Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes

doi:10.1016/j.insmatheco.2017.03.006

Insurance: Mathematics and Economics

Volume 74, May 2017, Pages 135-146

https://doi.org/10.1016/j.insmatheco.2017.03.006 Get rights and content

Abstract

In this paper, we investigate an optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. We assume that the periodic dividend strategy has exponential inter-dividend-decision times and continuous monitoring of solvency. Both proportional and fixed transaction costs from capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections until the time of ruin. By the fluctuation theory of Lévy processes in Albrecher et al. (2016), the optimal periodic dividend and capital injection strategies are derived. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Finally, numerical examples are studied to illustrate our results.

Introduction

The Lévy risk model with positive jumps (spectrally positive Lévy process), which is also called dual risk model, is proposed to offset continuous expenses by the stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, dividend optimization problems have attracted extensive attention. In Avanzi and Gerber (2008) and Avanzi et al. (2007), the authors studied how the expectation of the discounted dividends until ruin can be calculated in the dual compound Poisson risk model. In Bayraktar et al. (2013), Kyprianou et al. (2012), Yin and Wen (2013) and Zhao et al. (2015), the optimal dividend problems were studied in a general spectrally positive Lévy risk model.

In the above papers, the dividend decisions are made continuously, which usually leads to very irregular dividend payments. However, in practice, the companies that are capable of distributing dividends make dividend decisions on a periodic basis.Albrecher et al. (2011a) studied the random inter-dividend-decision times in the Cramér–Lundberg model, where the ruin cannot occur between dividend payment times. Continuous monitoring of solvency with periodic dividends was considered by Albrecher et al. (2011b) in the Brownian risk model, Avanzi et al. (2013) and Avanzi et al. (2014) in the dual compound Poisson model. Recently, when the inter-dividend-decision times of periodic dividend are exponential, Pérez and Yamazaki (2016) showed the optimality of periodic barrier strategy for spectrally positive Lévy processes.

When dividend payments are maximized, ruin is usually certain. In some cases, it may be profitable to rescue the company by capital injection. This idea goes back to Porteus (1977). Yao et al. (2011) considered an optimal dividend and capital injection problem in the dual compound Poisson risk model. Avanzi et al. (2011) discussed the same problem in the dual compound Poisson model with diffusion. Bayraktar et al. (2013) and Zhao et al. (2015) extended their work to general spectrally positive Lévy processes. In addition, transaction cost, which usually includes two parts: proportional cost and fixed cost, is an important factor in business activities. In Avanzi et al. (2011), the proportional transaction costs from dividend and capital injection were involved into an optimal dividend problem. In Yao et al. (2011) and Zhao et al. (2015), both proportional and fixed costs on capital injection were considered. Fixed costs on dividend were studied in Bayraktar et al. (2014).

In this paper, the optimal periodic dividend and capital injection problem is discussed for spectrally positive Lévy processes. For periodic dividend, we assume that the inter-dividend-decision times are exponential as in Avanzi et al. (2014) and Pérez and Yamazaki (2016), but different methods are used. For capital injection, we include the proportional and fixed transaction costs. We also assume that the ruin may occur even under the rescue of capital injection. Like Zhao et al. (2015), we decompose the optimal problem into two suboptimal problems. By the fluctuation theory of Lévy processes observed at Poisson arrival in Albrecher et al. (2016), we find the optimal strategy and the optimal return function. If the fixed transaction cost from capital injection tends to zero, we obtain the results in Pérez and Yamazaki (2016). When the positive jumps of the Lévy process are hyper-exponential compound Poisson distributed, the first suboptimal problem becomes that in Avanzi et al. (2014). If the dividend decision intensity goes to infinity, and meanwhile the fixed costs on capital injection tend to zero, the two suboptimal problems in this paper reduce to those in Bayraktar et al. (2013). Furthermore, for hyper-exponential compound Poisson positive jumps, our results are consistent with Avanzi et al. (2011).

This paper is organized as follows. Section 2 provides the formulations of the problem. Section 3 discusses the case without capital injection. The case with incorporated capital injection is considered in Section 4. The optimal periodic dividend and capital injection strategies are derived in Section 5. Section 6 gives some numerical examples.

Section snippets

Spectrally positive Lévy processes

Let $X = {X_{t}}$ be a spectrally positive Lévy process with non-monotone paths on a filtered probability space $(Ω, F, F, P)$ , where $F = {F_{t}}$ satisfies the usual conditions. The Lévy triplet of $X$ is given by $(c, σ, ν)$ , where $c > 0$ , $σ \geq 0$ , and $ν$ is a Lévy measure on $(0, \infty)$ satisfying the integrability condition $\int_{0}^{\infty} (1 \land x^{2}) ν (d x) < \infty$ . Let $E^{x}$ be the conditional expectation given the initial surplus $x$ , and in particular, denote $E^{0}$ by $E$ . The Laplace exponent of $X$ is given by $ψ (s) = \frac{1}{t} log E [e^{- s X_{t}}] = \frac{σ^{2}}{2} s^{2} + c s + \int_{0}^{\infty} (e^{- s x} - 1 + s x 1_{{0 < x \leq 1}})$

Preliminary discussions for the optimal problem without capital injection

We now study the optimal problem without capital injection. Let $Π_{p} = {π_{p} : π_{p} = (L^{π_{p}}; 0) \in Π} \subset Π$ denote the set of all admissible strategies for this suboptimal problem. The value function $V_{p} (x)$ is defined by $V_{p} (x) = sup_{π_{p} \in Π_{p}} V (x; π_{p}) = sup_{π_{p} \in Π_{p}} E^{x} [\int_{0}^{T^{π_{p}}} e^{- δ s} ϑ_{s}^{π_{p}} d N_{s}] .$ The objective is to find the optimal strategy $π_{p}^{*} \in Π_{p}$ such that $V_{p} (x) = V (x; π_{p}^{*})$ .

By the definition of $V_{p} (x)$ , we know $V_{p} (x)$ is an increasing function with $V_{p} (0) = 0$ . Similar to Theorem 2.1, we give the following lemma.

Lemma 3.1

Let $v_{p} (x)$ be an increasing and

Preliminary discussions for the optimal problem without ruin

We assume that the company survives forever by forced capital injections. Let $Π_{q}$ denote the set of admissible strategies of this suboptimal problem, i.e., $Π_{q} = {π_{q} = (L^{π_{q}}; G^{π_{q}}) : π_{q} \in Π such that X_{t}^{π_{q}} > 0 for all t \geq 0} \subset Π .$ The value function $V_{q} (x)$ is defined by $V_{q} (x) = sup_{π_{q} \in Π_{q}} V (x; π_{q}) = sup_{π_{q} \in Π_{q}} E^{x} [\int_{0}^{\infty} e^{- δ s} ϑ_{s}^{π_{q}} d N_{s} - \sum_{n = 1}^{\infty} e^{- δ τ_{n}^{π_{q}}} (K + ϕ ξ_{n}^{π_{q}})] .$ We will search for the optimal strategy $π_{q}^{*} \in Π_{q}$ and the associated value function $V_{q} (x) = V (x; π_{q}^{*})$ . By the similar proof of Theorem 2.1, we give the following lemma.

Lemma 4.1

Let $v_{q} (x)$ be

Optimal periodic dividend and capital injection strategy

From the definitions of $V_{p}$ , $V_{q}$ and $V$ , we can easily get the relationship $V (x) \geq max {V_{p} (x), V_{q} (x)}, x \geq 0 .$

Lemma 5.1

If the function $v (x)$ satisfies one of the following two hypotheses, we have $v (x) \geq V (x)$ ,

(i)
If $v (x)$ satisfies the conditions ofLemma 3.1 and $M v (0) \leq v (0)$ ;
(ii)
If $v (x)$ satisfies the conditions ofLemma 4.1 and $v (0) \geq 0$ .

Proof

(i) If $v^{'} (x) < ϕ$ for all $x > 0$ , we have $M v (x) = v (x) - K < v (x)$ . If there exists $η > 0$ such that $v^{'} (η) \geq ϕ$ , we let $\bar{η} = sup {y > 0 : v^{'} (y) \geq ϕ}$ , and then $v^{'} (\bar{η}) = ϕ$ . Similar to (4.3), we get $M v (x) \leq v (x)$ . By Theorem 2.1

Numerical examples

In Egami and Yamazaki (2014), the authors considered the spectrally negative phase-type Lévy process, whose scale function admits an analytical expression; they proposed an approach to approximate the scale function for a general spectrally negative Lévy process. The numerical results of this paper are based on their approximation method. For simplicity, we discuss the cases of the Lévy process $X$ with hyperexponential and Gamma distributed compound Poisson positive jumps, respectively.

Acknowledgments

The authors acknowledge financial support of National Natural Science Foundation of China (11501321, 11571198, 71671082, 11501319), China Postdoctoral Science Foundation (2016M592157) and Natural Science Foundation of Shandong Province in China (ZR2014AM021).

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View full text

Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes

Abstract

Introduction

Section snippets

Spectrally positive Lévy processes

Preliminary discussions for the optimal problem without capital injection

Preliminary discussions for the optimal problem without ruin

Optimal periodic dividend and capital injection strategy

Numerical examples

Acknowledgments

Insurance Math. Econom.

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Insurance Math. Econom.

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Randomized observation periods for the compound Poisson risk model: dividends

Astin Bull.

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