Optimal dividend strategies in a delayed claim risk model with dividends discounted by stochastic interest rates

Abstract

We consider a discrete-time risk model with delayed claims and a dividend payment strategy. It is assumed that every main claim will induce a by-claim which may be delayed for one time period with a certain probability. In the evaluation of the expected present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. Dividends are paid to the shareholders according to an admissible strategy. The company controls the amount of dividends in order to maximize the cumulative expected discounted dividends prior to ruin minus the expected discounted penalty value at ruin. We obtain some properties of the optimal dividend-payment strategy, and offer high efficiency algorithms for obtaining the optimal strategy, the optimal value function and the expectation of time of ruin under the optimal strategy. Our method is mainly to transform the value function. Numerical examples are presented to illustrate the transformation method and the impact of the penalty for ruin on the optimal strategy, the corresponding expected present value of dividends and the expectation of time of ruin.

Appendix

In order to prove Theorem 4.1, we need the following lemma.

Lemma 6.1

Assume that $\mathbf A$ is a strictly diagonally dominant matrix such that all of the diagonal elements are positive and all of the off-diagonal entries are non-positive. Then, the elements of its inverse $\mathbf A ^{-1}$ are all non-negative.

Proof of Lemma 6.1

The conclusion holds true due to the fact that $\mathbf {A}$ is an M-matrix (see Minc 1988). $\square$

Proof of Theorem 4.1

(i) From (4.2) and (4.3), it follows by Fatou’s Lemma that for any $i=1,2,\ldots , m$

$$\begin{aligned} \liminf _{u\rightarrow \infty }W^*_{i0}(u)\ge & {} v_i(1-q)\sum _{j=1}^mp_{ij}\left[ \liminf _{u\rightarrow \infty }W^*_{j0}(u)+\liminf _{u\rightarrow \infty }\phi _{j0}(u)+(1-v_j)\Lambda \right] \nonumber \\&+\,v_iq\theta \sum _{j=1}^mp_{ij}\left[ \liminf _{u\rightarrow \infty }W^*_{j0}(u)+\liminf _{u\rightarrow \infty }\phi _{j0}(u)+(1-v_j)\Lambda \right] \nonumber \\&+\,v_iq(1-\theta )\sum _{j=1}^mp_{ij}\left[ \liminf _{u\rightarrow \infty }W^*_{j1}(u)+\liminf _{u\rightarrow \infty }\phi _{j1}(u)+(1-v_j)\Lambda \right] \nonumber \\ \end{aligned}$$

(6.1)

and

$$\begin{aligned} \liminf _{u\rightarrow \infty }W^*_{i1}(u)\ge & {} v_i(1-q)\sum _{j=1}^mp_{ij}\left[ \liminf _{u\rightarrow \infty }W^*_{j0}(u)+\liminf _{u\rightarrow \infty }\phi _{j0}(u)+(1-v_j)\Lambda \right] \nonumber \\&\quad +\,v_iq\theta \sum _{j=1}^mp_{ij}\left[ \liminf _{u\rightarrow \infty }W^*_{j0}(u)+\liminf _{u\rightarrow \infty }\phi _{j0}(u)+(1-v_j)\Lambda \right] \nonumber \\&\quad +\,v_iq(1-\theta )\sum _{j=1}^mp_{ij}\left[ \liminf _{u\rightarrow \infty }W^*_{j1}(u)+\liminf _{u\rightarrow \infty }\phi _{j1}(u)+(1-v_j)\Lambda \right] \nonumber \\ \end{aligned}$$

(6.2)

Let

$$\begin{aligned}&{\underline{\mathbf{W }}}=\left( \liminf _{u\rightarrow \infty }W^*_{10},\liminf _{u\rightarrow \infty }W^*_{20},\ldots ,\liminf _{u\rightarrow \infty }W^*_{m0},\right. \\&\quad \quad \quad \quad \left. \liminf _{u\rightarrow \infty }W^*_{11},\liminf _{u\rightarrow \infty }W^*_{21},\ldots ,\liminf _{u\rightarrow \infty }W^*_{m1}\right) ^\top ;\\&p_0=1+q\theta -q; \ \ \ q_0=q(1-\theta );\\&\underline{L_i}=v_i\sum _{j=1}^m p_{ij}\left[ p_0\liminf _{u\rightarrow \infty }\phi _{j0}(u) +q_0\liminf _{u\rightarrow \infty }\phi _{j1}(u)\right] +v_i\sum _{j=1}^m p_{ij}(1-v_j)\Lambda ;\\&{\underline{\mathbf{L }}}=\left( \underline{L_1},\underline{L_2},\ldots ,\underline{L_m},\underline{L_1},\underline{L_2},\ldots ,\underline{L_m}\right) ^\top ; \end{aligned}$$

and $\mathbf A =$

$$\begin{aligned} \left( \begin{array}{llllllll} 1-v_1p_0p_{11} &{}\; -v_1p_0p_{12} &{} \;\cdots &{}\; -v_1p_0p_{1m} &{}\; -v_1q_0p_{11} &{}\; -v_1q_0p_{12} &{}\; \cdots &{}\; -v_1q_0p_{1m}\\ -v_2p_0p_{21} &{}\; 1-v_2p_0p_{22} &{} \;\cdots &{}\; -v_2p_0p_{2m} &{}\; -v_2q_0p_{21} &{}\; -v_2q_0p_{22} &{}\; \cdots &{}\; -v_2q_0p_{2m}\\ \cdots &{}\; \cdots &{}\; \cdots &{}\; \cdots &{}\; \cdots &{}\; \cdots &{}\; \cdots &{}\; \cdots \\ -v_mp_0p_{m1} &{}\; -v_mp_0p_{m2} &{}\; \cdots &{} \; 1-v_mp_0p_{mm} &{} \;-v_mq_0p_{m1} &{}\; -v_mq_0p_{m2} &{}\; \cdots &{}\; -v_mq_0p_{mm}\\ -v_1p_0p_{11} &{}\; -v_1p_0p_{12} &{}\; \cdots &{}\; -v_1p_0p_{1m} &{} \;1-v_1q_0p_{11} &{}\; -v_1q_0p_{12} &{} \;\cdots &{}\; -v_1q_0p_{1m}\\ -v_2p_0p_{21} &{}\; -v_2p_0p_{22} &{}\; \cdots &{}\; -v_2p_0p_{2m} &{} \;-v_2q_0p_{21} &{}\; 1-v_2q_0p_{22} &{} \;\cdots &{} \;-v_2q_0p_{2m}\\ \cdots &{}\; \cdots &{}\; \cdots &{}\; \cdots &{} \;\cdots &{}\; \cdots &{} \;\cdots &{}\; \cdots \\ -v_mp_0p_{m1} &{}\; -v_mp_0p_{m2} &{}\; \cdots &{}\; -v_mp_0p_{mm} &{}\; -v_mq_0p_{m1} &{}\; -v_mq_0p_{m2} &{} \cdots &{} 1-v_mq_0p_{mm}\\ \end{array} \right) . \end{aligned}$$

Then, the set of inequalities (6.1) and (6.2) can be rewritten as

$$\begin{aligned} \mathbf A {\underline{\mathbf{W }}}\ge {\underline{\mathbf{L }}}. \end{aligned}$$

(6.3)

Taking the limit superior as $u\rightarrow \infty$ in (4.2) and (4.3) yields, by Fatou’s Lemma,

$$\begin{aligned} \mathbf A \overline{\mathbf{W }}\le \overline{\mathbf{L }}, \end{aligned}$$

(6.4)

where

$$\begin{aligned}&\displaystyle \overline{\mathbf {W}}=\left( \limsup _{u\rightarrow \infty }W^{*}_{10},\limsup _{u\rightarrow \infty }W^{*}_{20},\ldots ,\limsup _{u\rightarrow \infty }W^{*}_{m0},\limsup _{u\rightarrow \infty }W^{*}_{11},\limsup _{u\rightarrow \infty }W^{*}_{21},\ldots ,\limsup _{u\rightarrow \infty }W^{*}_{m1}\right) ^\top ;\\&\displaystyle {\overline{\mathbf{L }}}=\left( \overline{L_1},\overline{L_2},\ldots ,\overline{L_m},\overline{L_1},\overline{L_2},\ldots ,\overline{L_m}\right) ^\top ;\\&\displaystyle \overline{L_i}=v_i\sum _{j=1}^m p_{ij}\left[ p_0\limsup _{u\rightarrow \infty }\phi _{j0}(u) +q_0\limsup _{u\rightarrow \infty }\phi _{j1}(u)\right] +v_i\sum _{j=1}^m p_{ij}(1-v_j)\Lambda . \end{aligned}$$

Obviously, $\mathbf A$ is a strictly diagonally dominant matrix. Hence, its inverse $\mathbf A ^{-1}$ exists, and it follows from (6.3) and (6.4) that

$$\begin{aligned} \mathbf A ^{-1}\underline{\mathbf{L }} \le \underline{\mathbf{W }} \le \overline{\mathbf{W }} \le \mathbf A ^{-1} \overline{\mathbf{L }}. \end{aligned}$$

(6.5)

Because $\mathbf A$ satisfies the conditions of Lemma 6.1, the elements of $\mathbf A ^{-1}$ are all non-negative. Therefore, when $\liminf _{u\rightarrow \infty } \phi _{ij}=c_0$ for all $i\in \{1,2,\ldots ,m\}$ and $j=0,1$, the elements of $\mathbf A ^{-1}\underline{\mathbf{L }}$ and $\mathbf A ^{-1}\overline{\mathbf{L }}$ are all maximal, which leads to that all the elements of $\underline{\mathbf{W }}$ or $\overline{\mathbf{W }}$ reach the maximum values, and

$$\begin{aligned} \mathbf A ^{-1}\underline{\mathbf{L }} = \underline{\mathbf{W }} = \overline{\mathbf{W }} = \mathbf A ^{-1} \overline{\mathbf{L }}. \end{aligned}$$

Hence, the limits of $W^*_{ij}(u)$ and $\phi _{ij}(u)$ exist, and (4.8) holds owing to the boundedness and uniqueness of the optimal image function.

(ii) Since the inequality signs in (6.1) and (6.2) can be replaced with equality signs, we come to the conclusion (4.9). $\square$