Robust Optimal Excess-of-Loss Reinsurance and Investment Problem with more General Dependent Claim Risks and Defaultable Risk

Abstract

This paper is devoted to investigating a robust optimal excess-of-loss reinsurance and investment problem with defaultable risk, in which the insurer’s wealth process is described by a more general dependent risk model with two classes of insurance business. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a risk-free asset, a defaultable bond and a risky asset whose price depends on a square-root stochastic factor process which makes the geometric Brownian motion, CEV model and Heston model as special cases. Our aim is to seek the optimal excess-of-loss reinsurance and investment strategy under the criterion of maximizing the expected exponential utility of the terminal wealth. Applying the stochastic control theory, the robust Hamilton-Jacobi-Bellman (HJB) equations for the post-default case and the pre-default case are first established, respectively. Then the explicit expressions of the optimal control strategy are obtained, moreover, we provide the verification theorem. Finally, some numerical examples are given to illustrate our results.

Appendix

Proof of Theorem 3.1

(i) If ${\xi }_{1}/(({\beta }_{0}+v){d}_{1})\ge 1$ or ${d}_{1}=0,$ then ${t}_{0}\le T$ holds.

For ${t}_{0}\le t\le T,$ we have $0<{d}_{1}\le {d}_{2}.$ Since ${h}_{1}(x)$ and ${h}_{3}(x)$ are increasing functions on $[{d}_{1},+\infty ),$ it’s easy to verify that $l(x)$ increases on $[{d}_{1},+\infty ).$ Moreover, based on the fact that ${h}_{1}({d}_{1})=0,$ we get $l({d}_{1})={h}_{3}({d}_{1})\le {h}_{3}({d}_{2})=0.$

If $l({D}_{1})\ge 0,$ the equation $l(x)=0$ has a unique solution ${d}_{0}(t)\in [{d}_{1},{D}_{1}],$ thus we have

$${a}_{1}^{*}(t)={d}_{0}(t).$$

(85)

Moreover, since ${a}_{2}(t)\le {D}_{2},$ we from (41) obtain that

$$\begin{aligned}{a}_{2}^{*}(t)&={h}_{2}^{-1}({h}_{1}({d}_{0}(t)))I\{l({D}_{1})\ge 0,{h}_{1}({d}_{0}(t))\le {h}_{2}({D}_{2})\}\\ &+{D}_{2}I\{l({D}_{1})\ge 0,{h}_{1}({d}_{0}(t))>{h}_{2}({D}_{2})\}\end{aligned}$$

(86)

If $l({D}_{1})<0,$ it implies the equation $l(x)=0$ has a unique solution on $({D}_{1},+\infty ).$ Notice that ${a}_{1}(t)\le {D}_{1},$ so at this time, we choose ${a}_{1}^{*}(t)={D}_{1}.$ Then substituting ${a}_{1}^{*}(t)={D}_{1}$ back into $g({a}_{1},{a}_{2})$ in (36), and taking the first derivative of $g({a}_{1},{a}_{2})$ with respect to ${a}_{2}$, we can obtain the minimizer of $g({a}_{1},{a}_{2})$ as follows

$${a}_{2}^{*}(t)=\frac{{\xi }_{2}}{{\beta }_{0}+v}{e}^{-r(T-t)}-\frac{\lambda p}{{\lambda }_{2}+\lambda p}{\widetilde{\mu }}_{X}({D}_{1}).$$

(87)

Based on (85), (86) and (87), consequently, ${a}_{1}^{*}(t)$ and ${a}_{2}^{*}(t)$ can be expressed as ${\widehat{a}}_{1}^{*}(t)$ and ${\widehat{a}}_{2}^{*}(t)$ given by (45).

For $0\le t<{t}_{0},$ we have ${d}_{1}>{d}_{2},$ then $l({a}_{l})={h}_{3}({d}_{1})>{h}_{3}({d}_{2})=0.$ Meanwhile, $l(x)$ strictly increases on $[{d}_{1},+\infty )$ and $l(x)>0,x\in [{d}_{1},+\infty ).$ As a result, the equation $l(x)=0$ has no solution on $[{d}_{1},+\infty )$. In other words, there does not exist the solution $({a}_{1}^{*},{a}_{2}^{*})$ satisfying (38) when ${a}_{1}\in ({d}_{1},+\infty )$ and ${a}_{2}\in [0,+\infty ).$ However, it doesn’t mean that (38) has no solution on ${a}_{1}\in [0,{d}_{1})$ and ${a}_{2}\in [0,+\infty ).$ It is not difficult to prove that ${h}_{1}(x)$ is a convex function on $[0,{d}_{1}).$ So for $0\le t<{t}_{0},$ if the solution of $l(x)=0$ exists, it will only be obtained on $x\in [0,{d}_{1})$, hence, it is indeed ${a}_{1}^{*}(t)$ that we try to find.

Due to the fact that ${h}_{1}(0)={h}_{1}({d}_{1})=0,$ and ${h}_{1}(x)$ is a convex function on $[0,{d}_{1}),$ we have ${h}_{1}({a}_{1})\le 0$ on ${a}_{1}\in [0,{d}_{1}).$ Moreover, notice that ${h}_{2}^{-1}(x)$ is strictly increasing and ${h}_{2}^{-1}(0)=0,$ so we obtain from (41) that ${a}_{2}={h}_{2}^{-1}({h}_{1}({a}_{1}))\le 0.$

Based on ${a}_{2}\ge 0,$ we get ${a}_{2}^{*}(t)=0.$ Substituting ${a}_{2}^{*}(t)=0$ into the function $g({a}_{1},{a}_{2})$ given by (36) and taking the first derivative of the function $g({a}_{1},0)$ with respect to ${a}_{1}$, we can obtain the minimizer of $g({a}_{1},0)$ which is expressed as

$${a}_{1}^{*}(t)=\frac{{\xi }_{1}}{{\beta }_{0}+v}{e}^{-r(T-t)}={d}_{2}\le {d}_{1}.$$

To sum up, we derive (46) when $0\le t<{t}_{0}$.

(iii)If ${\xi }_{1}/(({\beta }_{0}+v){d}_{1})<1$, then ${t}_{0}>T$ holds. Thus the inequality ${d}_{1}\ge {d}_{2}$ holds when $0\le t\le T.$ In this case, the optimal control problem is similar to that of $0\le t<{t}_{0}$ in case (i). Thus, the optimal reinsurance strategy with $0\le t\le T$ is the same as that with $0\le t<{t}_{0},$ which is given by (46).

Proof of Theorem 4.1

According to Kraft (2004, Corollary 1.2), Theorem 4.1 will hold if $({\varepsilon }^{*},{\theta }^{*})$ and the corresponding candidate value function $V(t,x,z,\alpha )$ satisfy the following three properties:

• (1) ${\varepsilon }^{*}$ is an admissible strategy and ${P}^{\theta ^{*}}$ is well-defined by ${\Lambda }^{\theta *}(t)$ with ${\theta }_{i}^{*},i=\mathrm{0,1},\mathrm{2,3};$

• (2) ${E}^{{{P^\theta }^{*}}}\left[\underset{t\in [0,T]}{\mathrm{sup}}{\left|V(t,{X}^{{\varepsilon }^{*}}(t),H(t),\alpha (t))\right|}^{4}\right]<\infty$;

• (3) ${E}^{{{P^\theta }^{*}}}\left[\underset{t\in [0,T]}{\mathrm{sup}}{\left|\Psi (t,{X}^{{\varepsilon }^{*}}(t),{\varepsilon }^{*}(t,\alpha (t)),{\theta }^{*}(t,\alpha (t)))\right|}^{2}\right]<\infty .$  

Next, we shall verify the properties (1)-(3), respectively. To understand the proof process easily and logically, we first prove property (2) and then prove property (1).

Proof of (2)

Substituting $({\varepsilon }^{*},{\theta }^{*})$ into (20), we get the following wealth process under $({\varepsilon }^{*},{\theta }^{*})$

$$\begin{aligned}{X}^{{\varepsilon }^{*}}(t)&={x}_{0}{e}^{rt}+{\int }_{0}^{t}{e}^{r(t-s)}({c}_{1}+{c}_{2})ds\\& +{\int }_{0}^{t}{e}^{r(t-s)}\frac{v{\delta }^{2}\alpha (s)}{{\beta }_{1}+v}{e}^{r(T-s)}(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}L(s))ds\\& +{\int }_{0}^{t}{e}^{r(t-s)}(1-H(s)\gamma {\pi }_{2}^{*}\text{(}s\text{)}ds\\& -{\int }_{0}^{t}{e}^{r(t-s)}{\theta }_{0}^{*}(s)\sqrt{{\gamma }_{1}^{2}+{\gamma }_{2}^{2}+{2}\lambda p{\widetilde{\mu }}_{X}({a}_{1}^{*}(s)){\widetilde{\mu }}_{Y}({a}_{2}^{*}(s\text{)})}ds\\& +{\int }_{0}^{t}{e}^{r(t-s)}{\pi }_{1}^{*}(s)\sigma (s){d}{W}_{1}^{{P}^{\theta ^{*} }{}}(s)\\& +{\int }_{0}^{t}{e}^{r(t-s)}\sqrt{{\gamma }_{1}^{2}+{\gamma }_{2}^{2}+{2}\lambda p{\widetilde{\mu }}_{X}({a}_{1}^{*}(s)){\widetilde{\mu }}_{Y}({a}_{2}^{*}(s\text{)})}{d}{W}_{0}^{{P}^{\theta ^{*}}{}}(s).\end{aligned}$$

(88)

Inserting (88) into (79), we obtain

$$\begin{array}{c}{\left|V(t,{X}^{{\varepsilon }^{*}}(t),H(t),\alpha (t))\right|}^{4}={\left|H(t)V(t,{X}^{{\varepsilon }^{*}}(t),0,\alpha (t))+(1-H(t))V(t,{X}^{{\varepsilon }^{*}}(t),1,\alpha (t))\right|}^{4}.\end{array}$$

Furthermore, we get the following estimate with appropriate constants ${M}_{1}<{M}_{2},$

$$\begin{array}{l}{\left|V(t,{X}^{{\varepsilon }^{*}}(t),0,\alpha (t))\right|}^{4}\\ =\frac{1}{{v}^{4}}\mathrm{exp}\{-4v{e}^{r(T-t)}{X}^{{\varepsilon }^{*}}(t)+4\tilde{L }(t)\alpha (t)+4\tilde{K }(t)]\}\\ \le {M}_{1}\mathrm{exp}\{-4v{e}^{r(T-t)}{X}^{{\varepsilon }^{*}}(t)\}\\ \le {M}_{2}\mathrm{exp}\{-4v{\int }_{0}^{t}\delta \sqrt{\alpha (s)}(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}\tilde{L }(s)){d}{W}_{1}^{{P}^{\theta }{}^{*}}(s)\}\\ \cdot \mathrm{exp}\{-4v{\int }_{0}^{t}\frac{v{\delta }^{2}}{{\beta }_{1}+v}\alpha (s)(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}\tilde{L }(s)){d}s\}\\ \cdot \mathrm{exp}\{-4v{\int }_{0}^{t}{e}^{r(T-s)}\sqrt{{\gamma }_{1}^{2}+{\gamma }_{2}^{2}+{2}\lambda p{\widetilde{\mu }}_{X}({a}_{1}^{*}(s)){\widetilde{\mu }}_{Y}({a}_{2}^{*}(s\text{)})}{d}{W}_{0}^{{P}^{\theta }{}^{*}}(s)\}.\\ \end{array}$$

$$={M}_{2}\prod_{i=1}^{4}\mathrm{exp}\{{E}_{i}(t)\},$$

(89)

where $\tilde{K }(t)$ is given in (75) and

$$\left\{\begin{array}{l}{E}_{1}(t)=-4v{\int }_{0}^{t}\delta \sqrt{\alpha (s)}(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}\tilde{L }(s)){d}{W}_{1}^{{P}^{\theta ^{*}}{}}(s)-{\int }_{0}^{t}32{v}^{2}{\delta }^{2}\alpha (s){(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}\tilde{L }(s))}^{2}{d}s,\\ {E}_{2}(t)={\int }_{0}^{t}\left(32{v}^{2}{\delta }^{2}{(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}\tilde{L }(s))}^{2}-\frac{4{v}^{2}{\delta }^{2}}{{\beta }_{1}+v}(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}\tilde{L }(s))\right)\alpha (s){d}s,\\ {E}_{3}(t)=-4v{\int }_{0}^{t}{e}^{r(T-s)}\sqrt{{\gamma }_{1}^{2}+{\gamma }_{2}^{2}+{2}\lambda p{\widetilde{\mu }}_{X}({a}_{1}^{*}(s)){\widetilde{\mu }}_{Y}({a}_{2}^{*}(s\text{)})}{d}{W}_{0}^{{P}^{\theta ^{*}}{}}(s)-{E}_{4}(t),\\ {E}_{4}(t)=32{v}^{2}{\int }_{0}^{t}{e}^{2r(T-s)}\left({\gamma }_{1}^{2}+{\gamma }_{2}^{2}+{2}\lambda p{\widetilde{\mu }}_{X}({a}_{1}^{*}(s)){\widetilde{\mu }}_{Y}({a}_{2}^{*}(s\text{)})\right){d}s.\end{array}\right.$$

(90)

Because $\tilde{L }(t)=L(t)<0$ and $\tilde{K }(t)$ are deterministic and bounded on $[0,T]$ and $\alpha (t)>\text{0,}$ we obtain that the first estimate in (89) is valid. The second inequality holds due to that ${a}_{1}^{*}(t),{a}_{2}^{*}{(}t{),\ }{\theta }_{0}^{*}{(}t\text{),}$ and ${\pi }_{2}^{*}{(}t{)},H(t),{e}^{r(T-t)}$ are deterministic and bounded on $[0,T].$

In what follows, we consider the four integrals about $\mathrm{exp}\{{E}_{i}(t)\},i=\mathrm{1,2},\mathrm{3,4}.$ Firstly, note that ${a}_{1}^{*}\text{(}t\text{)}$ and ${a}_{2}^{*}\text{(}t\text{)}$ are bounded on $[0,T],$ it easy to check that

$${E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{4}(t)\}]<\infty .$$

(91)

Secondly, by the Lemma 4.3 in Zeng and Taksar (2013), it is easily be seen that $\mathrm{exp}\{4{E}_{1}(t)\}$ and $\mathrm{exp}\{4{E}_{3}(t)\}$ are martingales under ${p}^{\theta }{}^{*},$ consequently,

$${E}^{{P}^{\theta ^{*} }{}}[\mathrm{exp}\{4{E}_{1}(t)\}]<\infty\;\mathrm{and}\;{E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{3}(t)\}]<\infty .$$

(92)

Thirdly, according to Theorem 5.1 in Zeng and Taksar (2013), we obtain a sufficient condition for

$${E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{2}(t)\}]<\infty ,$$

(93)

is

$$128{v}^{2}{\delta }^{2}{(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}L(s))}^{2}-\frac{16{v}^{2}{\delta }^{2}}{{\beta }_{1}+v}(\frac{1}{{\beta }_{1}+v}+\frac{{k}_{1}}{\delta v}L(s))\le \frac{{k}^{2}}{2({k}_{1}^{2}+{k}_{2}^{2})},$$

(94)

for $\forall s\in [0,T].$

Note that $\frac{{w}_{3}}{{w}_{1}+{w}_{2}}\le L(t)<0,$ based on (88), then (93) holds for $\forall t\in [0,T]$ and $T\in [0,\infty )$ due to the property of quadratic function. According to (91)-(93) and Cauchy–Schwarz inequality, we have

$$\begin{array}{l}{E}^{{P}^{\theta ^{*}}{}}{\left|V(t,{X}^{{\varepsilon }^{*}}(t),0,\alpha (t))\right|}^{4}\\ \le {M}_{2}{E}^{{P}^{\theta ^{*} }{}}[\mathrm{exp}\{{E}_{1}(t)+{E}_{2}(t)+{E}_{3}(t)+{E}_{4}(t)\}]\\ \le {M}_{2}{\left({E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{2{E}_{1}(t)+2{E}_{2}(t)\}]\right)}^\frac{1}{2}{\left({E}^{{P}^{\theta ^{*} }{}}[\mathrm{exp}\{2{E}_{3}(t)+2{E}_{4}(t)\}]\right)}^\frac{1}{2}\\ \le {M}_{2}{\left({E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{1}(t)\}]\cdot {E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{2}(t)\}]\right)}^\frac{1}{4}\cdot {\left({E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{1}(t)\}]\cdot {E}^{{P}^{\theta ^{*}}{}}[\mathrm{exp}\{4{E}_{2}(t)\}]\right)}^\frac{1}{4}\\<\infty .\end{array}$$

Similarly, ${E}^{{p}^{\theta ^{*}}}{|V(t,{X}^{{\varepsilon }^{*}}(t),0,\alpha (t))|}^{4}<\infty .$ Thus

$${E}^{{P}^{\theta ^{*}}{}}{\left|V(t,{X}^{{\varepsilon }^{*}}(t),H(t),\alpha (t))\right|}^{4}<\infty .$$

(95)

Proof of (1)

From the process of solving HJB equation, we know ${\varepsilon }^{*}(t,\alpha (t))$ is progressively measurable. Moreover, ${a}_{1}^{*}(t)$ and ${a}_{2}^{*}(t)$ are deterministic and bounded on $[0,T]$ and ${\varepsilon }^{*}(t,\alpha (t))$ depends on the state process $\alpha (t)$ which is a mean-reverting square root process. Notice that, for any fixed time, although $\alpha (t)$ is generally unbounded, its first and second order moments are bounded (see Mao 1997 and Kwok 1998), thus condition (ii) in Definition 2.1 holds. Finally, according to Property (2), we can see that Condition (iii) in Definition 2.1 is met.

Proof of (3)

Since ${E}^{{P}^{\theta ^{*}}{}}[{\alpha }^{\kappa }(t)]<\infty$ for any $\kappa >0,$ we obtain from (78) that

$$\begin{array}{l}{E}^{{P}^{\theta ^{*}}{}}\left[\mathrm{exp}\left\{{\int }_{0}^{T}(\frac{1}{2}{\theta }_{0}^{*2}(t)+\frac{1}{2}{\theta }_{1}^{*2}(t,\alpha (t))+\frac{1}{2}{\theta }_{2}^{*2}(t,\alpha (t))\right.\right.\\ \left.\left.+{h}^{P}({\theta }_{3}^{*}(t,\alpha (t))\mathrm{ln}{\theta }_{3}^{*}(t,\alpha (t))-{\theta }_{3}^{*}(t,\alpha (t))+1){d}t\right\}\right]\\ <\infty .\end{array}$$

Moreover, let

$$\Gamma (t)=\frac{v{\theta }_{0}^{*2}(t)}{2{\beta }_{0}}+\frac{v{\theta }_{1}^{*2}(t,\alpha (t))}{2{\beta }_{1}}+\frac{v{\theta }_{2}^{*2}(t,\alpha (t))}{2{\beta }_{2}}+v{h}^{P}(1-z)\frac{{\theta }_{3}^{*}\big{(}t,\alpha (t)\big{)}\mathrm{ln}{\theta }_{3}^{*}\big{(}t,\alpha (t)\big{)}-{\theta }_{3}^{*}\big{(}t,\alpha (t)\big{)}+1}{{\beta }_{3}},$$

then based on (78), we get ${E}^{{P}^{\theta ^{*}}{}}[{\Gamma }^{4}(t)]<\infty .$ From (27), we can see that $V(T,x,z,\alpha )=-\frac{{\beta }_{i}}{v{\phi }_{i}},i=\mathrm{0,1},2.$ Inserting ${\theta }^{*}$ and ${\varepsilon }^{*}$ into (27) arrives at

$$\begin{array}{l}{E}^{{{P ^{\theta ^{*}}}}}\left[\underset{t\in [0,T]}{\mathrm{sup}}{\left|\Psi (t,{X}^{{\varepsilon }^{*}}(t),{\varepsilon }^{*}(t,\alpha (t)),{\theta }^{*}(t,\alpha (t)))\right|}^{2}\right]\\ ={E}^{{{P ^{\theta^{*}}}}}\left[\underset{t\in [0,T]}{\mathrm{sup}}{\left|\Gamma (t)V(t,{X}^{{\varepsilon }^{*}}(t),H(t),\alpha (t))\right|}^{2}\right]\\ ={E}^{{{P ^{\theta ^{*}}}}}\left[\underset{t\in [0,T]}{\mathrm{sup}}{\left|\Gamma (t)\right|}^{2}\cdot {\left|V(t,{X}^{{\varepsilon }^{*}}(t),H(t),\alpha (t))\right|}^{2}\right]\end{array}$$

$$\le {\left[{E}^{{{{P}^{\theta ^{*}}}}}\left(\underset{t\in [0,T]}{\mathrm{sup}}{\left|V(t,{X}^{{\varepsilon }^{*}}(t),H(t),\alpha (t))\right|}^{4}\right)\right]}^{1/2}\cdot {\left[{E}^{{{{P}^{\theta ^{*}}}}}\left(\underset{t\in [0,T]}{\mathrm{sup}}{\left|\Gamma (t)\right|}^{4}\right)\right]}^{1/2}$$

$$<\infty ,$$

where the first estimate follows from the Cauchy–Schwarz inequality and the last estimate from (95) and ${E}^{{{{P}^{\theta ^{*}}}}}({|\Gamma (t)|}^{4})<\infty .$ Thus, property (3) holds.

Now we have proved all the properties, the result of Kraft (2004, Corollary 1.2) guarantees that $({\varepsilon }^{*},{\theta }^{*})$ is an optimal strategy and $V(t,x,z,\alpha )$ is the corresponding optimal value function.