Optimizing Dividends and Capital Injections Limited by Bankruptcy, and Practical Approximations for the Cramér-Lundberg Process

Abstract

The recent papers Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017) and Avram et al. (Mathematics 9(9):931, 2021) cost induced dichotomy for optimal dividends in the cramr-lundberg model. Avram et al. (Mathematics 9(9):931, 2021) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram et al. cost induced dichotomy for optimal dividends in the Cramér-Lundberg model (Avram et al. in Mathematics 9(9):931, 2021). Secondly, it extends the results to the case when a final penalty P is taken into consideration as well besides a proportional cost $k>1$ for capital injections. This requires amending the “scale and Gerber-Shiu functions” already introduced in Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017). Thirdly, in the exponential case, the results will be made even more explicit by employing the Lambert-W function. This tool has particular importance in computational aspects and can be employed in theoretical aspects such as asymptotics.

Apendix: The Proof of Theorem 1 when \(\sigma \equiv 0\)

To simplify our readers’ journey, we sketch here the main elements of proof, in the case $\sigma \equiv 0$, generalizing Avram et al. (2021). Please note that the main modification when diffusion is present applies to the computation of the term $I_y$ below, in which $\sigma ^2$ will appear accompanying a Dirac mass in the Gerber-Shiu measure.

We begin by applying the strong Markov property at the stopping time $\tau ^x:=\tau _{0-}^x\wedge \tau _{b+}^x=\inf \left\{ t\ge 0: \ X_t^\pi <0\right\} \wedge \inf \left\{ t\ge 0:\ X_t^\pi >b\right\}$. It follows that, for $0\le x\le b$,

$$\begin{aligned} {\begin{matrix} J_x =&{}\mathbb {E}_x\left[ e^{-q\tau _{b+}^x}\mathbf {1}_{\tau _{b+}^x<\tau _{0-}^x}\right] J_b+\mathbb {E}_x\left[ e^{-q\tau _{0-}^x}\mathbf {1}_{\tau _{b+}^x>\tau _{0-}^x}\left( J_0+kX_{\tau _{0-}^x}\right) \mathbf {1}_{X_{\tau _{0-}^x}\ge -a}\right] \\ {} &{}-P\mathbb {E}_x\left[ e^{-q\tau _{0-}^x}\mathbf {1}_{\tau _{b+}^x>\tau _{0-}^x}\mathbf {1}_{X_{\tau _{0-}^x}< -a}\right] =\frac{W_q(x)}{W_q(b)}\left( J_b-I_b\right) +I_x, \end{matrix}} \end{aligned}$$

(53)

where

$$\begin{aligned} I_y:=\mathbb {E}_y\left[ e^{-q\tau _{0-}}\left\{ \left( J_0+kX_{\tau _{0-}}\right) \mathbf {1}_{X_{\tau _{0-}}\ge -a}-P\mathbf {1}_{X_{\tau _{0-}}< -a}\right\} \right] . \end{aligned}$$

The term $I_y$ can be explicitly computed (using the Gerber-Shiu measure).

$$\begin{aligned}&I_y=\\ =&\int _{\mathbb {R}_+} \left[ \left( J_0-ku\right) \mathbf {1}_{0\le u\le a}-P\mathbf {1}_{u>a}\right] \int _{\mathbb {R}_+}\left( e^{-\Phi (q)v}W_q(y)-W_q(y-v)\right) \nu \left( du+v\right) dv\\ =&\int _{\mathbb {R}_+}\left( J_0\left( \bar{\nu }(v)-\bar{\nu }(a+v)\right) -km_a(v)-P\bar{\nu }(a+v)\right) \left( e^{-\Phi (q)v}W_q(y)-W_q(y-v)\right) dv\\ =&W_q(y)\int _{\mathbb {R}_+}\left( J_0\left( \bar{\nu }(v)-\bar{\nu }(a+v)\right) -km_a(v)-P\bar{\nu }(a+v)\right) e^{-\Phi (q)v}dv\\ {}&+G_a(y)-J_0\left( C(y)-C_a(y)\right) . \end{aligned}$$

Since the term accompanying $W_q(y)$ is a constant, by replacing this in Eq. (53), it follows that

$$\begin{aligned} {\begin{matrix} J_x-\left( G_a(x)-J_0\left( C(x)-C_a(x)\right) \right) =\frac{W_q(x)}{W_q(b)}\left\{ J_b-\left( G_a(b)-J_0\left( C(b)-C_a(b)\right) \right) \right\} . \end{matrix}} \end{aligned}$$

(54)

With the particular choice of $x=0$, by recalling that $C_a(0)=G_a(0)=C(0)=0$, the last equation yields

$$J_0=\frac{W_q(0)}{W_q(b)}\left\{ J_b-\left( G_a(b)-J_0\left( C(b)-C_a(b)\right) \right) \right\} ,$$

which, combined with $W_q(0)=\frac{1}{c}$ and $C(x)=cW_q(x)-Z_q(x)$, leads to

$$\begin{aligned} J_x=G_a(x)-J_0\left( C(x)-C_a(x)-cW_q(x)\right) =G_a(x)+J_0\left( C_a(x)+Z_q(x)\right) . \end{aligned}$$

(55)

Theorem 1 is now proven, in the case $\sigma \equiv 0$.