# Optimal Dividend and Capital Injection Problem with Transaction Cost and Salvage Value: The Case of Excess-of-Loss Reinsurance Based on the Symmetry of Risk Information

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation and the Control Problem

**Definition**

**1.**

- (i)
- $\left\{{m}_{t}^{\pi}\right\}$ is an ${\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$ adapted process with ${m}_{t}^{\pi}\in [0,M]$ for all $t\ge 0$.
- (ii)
- $\left\{{L}_{t}^{\pi}\right\}$ is an increasing, ${\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$-adapted cádlág process and $\mathsf{\Delta}{L}_{t}^{\pi}\le {X}_{t-}^{\pi}$.
- (iii)
- $\left\{{\tau}_{n}^{\pi}\right\}$ is a sequence stopping times with respect to ${\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$ and $0\le {\tau}_{1}^{\pi}\le \cdots \le {\tau}_{n}^{\pi}\le \cdots $, a.s.
- (iv)
- ${\eta}_{n}^{\pi}(n=1,\text{}2,\text{}3,\dots )$ is measurable and non-negative with respect to $\left\{{\mathcal{F}}_{{\tau}_{n}^{\pi}}\right\}$.
- (v)
- $\forall T>0$, it has $P(\underset{n\to \infty}{\mathrm{lim}}{\tau}_{n}^{\pi}<T)=0$.

**Remark**

**1.**

**Remark**

**2.**

## 3. The Solution to the Problem Where Bankruptcy is Not Allowed

**Theorem**

**1.**

- (i)
- For each admissible strategy ${\pi}_{c}$, there exists $g(x)\ge V(x,{\pi}_{c})$ and, therefore, $g(x)\ge {V}_{c}(x)$ for all $x\ge 0$.
- (ii)
- In case of the strategy, ${\pi}_{c}^{*}=\{{m}^{{\pi}_{c}^{*}};{L}^{{\pi}_{c}^{*}};{G}^{{\pi}_{c}^{*}}\}$ is constructed by Equations (13)–(18) with $g(x)=V(x,{\pi}_{c}^{*})$. Then $g(x)={V}_{c}(x)$ and ${\pi}_{c}^{*}$ is the optimal control strategy.

**Proof.**

**Lemma**

**1.**

**Proof.**

- (i)
- If $0<K\le \mathsf{\Phi}(0)$, it’s conjectured that $\mathcal{M}g(0)=g(0)$ holds under the condition that there exist some ${m}^{{\pi}_{c}^{*}}(0)\in [0,M]$ and ${\eta}^{*}({m}^{{\pi}_{c}^{*}}(0))>0$ in which the following equations are true:$${g}^{\prime}({\eta}^{*})={\beta}_{2},$$$$g(0)=g({\eta}^{*})-{\beta}_{2}{\eta}^{*}-K=\mathcal{M}g(0).$$$$K={\displaystyle {\int}_{0}^{{\eta}^{*}}[{g}^{\prime}(x)-{\beta}_{2}]dx}={\displaystyle {\int}_{0}^{{\eta}^{*}({m}^{{\pi}_{c}^{*}}(0))}[F(x,{m}^{{\pi}_{c}^{*}}(0))-{\beta}_{2}]\text{}}dx=\mathsf{\Phi}({m}^{{\pi}_{c}^{*}}(0)).$$By Lemma 1, it follows that ${m}^{{\pi}_{c}^{*}}(0)\in [0,{\widehat{m}}_{0})$ and ${\eta}^{*}$ exist if and only if $0<K\le \mathsf{\Phi}(0)$.
- (ii)
- If $K>\mathsf{\Phi}(0)$, the value ${\eta}^{*}({m}^{{\pi}_{c}^{*}}(0))>0$ satisfying Equations (40) and (41) doesn’t exist and $\mathcal{M}g(0)<g(0)$, which implies that ${G}^{{\pi}_{c}^{*}}\equiv 0.$ Therefore, in order to meet the boundary condition in Equation (12), we have $g(0)=0$ and ${m}^{{\pi}_{c}^{*}}(0)\equiv 0$.Now, summarizing the above discussions, we have the following result.

**Theorem**

**2.**

- (i)
- If $0<K\le \mathsf{\Phi}(0)$, ${m}^{{\pi}_{c}^{*}}(0)={m}_{0}\in [0,{\widehat{m}}_{0}]$ is the unique solution to the equation $\mathsf{\Phi}({m}_{0})=K$. The optimal injection strategy ${G}_{t}^{{\pi}_{c}^{*}}$ is given by Equations (13)–(15) and the optimal amount of capital injection ${\eta}^{*}$ is obtained by Equations (40) and (41). This means that, by injecting the capital, the insurance company’s surplus immediately jumps to ${\eta}^{*}$ when it hits the barrier 0. In this case, the boundary condition are $\mathcal{M}g(0)=g(0)$ and $g(0)\ge 0$.
- (ii)
- If $K>\mathsf{\Phi}(0)$, then ${m}^{{\pi}_{c}^{*}}(0)=0$. The optimal strategy of capital injection satisfies ${G}_{t}^{{\pi}_{c}^{*}}\equiv 0$, which means the capital injection never happens. It suggests that, if the insurance company’s surplus attains zero, it will cede all the potential risk to the reinsurance company and keep the surplus stay at 0. Therefore, the bankruptcy will never happen.

**Remark**

**3.**

**Remark**

**4.**

## 4. The Solution to the Problem without Capital Injection

**Theorem**

**3.**

- (i)
- For each ${\pi}_{d}\in {\mathsf{\Pi}}_{x}$, it shows that $f(x)\ge V(x,{\pi}_{d})$. Therefore, $f(x)\ge {V}_{d}(x)$ for all $x\ge 0$.
- (ii)
- If the strategy ${\pi}_{d}^{*}=\{{m}^{{\pi}_{d}^{*}};{L}^{{\pi}_{d}^{*}};0\}$ is constructed by Equations (49) and (50) so that $f(x)=V(x,{\pi}_{d}^{*})$ is constructed by Equations (49) and (50) so that , then $f(x)={V}_{d}(x)$ and ${\pi}_{d}^{*}$ is optimal.

**Proof.**

**Theorem**

**4.**

- (i)
- If $0\le P\le \frac{\theta \lambda}{\delta}{k}_{1}\left({\mu}^{(1)}-\frac{{\mu}^{(2)}}{2M}\right)$, then $f(x)$ has the form:$$f(x)=\{\begin{array}{ll}{k}_{1}{\displaystyle {\int}_{0}^{x}\mathrm{exp}\left[{\displaystyle {\int}_{z}^{{x}_{0}}\frac{\theta}{m(s)}ds}\right]}\text{}dz+P,\hfill & \hfill 0\le x{x}_{0d}^{*},\\ {k}_{3}{e}^{{r}_{+}(x-{x}_{1d}^{*})}+{k}_{4}{e}^{{r}_{-}(x-{x}_{1d}^{*})},\hfill & \hfill {x}_{0d}^{*}\le x{x}_{1d}^{*},\\ {\beta}_{1}(x-{x}_{1d}^{*})+\frac{\theta \lambda {\mu}^{(1)}{\beta}_{1}}{\delta},\hfill & \hfill x\ge {x}_{1d}^{*},\end{array}$$$${x}_{1d}^{*}={x}_{0d}^{*}+\frac{1}{{r}_{+}-{r}_{-}}\mathrm{ln}\left[\frac{M+\theta /{r}_{+}}{M+\theta /{r}_{-}}\right],$$$${x}_{0d}^{*}=G(M)-G({m}^{{\pi}_{d}^{*}}(0)).$$In addition, ${m}^{{\pi}_{d}^{*}}(0)$ is the solution to the following equation$$\frac{\theta \lambda}{\delta}{k}_{1}\left({\mu}^{(1)}({m}^{{\pi}_{d}^{*}}(0))-\frac{{\mu}^{(2)}({m}^{{\pi}_{d}^{*}}(0))}{2{m}^{{\pi}_{d}^{*}}(0)}\right)\mathrm{exp}\left({\displaystyle {\int}_{{m}^{{\pi}_{d}^{*}}(0)}^{M}\frac{{\mu}^{(2)}(y)}{2{y}^{2}\left(\frac{\delta}{{\theta}^{2}\lambda}y+{\mu}^{(1)}(y)-\frac{{\mu}^{(2)}(y)}{2y}\right)}dy}\right)=P.$$Accordingly, the optimal dividend strategy ${L}_{t}^{{\pi}_{d}^{*}}$ should satisfy the equation below:$${L}_{t}^{{\pi}_{d}^{*}}={(x-{x}_{1d}^{*})}^{+}+{\displaystyle {\int}_{0}^{t}{I}_{\{{X}_{s}^{{\pi}_{d}^{*}}={x}_{1d}^{*}\}}}d{L}_{s}^{{\pi}_{d}^{*}},\text{}\mathrm{for}\text{}\mathrm{all}\text{}t\ge 0.$$In addition, the optimal excess-of-loss reinsurance retention level ${m}^{{\pi}_{d}^{*}}(x)$ is shown below:$${m}^{{\pi}_{d}^{*}}(x)=\{\begin{array}{ll}{G}^{-1}(x+G({m}^{{\pi}_{d}^{*}}(0))),\hfill & \hfill 0\le x\le {x}_{0d}^{*},\\ M,\hfill & \hfill x\ge {x}_{0d}^{*}.\text{}\end{array}$$
- (ii)
- If $\frac{\theta \lambda}{\delta}{k}_{1}\left({\mu}^{(1)}-\frac{{\mu}^{(2)}}{2M}\right)<P\le \frac{\theta \lambda {\mu}^{(1)}}{\delta}$, then $f(x)$ has the form below:$$f(x)=\{\begin{array}{ll}{k}_{3}{e}^{{r}_{+}(x-{x}_{1d}^{*})}+{k}_{4}{e}^{{r}_{-}(x-{x}_{1d}^{*})},\hfill & \hfill 0\le x<{x}_{1d}^{*},\\ {\beta}_{1}(x-{x}_{1d}^{*})+\frac{\theta \lambda {\mu}^{(1)}{\beta}_{1}}{\delta},\hfill & \hfill x\ge {x}_{1d}^{*},\end{array}$$$${k}_{3}{e}^{-{r}_{+}{x}_{1d}^{*}}+{k}_{4}{e}^{-{r}_{-}{x}_{1d}^{*}}=P.$$Correspondingly, the optimal dividend strategy ${L}_{t}^{{\pi}_{d}^{*}}$ should satisfy the formula below:$${L}_{t}^{{\pi}_{d}^{*}}={(x-{x}_{1d}^{*})}^{+}+{\displaystyle {\int}_{0}^{t}{I}_{\{{X}_{s}^{{\pi}_{d}^{*}}={x}_{1d}^{*}\}}}d{L}_{s}^{{\pi}_{d}^{*}},\text{}\mathrm{for}\text{}\mathrm{all}\text{}t\ge 0.$$In addition, the optimal excess-of-loss reinsurance retention level is ${m}^{{\pi}_{d}^{*}}(x)\equiv M$.
- (iii)
- If $P>\frac{\theta \lambda {\mu}^{(1)}}{\delta}$, then $f(x)={\beta}_{1}x+P$. The optimal dividend strategy is to pay the whole initial surplus $x$ as the dividends and declare bankruptcy at once. Then, the salvage $P$ is realized.

**Remark**

**5.**

## 5. The Solution to the General Control Problem

**Theorem**

**5.**

- (i)
- For each $\pi \in {\mathsf{\Pi}}_{x}$, it shows that $v(x)\ge V(x,\pi )$. So $v(x)\ge V(x)$ for all $x\ge 0$.
- (ii)
- If there is a strategy of ${\pi}^{*}=\{{m}^{{\pi}^{*}};{L}^{{\pi}^{*}};{G}^{{\pi}^{*}}\}$ so that $v(x)=V(x,{\pi}^{*})$, then $v(x)=V(x)$ and ${\pi}^{*}$ is optimal.

**Proof.**

**Lemma**

**2.**

- (i)
- If $0<K\le \mathsf{\Phi}(0)$, $P\le \widehat{P}$ and ${m}^{{\pi}_{c}^{*}}(0)\le {m}^{{\pi}_{d}^{*}}(0)$, it has $\mathcal{M}{V}_{c}(0)-{V}_{c}(0)=0$ and $P-{V}_{c}(0)\ge 0$.
- (ii)
- If $0<K\le \mathsf{\Phi}(0)$, $P\le \widehat{P}$ and ${m}^{{\pi}_{c}^{*}}(0)>{m}^{{\pi}_{d}^{*}}(0)$, it has $\mathcal{M}{V}_{c}(0)-{V}_{c}(0)=0$ and $P-{V}_{c}(0)<0$.
- (iii)
- If $0<K\le \mathsf{\Phi}(0)$ and $P>\widehat{P}$, it has $\mathcal{M}{V}_{c}(0)-{V}_{c}(0)<0$.
- (iv)
- If $K>\mathsf{\Phi}(0)$, it has $\mathcal{M}{V}_{c}(0)-{V}_{c}(0)<0$.

**Proof.**

**Lemma**

**3.**

- (i)
- If $0<K\le \mathsf{\Phi}(0)$, $P\le \widehat{P}$, and ${m}^{{\pi}_{c}^{*}}(0)<{m}^{{\pi}_{d}^{*}}(0)$, we find that $\mathcal{M}{V}_{d}(0)-{V}_{d}(0)<0$.
- (ii)
- If $0<K\le \mathsf{\Phi}(0)$, $P\le \widehat{P}$, and ${m}^{{\pi}_{c}^{*}}(0)\ge {m}^{{\pi}_{d}^{*}}(0)$, we find that $\mathcal{M}{V}_{d}(0)-{V}_{d}(0)\ge 0$.
- (iii)
- If $0<K\le \mathsf{\Phi}(0)$ and $P>\widehat{P}$, we find that $\mathcal{M}{V}_{d}(0)-{V}_{d}(0)<0$.
- (iv)
- If $K>\mathsf{\Phi}(0)$, we find that $\mathcal{M}{V}_{d}(0)-{V}_{d}(0)<0$.

**Proof.**

**Theorem**

**6.**

**Case 1.**If $\mathcal{M}g(0)-g(0)=0$ and $P-g(0)\le 0$, the following equivalent condition is valid

**Case 2.**If $\mathcal{M}f(0)-f(0)<0$ and $P-f(0)=0$, one of the following equivalent conditions holds.

- (i)
- $0<K\le \mathsf{\Phi}(0)\uff0cP\le \widehat{P}$ and ${m}^{{\pi}_{c}^{*}}(0)<{m}^{{\pi}_{d}^{*}}(0)$;
- (ii)
- $0<K\le \mathsf{\Phi}(0)$ and $P>\widehat{P}$;
- (iii)
- $K>\mathsf{\Phi}(0)$.

**Proof.**

**Remark**

**6.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Aniunas, P.; Gipiene, G.; Valukonis, M.; Vijunas, M. Liquidity Risk Management Model for Local Banks. Transform. Bus. Econ.
**2017**, 16, 153–173. [Google Scholar] - Lakstutiene, A.; Witkowska, J.; Leskauskiene, E. Transformation of the EU Deposit Insurance System: Evaluation. Transform. Bus. Econ.
**2017**, 16, 147–170. [Google Scholar] - Kurach, R. International Diversification of Pension Funds Using the Cointegration Approach: The Case of Poland. Transform. Bus. Econ.
**2017**, 16, 42–55. [Google Scholar] - Asmussen, S.; Taksar, M. Controlled diffusion models for optimal dividend pay-out. Insur. Math. Econ.
**1997**, 20, 1–15. [Google Scholar] [CrossRef] - Gerber, H.U.; Shiu, E.S.W. On optimal dividends strategies in the compound Poisson model. N. Am. Actuar. J.
**2006**, 10, 76–93. [Google Scholar] [CrossRef] - Belhaj, M. Optimal dividend payments when cash reserves follow a jump-diffusion process. Math. Financ.
**2010**, 20, 313–325. [Google Scholar] [CrossRef] - Azcue, P.; Muler, N. Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates. Insur. Math. Econ.
**2012**, 51, 26–42. [Google Scholar] [CrossRef] - Meng, H.; Siu, T.K.; Yang, H. Optimal dividends with debts and nonlinear insurance risk processes. Insur. Math. Econ.
**2013**, 53, 110–121. [Google Scholar] [CrossRef] [Green Version] - Kulenko, N.; Schmidli, H. Optimal dividend strategies in a Cramer–Lundberg model with capital injections. Insur. Math. Econ.
**2008**, 43, 270–278. [Google Scholar] [CrossRef] - Yao, D.; Yang, H.; Wang, R. Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. Eur. J. Oper. Res.
**2011**, 211, 568–576. [Google Scholar] [CrossRef] - Zhao, Y.X.; Yao, D.J. Optimal dividend and capital injection problem with a random time horizon and a ruin penalty in the dual model. Appl. Math. A J. Chin. Univ.
**2015**, 30, 325–339. [Google Scholar] [CrossRef] - Yin, C.; Yuen, K.C. Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs. J. Ind. Manag. Optim.
**2015**, 11, 1247–1262. [Google Scholar] [CrossRef] [Green Version] - Højgaard, B.; Taksar, M. Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quant. Financ.
**2004**, 4, 315–327. [Google Scholar] [CrossRef] [Green Version] - Peng, X.; Chen, M.; Guo, J. Optimal dividend and equity issuance problem with proportional and fixed transaction costs. Insur. Math. Econ.
**2012**, 51, 576–585. [Google Scholar] [CrossRef] - Yao, D.; Yang, H.; Wang, R. Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle. Econ. Model.
**2014**, 37, 53–64. [Google Scholar] [CrossRef] [Green Version] - Yao, D.; Yang, H.; Wang, R. Optimal dividend and reinsurance strategies with financing and liquidation value. ASTIN Bull. J. IAA
**2016**, 46, 365–399. [Google Scholar] [CrossRef] - Yao, D.; Fan, K. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. J. Ind. Manag. Optim.
**2018**, 14, 1–15. [Google Scholar] [CrossRef] - Cadenillas, A.; Choulli, T.; Taksar, M.; Zhang, L. Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm. Math. Financ.
**2006**, 16, 181–202. [Google Scholar] [CrossRef] - Meng, H.; Siu, T. On optimal reinsurance, dividend and reinvestment strategies. Econ. Model.
**2011**, 28, 211–218. [Google Scholar] [CrossRef] - Xu, J.; Zhou, M. Optimal risk control and dividend distribution policies for a diffusion model with terminal value. Math. Comput. Model.
**2012**, 56, 180–190. [Google Scholar] [CrossRef] - Yao, D.; Wang, R.; Cheng, G. Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs. J. Ind. Manag. Optim.
**2017**, 13, 51. [Google Scholar] [CrossRef] - Chunxiang, A.; Lai, Y.; Shao, Y. Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model. J. Comput. Appl. Math.
**2018**, 342, 317–336. [Google Scholar] - Yao, D.; Wang, R.; Lin, X. Optimal dividend, capital injection and excess-of-loss reinsurance strategies for insurer with a terminal value of the bankruptcy. Sci. Sin. Math.
**2017**, 47, 969–994. [Google Scholar] - Paulsen, J. Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs. SIAM J. Control Optim.
**2008**, 47, 2201–2226. [Google Scholar] [CrossRef] - Bai, L.; Guo, J.; Zhang, H. Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes. Quant. Financ.
**2010**, 10, 1163–1172. [Google Scholar] [CrossRef] - Liu, W.; Hu, Y. Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy. Stat. Probab. Lett.
**2014**, 84, 121–130. [Google Scholar] [CrossRef] - Loeffen, R.L.; Renaud, J.F. De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insur. Math. Econ.
**2010**, 46, 98–108. [Google Scholar] [CrossRef] [Green Version] - Liang, Z.; Young, V.R. Dividends and reinsurance under a penalty for ruin. Insur. Math. Econ.
**2012**, 50, 437–445. [Google Scholar] [CrossRef] - Nguyen, V.H.; Vuong, Q.H.; Tran, M.N. Central limit theorem for functional of jump Markov processes. Vietnam J. Math.
**2005**, 33, 443–461. [Google Scholar] - Nguyen, V.H.; Vuong, Q.H. On the martingale representation theorem and approximate hedging a contingent claim in the minimum mean square deviation criterion. VNU J. Sci. Math. Phys.
**2007**, 23, 143–154. [Google Scholar] - Hoang, T.P.T.; Vuong, Q.H. A Merton Model of Credit Risk with Jumps. J. Stat. Appl. Probab. Lett.
**2015**, 2, 97–103. [Google Scholar]

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## Share and Cite

**MDPI and ACS Style**

Yan, Q.; Yang, L.; Baležentis, T.; Streimikiene, D.; Qin, C.
Optimal Dividend and Capital Injection Problem with Transaction Cost and Salvage Value: The Case of Excess-of-Loss Reinsurance Based on the Symmetry of Risk Information. *Symmetry* **2018**, *10*, 276.
https://doi.org/10.3390/sym10070276

**AMA Style**

Yan Q, Yang L, Baležentis T, Streimikiene D, Qin C.
Optimal Dividend and Capital Injection Problem with Transaction Cost and Salvage Value: The Case of Excess-of-Loss Reinsurance Based on the Symmetry of Risk Information. *Symmetry*. 2018; 10(7):276.
https://doi.org/10.3390/sym10070276

**Chicago/Turabian Style**

Yan, Qingyou, Le Yang, Tomas Baležentis, Dalia Streimikiene, and Chao Qin.
2018. "Optimal Dividend and Capital Injection Problem with Transaction Cost and Salvage Value: The Case of Excess-of-Loss Reinsurance Based on the Symmetry of Risk Information" *Symmetry* 10, no. 7: 276.
https://doi.org/10.3390/sym10070276