Equilibrium reinsurance-investment strategies with partial information and common shock dependence

Abstract

In this paper, we study an optimal reinsurance-investment problem with partial information and common shock dependence under the mean-variance criterion for an insurer. The insurer has two dependent classes of insurance business, which are subject to a common shock. We consider the optimal reinsurance-investment problem under complete information and partial information, respectively. We formulate the complete information problem within a game theoretic framework and seek the equilibrium reinsurance-investment strategy and equilibrium value function by solving an extended Hamilton–Jacobi–Bellman system of equations. For the partial information problem, we first transform it to a completely observable model by virtue of the filtering theory, then derive the equilibrium strategy and equilibrium value function by using the methods similar to those for the complete information problem. In addition, we illustrate the equilibrium reinsurance-investment strategies by numerical examples and discuss the impacts of model parameters on the equilibrium reinsurance-investment strategies for both the complete information and partial information cases.

Appendices

Appendix A: The proof of Theorem 2

Before giving the proof of Theorem 2, we first provide Proposition 1 and Lemma 1 as follows.

Proposition 1

The extended HJB system of Eq. (3.4) can be simplified as follows

$$\begin{aligned}&(V_{c})_{t}+ \mathop {\sup }_{\pi \in {\varPi }_{c}}\bigg \{\left[ rx+(\mu -r)u+c_{1}q_{1}+c_{2}q_{2}+c_{3}\right] (V_{c})_{x}+\kappa (\delta -\mu )(V_{c})_{\mu } +\frac{1}{2}\sigma _{1}^{2} u^{2}(V_{c})_{xx}\nonumber \\&\qquad +\frac{1}{2}(\sigma _{12}^{2}+\sigma _{2}^{2})(V_{c})_{\mu \mu }+\sigma _{1}\sigma _{12}u (V_{c})_{x\mu }-\frac{\gamma }{2}\sigma _{1}^{2} u^{2}[(g_{c})_{x}]^{2}-\frac{\gamma }{2}(\sigma _{12}^{2}+\sigma _{2}^{2})[(g_{c})_{\mu }]^{2}\nonumber \\&\qquad -\gamma \sigma _{1}\sigma _{12}u(g_{c})_{x}(g_{c})_{\mu }\nonumber \\&\qquad +\lambda _{1}{\mathbb {E}}\left[ V_{c}(t,x-q_{1}L,\mu )-V_{c}(t,x,\mu )-\frac{\gamma }{2}(g_{c}(t,x-q_{1}L,\mu )-g_{c}(t,x,\mu ) )^{2} \right] \nonumber \\&\qquad +\lambda _{2}{\mathbb {E}}\left[ V_{c}(t,x-q_{2}Y,\mu )-V_{c}(t,x,\mu )-\frac{\gamma }{2}(g_{c}(t,x-q_{2}Y,\mu )-g_{c}(t,x,\mu ) )^{2} \right] \nonumber \\&\qquad +\lambda {\mathbb {E}}\left[ V_{c}(t,x-q_{1}L-q_{2}Y,\mu )-V_{c}(t,x,\mu )-\frac{\gamma }{2}(g_{c}(t,x-q_{1}L-q_{2}Y,\mu )-g_{c}(t,x,\mu ) )^{2} \right] \bigg \} \nonumber \\&\quad =0, \nonumber \\&V_{c}(T,x,\mu )=x, \end{aligned}$$

(7.1)

$$\begin{aligned}&(g_{c})_{t} + \left[ rx+(\mu -r)u^{*}+c_{1}q_{1}^{*}+c_{2}q_{2}^{*}+c_{3}\right] (g_{c})_{x}+\kappa (\delta -\mu )(g_{c})_{\mu }\nonumber \\&\qquad +\frac{1}{2}\sigma _{1}^{2} {u^{*}}^{2}(g_{c})_{xx}+\frac{1}{2}(\sigma _{12}^{2}+\sigma _{2}^{2})(g_{c})_{\mu \mu } \nonumber \\&\qquad +\sigma _{1}\sigma _{12}u^{*} (g_{c})_{x\mu }+\lambda _{1}{\mathbb {E}}\left[ g_{c}(t,x-q_{1}^{*}L,\mu )-g_{c}(t,x,\mu )\right] \nonumber \\&\qquad +\lambda _{2}{\mathbb {E}}\left[ g_{c}(t,x-q_{2}^{*}Y,\mu )-g_{c}(t,x,\mu ) \right] \nonumber \\&\qquad +\lambda {\mathbb {E}}\left[ g_{c}(t,x-q_{1}^{*}L-q_{2}^{*}Y,\mu )-g_{c}(t,x,\mu ) \right] =0, \nonumber \\&g_{c}(T,x,\mu )=x. \end{aligned}$$

(7.2)

Proof

According to the model assumptions, after a straightforward but tedious calculation, we find that the extended HJB system of Eq. (3.4) has the simplified form (7.1–7.2). We omit the detailed calculation. $\square$

Lemma 1

For two-dimensional function $f(x,y)={\tilde{a}}x^{2}+{\tilde{b}}y^{2}+{\tilde{c}}x+{\tilde{d}}y+{\tilde{e}}xy$, if ${\tilde{a}}>0$ and $4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}>0$, f(x, y) attains its minimum value $f(x^{*},y^{*})=-\frac{{\tilde{b}}{\tilde{c}}^{2}+{\tilde{a}}{\tilde{d}}^{2}-{\tilde{c}}{\tilde{d}}{\tilde{e}}}{4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}}$ at $(x^{*},y^{*})=\left( \frac{{\tilde{e}}{\tilde{d}}-2{\tilde{b}}{\tilde{c}}}{4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}}, \frac{{\tilde{e}}{\tilde{c}}-2{\tilde{a}}{\tilde{d}}}{4{\tilde{a}}{\tilde{b}}-{\tilde{e}}^{2}}\right) .$

Proof

The lemma follows the standard solutions for an extreme value problem of a two-dimensional function. The detailed arguments are omitted. $\square$

The proof of Theorem 2

Since the wealth process has the linear structure, and in accordance with the forms of the boundary conditions, we conjecture that

$$\begin{aligned}&V_{c}(t,x,\mu )=A(t)x+B(t)\mu ^{2}+C(t)\mu +D(t),\\&g_{c}(t,x,\mu )=a(t)x+b(t)\mu ^{2}+c(t)\mu +d(t). \end{aligned}$$

Then, we have

$$\begin{aligned}&(V_{c})_{t}=A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t},\\&(V_{c})_{x}=A,\ (V_{c})_{xx}=0,\ (V_{c})_{\mu }=2B\mu +C, \ (V_{c})_{\mu \mu }=2B, \ (V_{c})_{x\mu }=0, \end{aligned}$$

and

$$\begin{aligned}&(g_{c})_{t}=a_{t}x+b_{t}\mu ^{2}+c_{t}\mu +d_{t},\\&(g_{c})_{x}=a, \ (g_{c})_{xx}=0,\ (g_{c})_{\mu }=2b\mu +c, \ (g_{c})_{\mu \mu }=2b, \ (g_{c})_{x\mu }=0. \end{aligned}$$

Substituting the above expressions into (7.1) and (7.2), we have

$$\begin{aligned}&A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t}+ \mathop {\sup }_{\pi \in {\varPi }_{c}}\bigg \{\left[ rx+(\mu -r)u+c_{1}q_{1}+c_{2}q_{2}+c_{3}\right] A+\kappa (\delta -\mu )(2B\mu +C) \nonumber \\&\quad +(\sigma _{12}^{2}+\sigma _{2}^{2})B-\frac{\gamma }{2}\sigma _{1}^{2} a^{2}u^{2}-\frac{\gamma }{2}(\sigma _{12}^{2}+\sigma _{2}^{2})(2b\mu +c)^{2}-\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)u\nonumber \\&\quad +\lambda _{1}\left( -A q_{1}\mu _{1L}-\frac{\gamma }{2}a^{2}q_{1}^{2}\mu _{2L} \right) +\lambda _{2}\left( -A q_{2}\mu _{1Y}-\frac{\gamma }{2}a^{2}q_{2}^{2}\mu _{2Y} \right) \nonumber \\&\quad +\lambda \left[ -A q_{1}\mu _{1L}-A q_{2}\mu _{1Y}-\frac{\gamma }{2}a^{2}(q_{1}^{2}\mu _{2L}+q_{2}^{2}\mu _{2Y}+2q_{1}q_{2}\mu _{1L}\mu _{1Y}) \right] \bigg \}=0, \nonumber \\&A(T)=1,~B(T)=C(T)=D(T)=0, \end{aligned}$$

(7.3)

and

$$\begin{aligned}&a_{t}x+b_{t}\mu ^{2}+c_{t}\mu +d_{t}+ \left[ rx+(\mu -r)u^{*}+c_{1}q_{1}^{*}+c_{2}q_{2}^{*}+c_{3}\right] a+\kappa (\delta -\mu )(2b\mu +c), \nonumber \\&\quad +(\sigma _{12}^{2}+\sigma _{2}^{2})b+\lambda _{1}(-aq_{1}^{*}\mu _{1L})+\lambda _{2}(-aq_{2}^{*}\mu _{1Y})+\lambda (-aq_{1}^{*}\mu _{1L}-aq_{2}^{*}\mu _{1Y})=0, \nonumber \\&a(T)=1,~b(T)=c(T)=d(T)=0. \end{aligned}$$

(7.4)

Rearrange (7.3), we have

$$\begin{aligned}&A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t}+ \mathop {\sup }_{\pi \in {\varPi }_{c}}\bigg \{-\frac{\gamma }{2}\sigma _{1}^{2} a^{2}u^{2}+\left[ (\mu -r)A-\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)\right] u -f(q_{1},q_{2}) \nonumber \\&\quad +\left[ rx+c_{3}\right] A+\kappa (\delta -\mu )(2B\mu +C)+(\sigma _{12}^{2}+\sigma _{2}^{2})B-\frac{\gamma }{2}(\sigma _{12}^{2} +\sigma _{2}^{2})(2b\mu +c)^{2} \bigg \}=0, \end{aligned}$$

(7.5)

where

$$\begin{aligned} f(q_{1},q_{2}) =&\frac{\gamma }{2}(\lambda +\lambda _{1})a^{2}\mu _{2L}q_{1}^{2}+\frac{\gamma }{2}(\lambda +\lambda _{2})a^{2}\mu _{2Y}q_{2}^{2} -\eta _{1}(\lambda +\lambda _{1})\mu _{1L}Aq_{1}\\&-\eta _{2}(\lambda +\lambda _{2})\mu _{1Y}Aq_{2}+\lambda \gamma a^{2} \mu _{1L}\mu _{1Y}q_{1}q_{2}. \end{aligned}$$

From (7.5), we have

$$\begin{aligned} u^{*}&=\frac{(\mu -r)A-\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)}{\gamma \sigma _{1}^{2}a^{2}}. \end{aligned}$$

(7.6)

Next we look for the equilibrium reinsurance strategy and the corresponding equilibrium value function. We firstly give the detailed discussion for Case 2.

For Case 2, using Lemma 1, we obtain the equilibrium reinsurance strategy

$$\begin{aligned}&q_{1}^{*}=\frac{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\eta _{1}\mu _{1L}\mu _{2Y}-\lambda (\lambda +\lambda _{2})\eta _{2}\mu _{1L}\mu _{1Y}^{2}}{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\mu _{2L}\mu _{2Y}-\lambda ^{2}\mu _{1L}^{2}\mu _{1Y}^{2}}\frac{A}{\gamma a^{2} }=m_{1} \frac{A}{\gamma a^{2} }, \end{aligned}$$

(7.7)

$$\begin{aligned}&q_{2}^{*}=\frac{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\eta _{2}\mu _{1Y}\mu _{2L}-\lambda (\lambda +\lambda _{1})\eta _{1}\mu _{1L}^{2}\mu _{1Y}}{(\lambda +\lambda _{1})(\lambda +\lambda _{2})\mu _{2L}\mu _{2Y}-\lambda ^{2}\mu _{1L}^{2}\mu _{1Y}^{2}}\frac{A}{\gamma a^{2} }=m_{2} \frac{A}{\gamma a^{2} }, \end{aligned}$$

(7.8)

and

$$\begin{aligned} f(q_{1}^{*},q_{2}^{*})= -\frac{A^{2}}{2\gamma a^{2} }\frac{a_{1}^{2}b_{2}^{2}\eta _{1}^{2}+a_{2}^{2}b_{1}^{2}\eta _{2}^{2}-2{\bar{\rho }} a_{1}a_{2} b_{1}b_{2}\eta _{1}\eta _{2}}{b_{1}^{2}b_{2}^{2}(1-{\bar{\rho }}^{2})} =-\frac{A^{2}}{2\gamma a^{2} }{\tilde{A}}_{2} \end{aligned}$$

with ${\tilde{A}}_{2}$ given by (3.9).

Then we look for the equilibrium value function. Inserting $u^{*}$, $q_{1}^{*}$ and $q_{2}^{*}$ into (7.5) yields

$$\begin{aligned}&A_{t}x+B_{t}\mu ^{2}+C_{t}\mu +D_{t}+ \frac{1}{2\gamma a^{2}}\frac{\left[ \mu (A-2\gamma \sigma _{1}\sigma _{12}ab)-(\gamma \sigma _{1}\sigma _{12}ac+rA)\right] ^{2}}{\sigma _{1}^{2} }+\frac{A^{2}}{2\gamma a^{2} }{\tilde{A}}_{2} \\&\quad +\left[ rx+c_{3}\right] A+\kappa (\delta -\mu )(2B\mu +C)+(\sigma _{12}^{2}+\sigma _{2}^{2})B-\frac{\gamma }{2}(\sigma _{12}^{2} +\sigma _{2}^{2})(2b\mu +c)^{2} =0. \end{aligned}$$

By separating variables, we obtain the following ordinary differential equations

$$\begin{aligned}&\left\{ \begin{aligned}&A_{t}+rA=0,\\&A(T)=1; \end{aligned}\right. \\&\left\{ \begin{aligned}&B_{t}-2\kappa B-2\gamma (\sigma _{12}^{2}+\sigma _{2}^{2}) b^{2}+\frac{1}{2\gamma a^{2}}\frac{(A-2\gamma \sigma _{1}\sigma _{12}ab)^{2}}{\sigma _{1}^{2} }=0,\\&B(T)=0; \end{aligned}\right. \\&\left\{ \begin{aligned}&C_{t}-\kappa C+2\kappa \delta B-2\gamma (\sigma _{12}^{2}+\sigma _{2}^{2}) bc-\frac{1}{\gamma a^{2}}\frac{(A-2\gamma \sigma _{1}\sigma _{12}ab)(\gamma \sigma _{1}\sigma _{12}ac+rA)}{\sigma _{1}^{2} }=0,\\&C(T)=0; \end{aligned}\right. \\&\left\{ \begin{aligned}&D_{t}+c_{3}A+(\sigma _{12}^{2}+\sigma _{2}^{2})B+\kappa \delta C-\frac{\gamma }{2}(\sigma _{12}^{2} +\sigma _{2}^{2})c^{2}+\frac{1}{2\gamma a^{2} }\frac{(\gamma \sigma _{1}\sigma _{12}ac+rA)^{2}}{\sigma _{1}^{2}}+\frac{A^{2}}{2\gamma a^{2} }{\tilde{A}}_{2}=0,\\&D(T)=0. \end{aligned}\right. \end{aligned}$$

Then we have $A(t)=e^{r(T-t)}.$ Substituting the expression of $u^{*}$, $q_{1}^{*}$ and $q_{2}^{*}$ given by (7.6), (7.7) and (7.8) into (7.4) yields

$$\begin{aligned}&a_{t}x\!+\!b_{t}\mu ^{2}\!+\!c_{t}\mu \!+\!d_{t}\!+ \! \left[ rx\!+\!\frac{(\mu -r)^{2}A\!-\!\gamma \sigma _{1}\sigma _{12}a(2b\mu +c)(\mu -r)}{\gamma \sigma _{1}^{2}a^{2}}\!+\!c_{1}m_{1} \frac{A}{\gamma a^{2} }\!+\!c_{2}m_{2} \frac{A}{\gamma a^{2} }\!+\!c_{3}\!\right] \!a\\&\quad +\kappa (\delta -\mu )(2b\mu +c)+(\sigma _{12}^{2}+\sigma _{2}^{2})b-\frac{A}{\gamma a}(\lambda _{1}m_{1}\mu _{1L}+\lambda _{2}m_{2}\mu _{1Y})-\frac{A}{\gamma a}\lambda (m_{1}\mu _{1L}+m_{2}\mu _{1Y})=0. \end{aligned}$$

By separating variables again, we obtain

$$\begin{aligned}&a_{t}+ra=0,~a(T)=1;\\&b_{t}-2\kappa b+\frac{ A-2\gamma \sigma _{1}\sigma _{12}ab}{\gamma \sigma _{1}^{2}a}=0,~b(T)=0;\\&c_{t}-\left( \kappa +\frac{\sigma _{12}}{\sigma _{1}}\right) c+2\left( \kappa \delta +\frac{r\sigma _{12}}{\sigma _{1}}\right) b-2\frac{rA}{\gamma \sigma _{1}^{2}a}=0,~c(T)=0;\\&d_{t}\!+\!c_{3}a\!+\!(\!\sigma _{12}^{2}+\sigma _{2}^{2})b\!+\!\left( \!\kappa \delta \!+\!\frac{r\sigma _{12}}{\sigma _{1}}\right) \!c\!+\!\frac{A}{a}\!\bigg [\!\frac{r^{2}}{\gamma \sigma _{1}^{2}}\!+\!\frac{c_{1}m_{1}\!+\!c_{2}m_{2}\!-\!(\!\lambda \!+\!\lambda _{1}\!)m_{1}\mu _{1L}\!-\!(\!\lambda \!+\!\lambda _{2})m_{2}\mu _{1Y}}{\gamma } \!\bigg ]\!=\!0,\!\\&\quad d(T)=0. \end{aligned}$$

Then, we have $a(t)=e^{r(T-t)},$ b(t) and c(t) are given by (3.13) and (3.14), respectively, and

$$\begin{aligned} d(t)&=\int _{t}^{T}\bigg [ c_{3}a(\tau )+(\sigma _{12}^{2}+\sigma _{2}^{2})b(\tau )+\left( \kappa \delta +\frac{r\sigma _{12}}{\sigma _{1}}\right) c(\tau )\\&\quad +\frac{r^{2}}{\gamma \sigma _{1}^{2}}+\frac{ m_{1}(\lambda +\lambda _{1})\eta _{1}\mu _{1L}+m_{2}(\lambda +\lambda _{2})\eta _{2}\mu _{1Y} }{\gamma } \bigg ]d\tau . \end{aligned}$$

Moreover, we can obtain the expressions of B(t), C(t) and $D_{2}(t)$ as (3.16)–(3.18). Note that in Case 2, we use $D_{2}(t)$ to denote D(t). Recall that in Case 2, $m_{1}>0,~m_{2}>0$, so the equivalent reinsurance strategy $q_{1}^{*},~q_{2}^{*}$ we have obtained in (7.7) and (7.8) satisfy the nonnegative constraint for the reinsurance strategy. The results in Case 1 and Case 3 can be derived similarly. This completes the proof. $\square$

Appendix B: The proof of Theorem 2

Proposition 2

The extended HJB system of Eq. (3.4) with ${\varPi }_{c}$, $V_{c}$ and $g_{c}$ replaced by ${\varPi }$, V and g, respectively, can be simplified as follows

$$\begin{aligned}&V_{t}+ \mathop {\sup }_{\pi \in {\varPi }}\bigg \{\left[ rx+({\widehat{\mu }}-r)u+c_{1}q_{1}+c_{2}q_{2}+c_{3}\right] V_{x}+\kappa (\delta -{\widehat{\mu }})V_{{\widehat{\mu }}} +\frac{1}{2}\sigma _{1}^{2} u^{2}V_{xx}\\&\quad +\frac{1}{2}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ^{2}V_{{\widehat{\mu }}{\widehat{\mu }}}\\&\quad +\sigma _{1}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) u V_{x{\widehat{\mu }}}-\frac{\gamma }{2}\sigma _{1}^{2} u^{2}g_{x}^{2}-\frac{\gamma }{2}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ^{2}g_{{\widehat{\mu }}}^{2}-\gamma \sigma _{1}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ug_{x}g_{{\widehat{\mu }}}\\&\quad +\lambda _{1}{\mathbb {E}}\left[ V(t,x-q_{1}L,{\widehat{\mu }})-V(t,x,{\widehat{\mu }})-\frac{\gamma }{2}(g(t,x-q_{1}L,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) )^{2} \right] \\&\quad +\lambda _{2}{\mathbb {E}}\left[ V(t,x-q_{2}Y,{\widehat{\mu }})-V(t,x,{\widehat{\mu }})-\frac{\gamma }{2}(g(t,x-q_{2}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) )^{2} \right] \\&\quad +\lambda {\mathbb {E}}\left[ V(t,x-q_{1}L-q_{2}Y,{\widehat{\mu }})-V(t,x,{\widehat{\mu }})-\frac{\gamma }{2}(g(t,x-q_{1}L-q_{2}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) )^{2} \right] \bigg \}=0, \\&V(T,x,{\widehat{\mu }})=x,\\&g_{t}+ \left[ rx+({\widehat{\mu }}-r){\widehat{u}}^{*}+c_{1}{\widehat{q}}_{1}^{*}+c_{2}{\widehat{q}}_{2}^{*}+c_{3}\right] g_{x}+\kappa (\delta -{\widehat{\mu }})g_{{\widehat{\mu }}} +\frac{1}{2}\sigma _{1}^{2} ({\widehat{u}}^{*})^{2}g_{xx}+\frac{1}{2}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) ^{2}g_{{\widehat{\mu }}{\widehat{\mu }}} \\&\quad +\sigma _{1}\left( \frac{{\varSigma }}{\sigma _{1}}+\sigma _{12}\right) {\widehat{u}}^{*} g_{x{\widehat{\mu }}}+\lambda _{1}{\mathbb {E}}\left[ g(t,x-{\widehat{q}}_{1}^{*}L,{\widehat{\mu }})-g(t,x,{\widehat{\mu }})\right] \\&\quad +\lambda _{2}{\mathbb {E}}\left[ g(t,x-{\widehat{q}}_{2}^{*}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) \right] \\&\quad +\lambda {\mathbb {E}}\left[ g(t,x-{\widehat{q}}_{1}^{*}L-{\widehat{q}}_{2}^{*}Y,{\widehat{\mu }})-g(t,x,{\widehat{\mu }}) \right] =0, \\&g(T,x,{\widehat{\mu }})=x. \end{aligned}$$

Proof

The proof of this proposition is similar to that of Proposition 1 and thus is omitted. $\square$