *Risks*. 2019; 7(3):83.
https://doi.org/10.3390/risks7030083

**Chicago/Turabian Style**

Dȩbicki, Krzysztof, Lanpeng Ji, and Tomasz Rolski.
2019. "Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model" *Risks* 7, no. 3: 83.
https://doi.org/10.3390/risks7030083