Optimal Dividends and Capital Injections in the Dual Model with a Random Time Horizon

Abstract

This paper investigates an optimal dividend and capital injection problem in the dual model with a random horizon. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. By the fluctuation theory of Lévy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Besides, numerical examples are studied to illustrate our results.

Appendices

Appendix 1: Heuristic Arguments for the QVI (7)

We can rewrite the QVI (7) as

$$\begin{aligned}&\mathcal {L}V(x)+\gamma x\le 0;~1-V'(x)\le 0;~\mathcal {M}V(x)-V(x)\le 0;\\&\qquad \qquad \qquad \left( \mathcal {L}V(x)+\gamma x\right) \left( 1-V'(x)\right) \left( \mathcal {M}V(x)-V(x)\right) =0. \end{aligned}$$

Then we give the heuristic arguments of the dynamic programming principle for the QVI (7). If the surplus process starts at $x$ and follows the optimal strategy, the performance function associated with this optimal strategy is $V(x)$. On the other hand, if the process starts at $x$, selects the best immediate capital injection, and then follows an optimal strategy, the associated performance function is $\mathcal {M}V(x)$. Since the first strategy is optimal, its performance function is larger than the performance function associated with the second strategy. Furthermore, if it is optimal to immediately inject capital, these two performance functions are equal. Hence, $\mathcal {M}V(x)\le V(x)$, and in the capital injection region $\mathcal {M}V(x)=V(x)$. Similarly, $V'(x)\ge 1$ and in the dividend region $V'(x)=1$. Furthermore, we have $\mathcal {L}V(x)+\gamma x\le 0$. In the continuation region, that is, when the manager does not intervene, we know $\mathcal {L}V(x)+\gamma x=0$. Therefore, the QVI (7) holds.

Appendix 2: Introduction of Scale Function

We now recall the definition of the q-scale function for the spectrally positive Lévy process $X$, whose Laplace exponent $\psi$ is given by (1). For $q>0$, there exists a continuous and strictly increasing function $W^{(q)}:\mathbb {R}\mapsto [0,\infty [$, called the q-scale function defined in such a way that $W^{(q)}(x)=0$ for all $x<0$ and on $[0,\infty [$ its Laplace transform is given by

$$\begin{aligned} \int \nolimits _0^\infty e^{-sx}W^{(q)}(x)\mathrm{{d}}x=\frac{1}{\psi (s)-q},\quad s>\Phi (q), \end{aligned}$$

where $\Phi (q):=\sup \{s\ge 0:\psi (s)=q\}$. Note that the Laplace exponent $\psi$ in (1) is known to be zero at the origin, convex on $\mathbb {R}_+$. Then $\varPhi (q)$ is well-defined and is strictly positive as $q>0$. We give the function $Z^{(q)}(x)$, closely related to $W^{(q)}(x)$, by

$$\begin{aligned} Z^{(q)}(x)=1+q\int \nolimits _0^x W^{(q)}(y)\mathrm{{d}}y,\quad x\in \mathbb {R}, \end{aligned}$$

and its anti-derivative

$$\begin{aligned} \overline{Z}^{(q)}(x)=\int \nolimits _0^xZ^{(q)}(y)\mathrm{{d}}y=x+q\int \nolimits _0^x\int \nolimits _0^y W^{(q)}(z)\mathrm{{d}}z,\quad x\in \mathbb {R}. \end{aligned}$$

Noting that $W^{(q)}(x)$ is uniformly zero on the negative half line, we have $Z^{(q)}(x)=1$ and $\overline{Z}^{(q)}(x)=x$ for $x\le 0$.

From [18], we know the following facts about the scale function. If $X$ has paths of bounded variation, we have that $W^{(q)}\in \mathcal {C}^1(]0,\infty [)$ if and only if the Lévy measure $\nu$ has no atoms. Particularly, if $\nu$ is absolutely continuous with respect to Lebesgue measure, $W^{(q)}\in \mathcal {C}^1(]0,\infty [)$. In the case that $X$ has paths of unbounded variation, it is known that $W^{(q)}\in \mathcal {C}^1(]0,\infty [)$. Moreover, if $\sigma >0$, $\mathcal {C}^1(]0,\infty [)$ may be replaced by $\mathcal {C}^2(]0,\infty [)$. Hence, $Z^{(q)}\in \mathcal {C}^1(]0,\infty [)$, $\overline{Z}^{(q)}\in \mathcal {C}^1(\mathbb {R})$ and $\overline{Z}^{(q)}\in \mathcal {C}^2(]0,\infty [)$ for the bounded variation case, while $Z^{(q)}\in \mathcal {C}^1(\mathbb {R})$, $Z^{(q)}\in \mathcal {C}^2(]0,\infty [)$, $\overline{Z}^{(q)}\in \mathcal {C}^2(\mathbb {R})$ and $\overline{Z}^{(q)}\in \mathcal {C}^3(]0,\infty [)$ for the unbounded variation case. Considering the asymptotic behavior near zero, we have

$$\begin{aligned} W^{(q)}(0+)=\left\{ \begin{array}{ll} 0, &{} \mathrm{if} X \mathrm{\;is\; of \;unbounded \;variation}, \\ \frac{1}{c_{_{0}}}, &{} \mathrm{if} X \mathrm{\; is \;of\; bounded \;variation}. \end{array} \right. \end{aligned}$$

(50)

and

$$\begin{aligned} {W^{(q)}}'(0+)=\left\{ \begin{array}{ll} \frac{2}{\sigma ^2}, &{} \text {if}~\sigma >0,\\ \infty , &{} \text {if}~\sigma =0 ~\text {and}~ \nu (0,\infty )=\infty ,\\ \frac{q+\nu (0,\infty )}{c_{_{0}}^2}, &{} \mathrm{if} X \mathrm{\;is\; of\; bounded \;variation}. \end{array} \right. \end{aligned}$$

(51)