Optimal implementation delay of taxation with trade-off for spectrally negative Lévy risk processes

Abstract

In this paper we consider two cases of optimal implementation delay of taxation with trade-off under spectrally negative Lévy insurance risk processes. In the first case, we assume that the insurance company starts to pay tax only when its surplus level reaches a certain level, and at the termination time of the business there is a terminal value incurred to the company. A method is developed to determine the optimal starting-tax surplus level at which the total expected discounted value of all tax payments up to the termination time plus the discounted terminal value is maximized. In the second case, the company still pays tax subject to a starting-tax surplus level, but with capital injections to prevent bankruptcy. The total expected discounted value of tax payments minus the total discounted capital injection costs is maximized to determine the optimal starting-tax surplus level. Numerical examples are given at the end to illustrate the existence of positive optimal starting-tax surplus levels for both cases considered in this paper.

Appendices

Appendix A

In this section, we briefly review some preliminaries of the spectrally negative Lévy process, namely a Lévy process with Lévy measure having support in $(-\infty , 0]$. Let $X=\{X(t);t\ge 0\}$ with probability laws $\{{\mathbb {P}}_{x};x\in {\mathbb {R}}\}$ and natural filtration $\{{\mathcal {F}}_{t};t\ge 0\}$ be a spectrally negative Lévy process, with the usual exclusion of pure increasing linear drift and the negative of a sub-ordinator. Denote by ${\overline{X}}$ and ${\underline{X}}$, i.e.

$$\begin{aligned} {\overline{X}}(t)=\sup \limits _{0\le s\le t}X(s),\quad {\underline{X}}(t)=\inf \limits _{0\le s\le t}X(s),\quad t\ge 0, \end{aligned}$$

the running supremum and running infimum process of X, respectively.

Define the Laplace exponent of X by

$$\begin{aligned} \Psi (\theta ):=\log {\mathbb {E}}_{x}\Big [\mathrm {e}^{\theta (X(1)-x)}\Big ]=\gamma \theta +\frac{1}{2}\sigma ^{2}\theta ^{2}-\int _{(0,\infty )}\left( 1-\mathrm {e}^{-\theta x}-\theta x{\mathbf {1}}_{(0,1)}(x)\right) \nu (\mathrm {d}x), \end{aligned}$$

where the Lévy measure $\upsilon$ satisfies $\int _{(0,\infty )}\left( 1\wedge x^{2}\right) \nu (\mathrm {d}x)<\infty$. It is known that $\Psi (\theta )$ is finite for $\theta \in [0,\infty )$ where it is strictly convex and infinitely differentiable. As defined in Theorem 8 on page 194 in [10], for each $q\ge 0$, the scale function $W_{q}:\,[0,\infty )\rightarrow [0,\infty )$ is the unique strictly increasing and continuous function with Laplace transform

$$\begin{aligned} \int _{0}^{\infty }\mathrm {e}^{-\theta x}W_{q}(x)\mathrm {d}x=\frac{1}{\Psi (\theta )-q},\quad \theta >\Phi (q), \end{aligned}$$

where $\Phi (q)$ is the largest solution of the equation $\Psi (\theta )=q$. For convenience, we extend the domain of $W_{q}$ to the whole real line by setting $W_{q}(x) = 0$ for all $x < 0$. In particular, write $W=W_0$ for simplicity. When X has sample paths of unbounded variation, or when X has sample paths of bounded variation and the Lévy measure has no atoms, the scale function $W_{q}$ is continuously differentiable over $(0, \infty )$. See [11] for more detailed discussions on the smoothness of scale functions.

We define

$$\begin{aligned} Z_{q}(x)=1+q\int _{0}^{x}W_{q}(z)\mathrm {d}z \quad \text{ and } \quad {\overline{Z}}_{q}(x)=\int _{0}^{x}Z_{q}(z)\mathrm {d}z,\quad q\ge 0, \,x\ge 0, \end{aligned}$$

with $Z_{q}(x)=1$ for $x<0$. For the process X, let its first down-crossing time of level 0 and first up-crossing time of level $b\in (0,\infty )$ be

$$\begin{aligned} \tau _{0}^{-}:=\inf \{t\ge 0: X(t)<0\}\,\,\,\text {and}\,\,\,\tau ^+_{b}:=\inf \{t\ge 0:X(t)>b\}. \end{aligned}$$

Then, Chapter 8 in [16] gives, for $q\ge 0$ and $b\in (0,\infty )$,

$$\begin{aligned} {\mathbb {E}}_{x}\Big [e^{-q\tau ^{+}_{b}} {\mathbf {1}}_{\{\tau ^{+}_{b}<\tau _{0}^{-}\}}\Big ] =\frac{W_{q}(x)}{W_{q}(b)},\quad x\in (0, b]. \end{aligned}$$

(18)

Let $x\wedge y$ denote $\min (x, y)$ and let $x\vee y$ denote $\max (x, y)$. We now define a process $\{Y(t);t\ge 0\}$ as

$$\begin{aligned} Y(t):=X(t)-{\underline{X}}(t)\wedge 0,\quad t\ge 0, \end{aligned}$$

which is the Lévy process X reflected at its infimum. In risk theory, if $X_t$ denotes the surplus level of an insurance company at time t, then the term $-{\underline{X}}(t)\wedge 0$ could be used to represent the cumulative capital injection up to time t, and hence $Y_t$ is the surplus process with capital injections such that $Y(t)\ge 0$ for $t\ge 0$, i.e., ruin never occurs. Let

$$\begin{aligned} \rho _{a}^{+}:=\inf \{t\ge 0;Y(t)>a\},\quad a\in (0,\infty ), \end{aligned}$$

which is the first up-crossing time of level a for the risk process Y. Then, for $q\ge 0$ and $b\in (0,\infty )$, by Proposition 2 of [23], it holds that

$$\begin{aligned} \mathrm {E}_x\big [\mathrm {e}^{-q\rho _{a}^{+}}\big ]=Z_{q}(x)/Z_{q}(a),\quad x\in [0,a]. \end{aligned}$$

(19)

In addition, by the proof of Theorem 1 of [8] (see the 1st and 2nd blocks of equations on Page 167), we have

$$\begin{aligned} {\mathbb {E}}\Big [\int _{0}^{\rho _{a}^{+}}\mathrm {e}^{-q t}\mathrm {d}(-{\underline{X}}(t)\wedge 0)\Big ] =-\frac{\Psi ^{\prime }(0+)}{q}+\frac{{\overline{Z}}_{q}(a)+\frac{\Psi ^{\prime }(0+)}{q}}{Z_{q}(a)},\quad a\in (0,\infty ), \end{aligned}$$

(20)

which is the expected total discounted capital injection made from time 0 until the risk process Y hits a.

By Remark 4.5 of [30] and the proof of Theorem 3.10 in Page 136 of [13], we see that the function $\frac{W_{q}(x)}{W_{q}^{\prime }(x)}$ is non-decreasing with respect to x, i.e.,

$$\begin{aligned}&W_{q}(x)W_{q}^{\prime \prime }(x)<\left( W_{q}^{\prime }(x)\right) ^{2},\quad x\in (0,\infty ). \end{aligned}$$

(21)

From Proposition 2 (ii) of [23], one has

$$\begin{aligned}&\mathrm {E}_{0}\big [\mathrm {e}^{-q{\overline{\tau }}_{a}^{+}}\big ]= Z_{q}(a)- \frac{q\left( W_{q}(a)\right) ^{2}}{W_{q}^{\prime }(a)} , \end{aligned}$$

where

$$\begin{aligned} {\overline{\tau }}_{a}^{+}:=\inf \{t\ge 0; {\overline{X}}(s)\vee 0-X(t)>a\}, \end{aligned}$$

is the first passage time of X reflected at its supremum. Because $\lim \limits _{a\rightarrow \infty }{\overline{\tau }}_{a}^{+}=\infty$, one knows

$$\begin{aligned}&\lim _{a \rightarrow \infty }\Big (Z_{q}(a)-q\frac{\left( W_{q}(a)\right) ^{2}}{W_{q}^{\prime }(a)}\Big )=0. \end{aligned}$$

(22)

In addition, it is found in [35] that

$$\begin{aligned}&\lim _{a \rightarrow \infty }\frac{W_{q}^{\prime }(a)}{W_{q}(a)}=\Phi (q), \end{aligned}$$

(23)

which together with the L’Hôpital’s rule gives

$$\begin{aligned} \lim \limits _{a\rightarrow \infty }\frac{Z_{q}(a)}{Z_{q}^{\prime }(a)}= \frac{1}{\Phi (q)}. \end{aligned}$$

(24)

Appendix B

1.1 Theorem 2.1

Proof

By (22), (23) and the L’Hôpital’s rule, one has

$$\begin{aligned}&\lim _{b \rightarrow \infty }\int _b^{\infty }\left( \frac{W_{q}(b)}{W_{q}(z)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}z =\frac{1-\ell }{\Phi (q)}, \end{aligned}$$

(25)

and

$$\begin{aligned}&\lim _{b \rightarrow \infty }\int _b^{\infty }\left( \frac{W_{q}(b)}{W_{q}(z)}\right) ^{\frac{1}{1-\ell }} \left( \frac{W_{q}^{\prime }(z)}{W_{q}(z)}Z_{q}(z)-qW_{q}(z)\right) \mathrm {d}z=0. \end{aligned}$$

(26)

Combining (23), (25) and (26) yields

$$\begin{aligned}&\lim _{b \rightarrow \infty }\left[ \upsilon (b)-V(b)\left( 1-SqW_{q}(b)\right) \right] \nonumber \\&\quad = \lim _{b \rightarrow \infty }\left[ \frac{S}{1-\ell }\int _b^{\infty }\left( \frac{W_{q}(b)}{W_{q}(z)}\right) ^{\frac{1}{1-\ell }} \left( \frac{W_{q}^{\prime }(z)}{W_{q}(z)}Z_{q}(z)-qW_{q}(z)\right) \mathrm {d}z +S\Big (q\frac{[W_{q}(b)]^{2}}{W_{q}^{\prime }(b)}-Z_{q}(b)\Big )\right. \nonumber \\&\qquad \left. +\frac{\ell }{1-\ell }\int _b^{\infty }\left( \frac{W_{q}(b)}{W_{q}(z)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}z-\frac{W_{q}(b)}{W_{q}^{\prime }(b)}\right] \nonumber \\&\quad = \frac{\ell -1}{\Phi (q)}<0. \end{aligned}$$

(27)

By (8), $\phi (x;b)$ can be rewritten as

$$\begin{aligned} \phi (x;b)= & {} SZ_{q}(x)+\frac{W_{q}(x)}{W_{q}(b)}\,\upsilon (b), \end{aligned}$$

from which one gets

$$\begin{aligned} \frac{\partial \phi (x;b)}{\partial b}= & {} \frac{W_{q}(x)}{[W_{q}(b)]^{2}}\left[ W_{q}(b)\upsilon ^{\prime }(b)-W_{q}^{\prime }(b) \,\upsilon (b)\right] \nonumber \\= & {} \frac{\ell }{1-\ell }\frac{W_{q}(x)W_{q}^{\prime }(b)}{[W_{q}(b)]^{2}} \left[ \upsilon (b)-V(b)\left( 1-SZ_{q}^{\prime }(b)\right) \right] , \end{aligned}$$

(28)

where the following expression for $\upsilon ^{\prime }(b)$ is used to derive the second equality in (28)

$$\begin{aligned} \upsilon ^{\prime }(b)= & {} \frac{1}{1-\ell }\frac{\upsilon (b)}{V(b)}+\frac{\ell }{1-\ell }\left( SZ_{q}^{\prime }(b)-1\right) . \end{aligned}$$

According to (28), $\frac{\partial \phi (x;b)}{\partial b}=0$ is equivalent to (10), i.e., $b_{0}$ is a critical point of $b\mapsto \phi (x;b)$ if $b_{0}$ solves (10). To specify the nature of this critical point, we examine the second partial derivative of $\phi (x; b)$ with respect to b,

$$\begin{aligned} \left. \frac{\partial ^{2}\phi (x;b)}{\partial b^{2}}\right| _{b=b_{0}}= & {} \frac{\ell }{1-\ell }\frac{W_{q}(x){W_{q}^{\prime \prime }(b_{0})\upsilon (b_{0})}}{[W_{q}(b_{0})]^{2}} +\frac{\ell }{1-\ell }\frac{SqW_{q}(x)W_{q}^{\prime }(b_{0})}{W_{q}(b_{0})} \nonumber \\= & {} \frac{\ell }{1-\ell }\frac{W_{q}(x)}{W_{q}(b_{0})}\left( \frac{1-SqW_{q}(b_{0})}{W_{q}^{\prime }(b_{0})}W_{q}^{\prime \prime }(b_{0})+SqW_{q}^{\prime }(b_{0})\right) \nonumber \\= & {} \frac{\ell }{1-\ell }\frac{W_{q}(x)}{W_{q}(b_{0})}f(b_0), \end{aligned}$$

(29)

where

$$\begin{aligned} f(b):=\frac{1-SqW_{q}(b)}{W_{q}^{\prime }(b)}W_{q}^{\prime \prime }(b)+SqW_{q}^{\prime }(b). \end{aligned}$$

(30)

In what follows, we are devoted to ruling out the possibility that the map $b\mapsto \phi (x;b)$ has local minimum points or saddle points in $[0,\infty )$, by separately considering two opposite cases, i.e., Case A: $S\le 0$ and Case B: $S>0$.

Case A. $S\le 0$. In this case, we have $1-SqW_{q}(b)>0$ for all $b\in [0,\infty )$, and the following two claims A1 and A2 hold true.

A1.:

The map $b\mapsto \phi (x;b)$ cannot have a local minimum point in $[0,\infty )$.

Indeed, by (27) and (28), there must exist a sufficiently large ${\overline{b}}>0$ such that $\phi (x;b)$ is decreasing in b over $[{\overline{b}},\infty )$. Hence, if $b_{1}$ is a local minimum point of $b\mapsto \phi (x;b)$, then there would have to exist a local maximum $b_{2}\in (b_{1},{\overline{b}}\,]$, i.e.

$$\begin{aligned}&\left. \frac{\partial ^{2}\phi (x;b)}{\partial b^{2}}\right| _{b=b_{1}}\ge 0 \quad \mathrm {and} \quad \left. \frac{\partial ^{2}\phi (x;b)}{\partial b^{2}}\right| _{b=b_{2}}\le 0, \end{aligned}$$

or equivalently,

$$\begin{aligned}&f(b_{1})\ge 0 \quad \mathrm {and} \quad f(b_{2})\le 0. \end{aligned}$$

(31)

However, as we assumed that the Lévy measure of X has a completely monotone density, by Corollary 1 of [21], we know

$$\begin{aligned}&W_{q}^{\prime }(b)W_{q}^{\prime \prime \prime }(b)-\left( W_{q}^{\prime \prime }(b)\right) ^{2}>0,\quad b\in (0,\infty ), \end{aligned}$$

(32)

which gives

$$\begin{aligned}&f^{\prime }(b)= \left( 1-SqW_{q}(b)\right) \frac{W_{q}^{\prime }(b)W_{q}^{\prime \prime \prime }(b) -[W_{q}^{\prime \prime }(b)]^{2}}{[W_{q}^{\prime }(b)]^{2}} >0,\quad b\in (0,\infty ). \end{aligned}$$

(33)

Clearly, (33) contradicts to (31), therefore claim A1 is true.

A2.:

The map $b\mapsto \phi (x;b)$ cannot have a saddle point $b_{0}$ in $[0,\infty )$. Here, we say that $b_{0}$ is a saddle point of a function if both the first as well as the second derivative at $b_{0}$ of the function is 0.

Assume $b_{0}$ is such that

$$\begin{aligned}&\upsilon (b_{0})=V(b_{0})\left( 1-SqW_{q}(b_{0})\right) \quad \mathrm {and} \quad \left. \frac{\partial ^{2}\phi (x;b)}{\partial b^{2}}\right| _{b=b_{0}}=0, \end{aligned}$$

one can verify that

$$\begin{aligned}&W_{q}^{\prime \prime }(b_{0})=-\frac{Sq[W_{q}^{\prime }(b_{0})]^{2}}{1-SqW_{q}(b_{0})} \quad \mathrm {and} \quad V^{\prime }(b_{0})=\frac{1}{1-SqW_{q}(b_{0})}. \end{aligned}$$

For convenience, let

$$\begin{aligned} h(b):=\upsilon (b)-V(b)\left( 1-SqW_{q}(b)\right) . \end{aligned}$$

Clearly, $h(b_0)=0$. We will show that $b_{0}$ is a local minimum point of h. The first and second derivatives of h are

$$\begin{aligned} h^{\prime }(b)= & {} \upsilon ^{\prime }(b)-V^{\prime }(b)\left( 1-SqW_{q}(b)\right) +SqW_{q}^{\prime }(b)V(b) \\= & {} \frac{\upsilon (b)}{(1-\ell )V(b)}+\frac{\ell }{1-\ell }\left( SqW_{q}(b)-1\right) \\&-\Big (1-\frac{W_{q}(b)W_{q}^{\prime \prime }(b)}{[W_{q}^{\prime }(b)]^{2}}\Big )\left( 1-SqW_{q}(b)\right) +SqW_{q}(b) \\= & {} \frac{h(b)}{(1-\ell )}+SqW_{q}(b) +\frac{W_{q}(b)W_{q}^{\prime \prime }(b)}{[W_{q}^{\prime }(b)]^{2}}\left( 1-SqW_{q}(b)\right) , \end{aligned}$$

and

$$\begin{aligned} h^{\prime \prime }(b)= & {} \frac{h'(b)}{(1-\ell )}+SqW_{q}^{\prime }(b)-\frac{W_{q}(b)W_{q}^{\prime \prime }(b)}{[W_{q}^{\prime }(b)]^{2}}SqW_{q}^{\prime }(b) \\&+\frac{\left( W_{q}^{\prime }(b)W_{q}^{\prime \prime }(b)+W_{q}(b)W_{q}^{\prime \prime \prime }(b)\right) W_{q}^{\prime }(b)- 2W_{q}(b)[W_{q}^{\prime \prime }(b)]^{2}}{[W_{q}^{\prime }(b)]^3}\left( 1-SqW_{q}(b)\right) \\= & {} \frac{h'(b)}{(1-\ell )}+SqW_{q}^{\prime }(b) +\frac{W_{q}^{\prime \prime }(b)}{W_{q}^{\prime }(b)}\left( 1-SqW_{q}(b)\right) +\frac{W_{q}(b)W_{q}^{\prime \prime \prime }(b)}{[W_{q}^{\prime }(b)]^{2}}\left( 1-SqW_{q}(b)\right) \\&-\frac{2W_{q}(b)[W_{q}^{\prime \prime }(b)]^{2}}{[W_{q}^{\prime }(b)]^{3}}\left( 1-SqW_{q}(b)\right) -\frac{SqW_{q}(b)W_{q}^{\prime \prime }(b)}{W_{q}^{\prime }(b)}. \end{aligned}$$

One can see that $h^{\prime }(b_{0})=0$, and

$$\begin{aligned}&SqW_{q}^{\prime }(b_0) +\frac{W_{q}^{\prime \prime }(b_0)}{W_{q}^{\prime }(b_0)}\left( 1-SqW_{q}(b_0)\right) =0. \end{aligned}$$

Also, we have

$$\begin{aligned} -\frac{2W_{q}(b_{0})[W_{q}^{\prime \prime }(b_{0})]^{2}}{[W_{q}^{\prime }(b_{0})]^{3}}\left( 1-SqW_{q}(b_{0})\right) -\frac{SqW_{q}(b_{0})W_{q}^{\prime \prime }(b_{0})}{W_{q}^{\prime }(b_{0})}= & {} -\frac{S^{2}q^{2}W_{q}(b_{0})W_{q}^{\prime }(b_{0})}{1-SqW_{q}(b_{0})}. \end{aligned}$$

By (32) we obtain

$$\begin{aligned} h^{\prime \prime }(b_{0})= & {} \frac{W_{q}(b_{0})W_{q}^{\prime \prime \prime }(b_{0})}{[W_{q}^{\prime }(b_{0})]^{2}}\left( 1-SqW_{q}(b_{0})\right) -\frac{S^{2}q^{2}W_{q}(b_{0})W_{q}^{\prime }(b_{0})}{1-SqW_{q}(b_{0})} \\> & {} \frac{W_{q}(b_{0})[W_{q}^{''}(b_{0})]^2}{[W_{q}^{\prime }(b_{0})]^3}\left( 1-SqW_{q}(b_{0})\right) -\frac{S^{2}q^{2}W_{q}(b_{0})W_{q}^{\prime }(b_{0})}{1-SqW_{q}(b_{0})} \\= & {} \frac{W_{q}(b_{0})\left( 1-SqW_{q}(b_{0})\right) }{[W_{q}^{\prime }(b_{0})]^3} \Big [(W_{q}^{''}(b_{0}))^2-\frac{S^{2}q^{2}(W_{q}^{\prime }(b_{0}))^4}{(1-SqW_{q}(b_{0}))^2}\Big ]=0. \end{aligned}$$

As a result, h(b) reaches a local minimum value of 0 at $b_{0}$. Hence, $h(b)>0$ holds true for all $b\in (b_{0},b_{0}+\epsilon _{0})$ for some $\epsilon _{0}>0$, which combined with (28) implies $\frac{\partial \phi (x;b)}{\partial b}>0$ for all $b\in (b_{0},b_{0}+\epsilon _{0})$. Therefore, the saddle point $b_{0}$ of the map $b\mapsto \phi (x;b)$, has to be followed by a maximum or another saddle point $b_{1}\in (b_{0},\infty )$ of the map $b\mapsto \phi (x;b)$, i.e., $f(b_{0})=0$ and $f(b_{1})\le 0$ (c.f., (29) and (30)), contradicting to (33). Hence the claim A2 is true.

Case B. $S>0$. The following two claims B1 and B2 are true.

B1.:

The map $b\mapsto \phi (x;b)$ cannot have a local minimum in $[0,\infty )$.

If there exists a local minimum point $b_{1}$ of the map $b\mapsto \phi (x;b)$, using (27) and (28) once again we can see that there exists a local maximum point $b_{2}\in (b_{1},\infty )$, and then we obtain (c.f., (29) and (30))

$$\begin{aligned}&f(b_{1})>0 \quad \mathrm {and} \quad f(b_{2})<0. \end{aligned}$$

(34)

Combining (30) with (21) yields

$$\begin{aligned} f(b)= & {} \frac{1-SqW_{q}(b)}{W_{q}^{\prime }(b)}W_{q}^{\prime \prime }(b)+SqW_{q}^{\prime }(b) \\= & {} \frac{W_{q}^{\prime \prime }(b)}{W_{q}^{\prime }(b)}-\frac{SqW_{q}(b)W_{q}^{\prime \prime }(b)}{W_{q}^{\prime }(b)}+SqW_{q}^{\prime }(b) > \frac{W_{q}^{\prime \prime }(b)}{W_{q}^{\prime }(b)}, \end{aligned}$$

which, together with (34), implies that

$$\begin{aligned} W_{q}^{\prime \prime }(b_{2})<0. \end{aligned}$$

(35)

• If $1-SqW_{q}(b_{2})\ge 0$, the strict increasing property of the scale function $W_q(b)$ gives $1-SqW_{q}(b)\ge 0$ for $b \in [b_{1},b_{2}]$. Similar to the derivation of result (33), we can show that $f'(b)>0$ for $b \in [b_{1},b_{2}]$, which contradicts to the result of $f(b_2)<0$ given in (34).

• Otherwise, if $1-SqW_{q}(b_{2})<0$, from $f(b_{2})=\frac{1-SqW_{q}(b_{2})}{W_{q}^{\prime }(b_{2})}W_{q}^{\prime \prime }(b_{2})+SqW_{q}^{\prime }(b_{2})<0$, it holds that $W_{q}^{\prime \prime }(b_{2})>\frac{-Sq[W_{q}^{\prime }(b_{2})]^{2}}{1-SqW_{q}(b_{2})}>0$, which contradicts to (35).

Therefore, the claim B1 is true.

B2.:

The map $b\mapsto \phi (x;b)$ cannot have a saddle point $b_{0}$ in $[0,\infty )$.

Assume that $b_0$ is a saddle point in $[0,\infty )$.

• If $1-SqW_{q}(b_{0})>0$, adopting the same arguments as in the claim A$_{2}$, we can show that h(b) reaches a local minimum value of 0 at $b_{0}$, which, together with (27) and (28), implies that there must be a local maximum point or another saddle point $b_{1}\in (b_{0},\infty )$ of the map $b\mapsto \phi (x;b)$, i.e.

  $$\begin{aligned} f(b_{1})\le 0=f(b_{0}). \end{aligned}$$

  (36)

  If $1-SqW_{q}(b_{1})\ge 0$, (33) holds true for all $b \in [b_{0},b_{1})$, contradicting to (36). Otherwise, if $1-SqW_{q}(b_{1})<0$, by $\frac{1-SqW_{q}(b_{1})}{W_{q}^{\prime }(b_{1})}W_{q}^{\prime \prime }(b_{1})+SqW_{q}^{\prime }(b_{1}) =f(b_{1})\le 0$ we get $W_{q}^{\prime \prime }(b_{1})\ge \frac{-Sq[W_{q}^{\prime }(b_{1})]^{2}}{1-SqW_{q}(b_{1})}>0$, contradicting to $W_{q}^{\prime \prime }(b_{1})\le 0$ (c.f., arguments used in verifying (35)).

• If $1-SqW_{q}(b_{0})\le 0$, adopting the same arguments as in verifying (35) we have $W_{q}^{\prime \prime }(b_{0})<0$. However, by $f(b_{0})=0$, one can see that $W_{q}^{\prime \prime }(b_{0})=-\frac{Sq[W_{q}^{\prime }(b_{0})]^{2}}{1-SqW_{q}(b_{0})}>0$, which is a contradiction.

Hence, the claim B2 is true.

Summing up the arguments concerning Case A and Case B, we conclude that (10) has at most one positive solution.

• When $\upsilon (0)>V(0)(1-SqW_{q}(0))$, by (27), one concludes that a positive solution of (10), denoted by $b^{+}\in (0,\infty )$, exists and satisfies

  $$\begin{aligned}&\upsilon (b)>V(b)(1-SqW_{q}(b)),\,\,b\in [0,b^{+})\quad \text{ and } \quad \\&\quad \upsilon (b)<V(b)(1-SqW_{q}(b)),\,\,b\in (b^{+},\infty ), \end{aligned}$$

  i.e., the map $b\mapsto \phi (x;b)$ reaches its global maximum value at $b^{*}=b^{+}$.

• When $\upsilon (0)<V(0)(1-SqW_{q}(0))$, the equation (10) has no positive solution and

  $$\begin{aligned} \upsilon (b)<V(b)(1-SqW_{q}(b)),\,\,b\in [0,\infty ), \end{aligned}$$

  (37)

  i.e., the map $b\mapsto \phi (x;b)$ reaches its global maximum value at $b^{*}=0$.

• When $\upsilon (0)=V(0)(1-SqW_{q}(0))$, then (37) holds true for $b\in (0,\infty )$ and the map $b\mapsto \phi (x;b)$ reaches its global maximum value at $b^{*}=0$.

The proof of Theorem 2.1 is completed. $\square$

1.2 Proposition 3.1

Proof

From (2), (5) and the Markov property, it is seen that, for $a\in (0,\infty )$, $x\in (0,a]$,

$$\begin{aligned} f_{a}(x)= & {} {\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{a}^{+}};\sigma _{a}^{+}<\sigma _{0}^{-}\right] +{\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{0}^{-}}\left( \left. {\mathbb {E}}_{0}\Big [\mathrm {e}^{-q \rho _{w}^{+}}\Big ]\right| _{w=\mathrm {H}_{1}}\right) {\mathbb {E}}_{\mathrm {H}_{1}} \left[ \mathrm {e}^{-q \kappa _{a}^{+}}\right] ;\sigma _{0}^{-}<\sigma _{a}^{+}\right] \nonumber \\= & {} \left( \frac{W_{q}(x)}{W_{q}(a)}\right) ^{\frac{1}{1-\ell }} +{\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{0}^{-}} \frac{f_{a}(\mathrm {H}_{1})}{Z_{q}(\mathrm {H}_{1})} ;\sigma _{0}^{-}<\sigma _{a}^{+}\right] \nonumber \\= & {} \left( \frac{W_{q}(x)}{W_{q}(a)}\right) ^{\frac{1}{1-\ell }} + \frac{1}{1-\ell }\int _{x}^{a}\frac{f_{a}\left( s\right) }{Z_{q}(s)} \left( \frac{W_{q}(x)}{W_{q}(s)}\right) ^{\frac{1}{1-\ell }} \left( \frac{W_{q}^{\prime }(s)}{W_{q}(s)}Z_{q}(s) -qW_{q}(s)\right) \mathrm {d}s. \end{aligned}$$

(38)

Differentiating both sides of (38) with respect to x gives the following differential equation, for $x\in (0, \infty )$,

$$\begin{aligned} f_{a}^{\prime }(x)= & {} \frac{1}{1-\ell } \frac{W_{q}^{\prime }(x)}{W_{q}(x)} \left( \frac{W_{q}(x)}{W_{q}(a)}\right) ^{\frac{1}{1-\ell }} +\frac{1}{1-\ell }\frac{W_{q}^{\prime }(x)}{W_{q}(x)}\left( f_{a}(x)- \left( \frac{W_{q}(x)}{W_{q}(a)}\right) ^{\frac{1}{1-\ell }}\right) \nonumber \\&-\frac{f_{a}\left( x\right) }{1-\ell } \left( \frac{W_{q}^{\prime }(x)}{W_{q}(x)}-\frac{qW_{q}(x)}{Z_{q}(x)}\right) \nonumber \\= & {} \frac{1}{1-\ell } \frac{qW_{q}(x)}{Z_{q}(x)}f_{a}\left( x\right) , \end{aligned}$$

(39)

with boundary condition $f_{a}(a)=1$. Solving (39), we obtain the first identity in (11). By (4) and an argument similar to the one used in deriving (38), we have

$$\begin{aligned} g_{a}(x)= & {} {\mathbb {E}}_{x}\left[ \int _{0}^{\sigma _{a}^{+}\wedge \sigma _{0}^{-}}\mathrm {e}^{-q t}\ell \,\mathrm {d}{\overline{X}}(t)\right] +{\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{0}^{-}} \frac{g(\mathrm {H}_{1})}{Z_{q}(\mathrm {H}_{1})} ;\sigma _{0}^{-}<\sigma _{a}^{+}\right] \nonumber \\= & {} \frac{\ell }{1-\ell }\int _{x}^{a} \left( \frac{W_{q}(x)}{W_{q}(s)}\right) ^{\frac{1}{1-\ell }} \mathrm {d}s + \frac{1}{1-\ell }\int _{x}^{a}\frac{g\left( s\right) }{Z_{q}(s)} \left( \frac{W_{q}(x)}{W_{q}(s)}\right) ^{\frac{1}{1-\ell }} \nonumber \\&\times \left( \frac{W_{q}^{\prime }(s)}{W_{q}(s)}Z_{q}(s) -qW_{q}(s)\right) \mathrm {d}s ,\qquad x\in (0,\infty ),\quad a\in (x,\infty ). \end{aligned}$$

(40)

Differentiating both sides of (40) with respect to x gives the following differential equation

$$\begin{aligned} g_{a}^{\prime }(x)= & {} \frac{1}{1-\ell }\frac{W_{q}^{\prime }(x)}{W_{q}(x)}g(x) -\frac{\ell }{1-\ell } - \frac{g\left( x\right) }{1-\ell } \left( \frac{W_{q}^{\prime }(x)}{W_{q}(x)}-\frac{qW_{q}(x)}{Z_{q}(x)}\right) \nonumber \\= & {} \frac{1}{1-\ell } \frac{qW_{q}(x)}{Z_{q}(x)}g\left( x\right) -\frac{\ell }{1-\ell } ,\qquad x\in (0,\infty ), \end{aligned}$$

(41)

with boundary condition $g_{a}(a)=0$. Solving (41) gives the second identity in (11).

Combining (20), (5) and (6) yields, for $a\in (0,\infty )$,

$$\begin{aligned} r_{a}(x)= & {} {\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{0}^{-}}\left| U(\sigma _{0}^{-})\right| ;\sigma _{0}^{-}<\sigma _{a}^{+}\right] \nonumber \\&+{\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{0}^{-}} {\mathbb {E}}\Big [\int _{0}^{\rho _{\mathrm {H}_{1}}^{+}}\mathrm {e}^{-q t}\mathrm {d}(-{\underline{X}}(t)\wedge 0))\Big ] ;\sigma _{0}^{-}<\sigma _{a}^{+}\right] \nonumber \\&+{\mathbb {E}}_{x}\left[ \mathrm {e}^{-q \sigma _{0}^{-}} \frac{r_{a}(\mathrm {H}_{1})}{Z_{q}(\mathrm {H}_{1})} ;\sigma _{0}^{-}<\sigma _{a}^{+}\right] \nonumber \\= & {} \frac{1}{1-\ell }\int _{x}^{a} \left( \frac{W_{q}(x)}{W_{q}(s)}\right) ^{\frac{1}{1-\ell }} \left( Z_{q}(s)-\Psi ^{\prime }(0+)W_{q}(s)-\frac{{\overline{Z}}_{q}(s)-\Psi ^{\prime }(0+) {\overline{W}}_{q}(s)}{W_{q}(s)}W_{q}^{\prime }(s)\right) \mathrm {d}s \nonumber \\&+\frac{1}{1-\ell }\int _{x}^{a} \left( \frac{W_{q}(x)}{W_{q}(s)}\right) ^{\frac{1}{1-\ell }} \left( -\frac{\Psi ^{\prime }(0+)}{q}+\frac{{\overline{Z}}_{q}(s)+\frac{\Psi ^{\prime }(0+)}{q}}{Z_{q}(s)} +\frac{r_{a}(s)}{Z_{q}(s)} \right) \nonumber \\&\times \left( \frac{W_{q}^{\prime }(s)}{W_{q}(s)}Z_{q}(s)-qW_{q}(s)\right) \mathrm {d}s,\qquad x\in (0,a]. \end{aligned}$$

(42)

Differentiating both sides of (42) with respect to x gives the following differential equation

$$\begin{aligned} r_{a}^{\prime }(x)= & {} \frac{1}{1-\ell }\frac{W_{q}^{\prime }(x)}{W_{q}(x)}r_{a}(x) \nonumber \\&- \frac{1}{1-\ell } \left( Z_{q}(x)-\Psi ^{\prime }(0+)W_{q}(x)-\frac{{\overline{Z}}_{q}(x)-\Psi ^{\prime }(0+) {\overline{W}}_{q}(x)}{W_{q}(x)}W_{q}^{\prime }(x)\right) \nonumber \\&- \frac{1}{1-\ell } \left( -\frac{\Psi ^{\prime }(0+)}{q}+\frac{{\overline{Z}}_{q}(x)+\frac{\Psi ^{\prime }(0+)}{q}}{Z_{q}(x)} +\frac{r_{a}(x)}{Z_{q}(x)} \right) \left( \frac{W_{q}^{\prime }(x)}{W_{q}(x)}Z_{q}(x)-qW_{q}(x)\right) \nonumber \\= & {} \frac{1}{1-\ell } \frac{qW_{q}(x)}{Z_{q}(x)}r_{a}\left( x\right) \nonumber \\&-\frac{1}{1-\ell }\left( Z_{q}(x)-\left( {\overline{Z}}_{q}(x)+\frac{\Psi ^{\prime }(0+)}{q}\right) \frac{qW_{q}(x)}{Z_{q}(x)}\right) ,\quad x\in (0,a], \end{aligned}$$

(43)

with boundary condition $r_{a}(a)=0$. Solving (43) gives (12). The proof is completed. $\square$

1.3 Theorem 3.1

Proof

Firstly, we show a result that is useful in our main proof:

$$\begin{aligned} \lim \limits _{a\rightarrow \infty } {\overline{h}}(a)= \frac{\ell -1}{\Phi (q)}<0. \end{aligned}$$

(44)

Using integration by parts, one can get

$$\begin{aligned}&-\frac{1}{1-\ell }\int _a^{\infty } \left( {\overline{Z}}_{q}(w)+\frac{\Psi ^{\prime }(0+)}{q}\right) \frac{qW_{q}(w)}{Z_{q}(w)} \left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}w \\&\quad = \left. \left( {\overline{Z}}_{q}(w)+\frac{\Psi ^{\prime }(0+)}{q}\right) \left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }}\right| _{w=a}^{\infty } - \int _a^{\infty } Z_{q}(w) \left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}w \\&\quad = -{\overline{Z}}_{q}(a)-\frac{\Psi ^{\prime }(0+)}{q} - \int _a^{\infty } Z_{q}(w) \left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}w, \quad a\in (0,\infty ), \end{aligned}$$

and

$$\begin{aligned}&-\frac{\ell }{1-\ell }\int _a^{\infty } Z_{q}(w)\left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}w \\&\quad = \int _a^{\infty } \frac{Z_{q}(w)\left( Z_{q}(a)\right) ^{\frac{1}{1-\ell }}}{Z_{q}^{\prime }(w)} \mathrm {d}\left( Z_{q}(w)\right) ^{-\frac{1}{1-\ell }+1} \\&\quad = -{\overline{V}}(a) Z_{q}(a)- \int _a^{\infty }Z_{q}(w) \left( \frac{Z_{q}(w)}{Z_{q}^{\prime }(w)}\right) ^{\prime }\left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }} \mathrm {d}w , \quad a\in (0,\infty ). \end{aligned}$$

Hence, by (13) and (15) we have

$$\begin{aligned} {\overline{h}}(a)= & {} \frac{\ell }{1-\ell }\int _a^{\infty }\left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }}\mathrm {d}w-\frac{Z_{q}(a)}{Z_{q}^{\prime }(a)} \nonumber \\&- \varphi \int _a^{\infty }Z_{q}(w) \frac{q\left( W_{q}(w)\right) ^{2}/W_{q}^{\prime }(w)-Z_{q}(w)}{q\left( W_{q}(w)\right) ^{2}/W_{q}^{\prime }(w)}\left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }} \mathrm {d}w. \end{aligned}$$

(45)

Using (22), (24) and L’Hôpital’s rule, we get

$$\begin{aligned} \lim \limits _{a\rightarrow \infty }\frac{\ell }{1-\ell }\int _a^{\infty } \left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }} \mathrm {d}w=\frac{\ell }{\Phi (q)}, \end{aligned}$$

and

$$\begin{aligned} \lim \limits _{a\rightarrow \infty }\int _a^{\infty }Z_{q}(w) \frac{q\left( W_{q}(w)\right) ^{2}/W_{q}^{\prime }(w)-Z_{q}(w)}{q\left( W_{q}(w)\right) ^{2}/W_{q}^{\prime }(w)}\left( \frac{Z_{q}(a)}{Z_{q}(w)}\right) ^{\frac{1}{1-\ell }} \mathrm {d}w=0, \end{aligned}$$

which, together with (24) and (45), gives (44).

Then, we show that the function ${\bar{h}}(a)$ can be used to find critical points of $a\mapsto {\overline{\phi }}(x;a)$. By the expression of ${\overline{\phi }}(x;a)$ given by (14), we have

$$\begin{aligned} \frac{\partial {\overline{\phi }}(x;a)}{\partial a}= & {} \frac{Z_{q}(x)}{\left( Z_{q}(a)\right) ^{2}}\left( Z_{q}(a){\overline{\upsilon }}^{\prime }(a)-Z_{q}^{\prime }(a) \,{\overline{\upsilon }}(a)\right) \nonumber \\= & {} \frac{\ell }{1-\ell }\frac{Z_{q}(x)Z_{q}^{\prime }(a)}{\left( Z_{q}(a)\right) ^{2}} \left[ {\overline{\upsilon }}(a)-{\overline{V}}(a)\left( 1-\varphi Z_{q}(a)\right) \right] \nonumber \\= & {} \frac{\ell }{1-\ell }\frac{Z_{q}(x)Z_{q}^{\prime }(a)}{\left( Z_{q}(a)\right) ^{2}} {\overline{h}}(a) , \end{aligned}$$

(46)

where the following expression for ${\overline{\upsilon }}^{\prime }(a)$ is employed

$$\begin{aligned} {\overline{\upsilon }}^{\prime }(a)= & {} \frac{1}{1-\ell }\frac{{\overline{\upsilon }}(a)}{{\overline{V}}(a)}+\frac{\ell }{1-\ell }\left( \varphi Z_{q}(a)-1\right) . \end{aligned}$$

According to (46), $\frac{\partial {\overline{\phi }}(x;a)}{\partial a}=0$ is equivalent to (16), i.e., $a_{0}$ is a critical point of $a\mapsto {\overline{\phi }}(x;a)$ if $a_{0}$ solves (16). To specify the nature of this critical point $a_{0}$, we examine the second partial derivative of ${\bar{\phi }}(x; a)$,

$$\begin{aligned} \left. \frac{\partial ^{2} {\overline{\phi }}(x;a)}{\partial a^{2}}\right| _{a=a_{0}}= & {} \frac{\ell }{1-\ell }\frac{W_{q}(x){W_{q}^{\prime \prime }(a_{0}){\overline{\upsilon }}(a_{0})}}{[W_{q}(a_{0})]^{2}} +\frac{\ell }{1-\ell }\frac{SqW_{q}(x)W_{q}^{\prime }(a_{0})}{W_{q}(a_{0})} \nonumber \\= & {} \frac{\ell }{1-\ell }\frac{Z_{q}(x)}{Z_{q}(a_{0})}\left( \frac{1-\varphi Z_{q}(a_{0})}{Z_{q}^{\prime }(a_{0})}Z_{q}^{\prime \prime }(a_{0})+\varphi Z_{q}^{\prime }(a_{0})\right) =\frac{\ell }{1-\ell }\frac{Z_{q}(x)}{Z_{q}(a_{0})}{\bar{f}}(a_0),\nonumber \\ \end{aligned}$$

(47)

where

$$\begin{aligned} {\overline{f}}(a):=\frac{1-\varphi Z_{q}(a)}{Z_{q}^{\prime }(a)}Z_{q}^{\prime \prime }(a)+\varphi Z_{q}^{\prime }(a),\quad a\in (0,\infty ). \end{aligned}$$

By (21), we have

$$\begin{aligned} {\overline{f}}^{\prime }(a)=\left( 1-\varphi Z_{q}(a)\right) \frac{W_{q}(a)W_{q}^{\prime \prime }(a)-\left( W_{q}^{\prime }(a)\right) ^{2}}{\left( W_{q}(a)\right) ^{2}}>0,\quad a\in (0,\infty ). \end{aligned}$$

(48)

Hence, by (44) and a similar argument as the one used in the proof of case A of Theorem 2.1, one can verify that the map $a\mapsto {\overline{\phi }}(x;a)$ cannot have a local minimum point or saddle point in $[0,\infty )$. Indeed,

• if $a_{0}$ is a local minimum point of the map $a\mapsto {\overline{\phi }}(x;a)$, then there should be $a_{0}^{\prime }\in (a_{0},\infty )$ such that ${\overline{f}}(a_{0})\ge 0$ and ${\overline{f}}(a_{0}^{\prime })\le 0$, which contradicts to (48);

• if $a_{0}$ is a saddle point of the map $a\mapsto {\overline{\phi }}(x;a)$, then by (21) and

  $$\begin{aligned} Z_{q}^{\prime \prime }(a_{0})=-\frac{\varphi \left( Z_{q}^{\prime }(a_{0})\right) ^{2}}{1-\varphi Z_{q}(a_{0})},\qquad {\overline{V}}^{\prime }(a_{0})=\frac{1}{1-\varphi Z_{q}(a_{0})}, \end{aligned}$$

  one can obtain ${\overline{h}}^{\prime }(a_{0})=0$ and

  $$\begin{aligned} {\overline{h}}^{\prime \prime }(a_{0})= & {} \frac{Z_{q}^{\prime }(a_{0})Z_{q}^{\prime \prime \prime }(a_{0})\left( 1-\varphi Z_{q}(a_{0})\right) ^{2}-\varphi ^{2}[Z_{q}^{\prime }(a_{0})]^{4}}{\left( 1-\varphi Z_{q}(a_{0})\right) [Z_{q}^{\prime }(a_{0})]^{3}/Z_{q}(a_{0})} \\> & {} \frac{\left( Z_{q}^{\prime \prime }(a_{0})\right) ^{2}\left( 1-\varphi Z_{q}(a_{0})\right) ^{2}-\varphi ^{2}[Z_{q}^{\prime }(a_{0})]^{4}}{\left( 1-\varphi Z_{q}(a_{0})\right) [Z_{q}^{\prime }(a_{0})]^{3}/Z_{q}(a_{0})}=0, \end{aligned}$$

  implying that ${\overline{h}}(a)$ reaches a local minimum value of 0 at $a_{0}$. The above inequality holds because that $Z_{q}^{\prime }(a_{0})Z_{q}^{\prime \prime \prime }(a_{0})>\left( Z_{q}^{\prime \prime }(a_{0})\right) ^{2}$. In this case there must exist $a_{0}^{\prime }\in (a_{0},\infty )$ such that $a_{0}$ is a maximum point or another saddle point of the map $a\mapsto {\overline{\phi }}(x;a)$, i.e., ${\overline{f}}(a_{0})=0$ and ${\overline{f}}(a_{0}^{\prime })\le 0$, which again contradicts to (48).

We thus conclude that the equation (16) has at most one positive solution.

Finally, we shall examine the existence of any positive solution of the equation (16).

• When ${\overline{\upsilon }}(0)>{\overline{V}}(0)(1-\varphi Z_{q}(0))$, by (44) one concludes that a positive solution $a^{+}\in (0,\infty )$ of (16) exists such that

  $$\begin{aligned}&{\overline{\upsilon }}(a)>{\overline{V}}(a)(1-\varphi Z_{q}(a)),\,\,a\in [0,a^{+})\quad \text{ and } \\&\quad {\overline{\upsilon }}(a)<{\overline{V}}(a)(1-\varphi Z_{q}(a)),\,\,a\in (a^{+},\infty ), \end{aligned}$$

  i.e., the map $a\mapsto {\overline{\phi }}(x;a)$ reaches its global maximum value at $a^{*}=a^{+}$.

• When ${\overline{\upsilon }}(0)<{\overline{V}}(0)(1-\varphi Z_{q}(0))$, then (16) has no positive solution and

  $$\begin{aligned} {\overline{\upsilon }}(a)<{\overline{V}}(a)(1-\varphi Z_{q}(a)),\qquad a\in [0,\infty ), \end{aligned}$$

  (49)

  i.e., the map $a\mapsto {\overline{\phi }}(x;a)$ reaches its global maximum value at $a^{*}=0$.

• When ${\overline{\upsilon }}(0)={\overline{V}}(0)(1-\varphi Z_{q}(0))$, (49) holds true for all $a\in (0,\infty )$ and the map $a\mapsto {\overline{\phi }}(x;a)$ reaches its global maximum value at $a^{*}=0$.

The proof is completed. $\square$