Optimal mean–variance investment/reinsurance with common shock in a regime-switching market

Abstract

In this paper, we consider the problem of optimal investment-reinsurance with two dependent classes of insurance risks in a regime-switching financial market. In our model, the two claim-number processes are correlated through a common shock component, and the market mode is classified into a finite number of regimes. We also assume that the insurer can purchase proportional reinsurance and invest its surplus in a financial market, and that the values of the model parameters depend on the market mode. Using the techniques of stochastic linear-quadratic control, under the mean–variance criterion, we obtain analytic expressions for the optimal investment and reinsurance strategies, and derive closed-form expressions for the efficient strategies and the efficient frontiers which are based on the solutions to some systems of linear ordinary differential equations. Finally, we carry out a numerical study for illustration purpose.

Appendices

Appendix A: The proof of Theorem 4.1

Let $\pi$ be any strategy from ${\widehat{\Pi }}$ and satisfy (2.7). We first define a number of functions with the form

$$\begin{aligned} \varphi (t, x, i)=P(t, i)[x+({\tilde{\lambda }}-z)H(t, i)]^2, \quad i=1,2,\ldots l. \end{aligned}$$

Then we have

$$\begin{aligned} \varphi _{t}(t,x,i)= & {} {\dot{P}}(t,i)[x+({\tilde{\lambda }}-z)H(t,i)]^2+2({\tilde{\lambda }}-z)P(t,i){\dot{H}}(t, i)[x+({\tilde{\lambda }}-z)H(t, i)], \\ \varphi _{x}(t,x,i)= & {} 2P(t,i)[x+({\tilde{\lambda }}-z)H(t,i)], \\ \varphi _{xx}(t,x,i)= & {} 2P(t,i). \end{aligned}$$

Applying the generalized Itô’s formula for Markov-modulated processes to $\varphi (t,X(t),\alpha (t))$ yields

$$\begin{aligned}&d\{P(t, \alpha (t))[X(t)+({\tilde{\lambda }}-z)H(t, \alpha (t))]^2\}\\&\quad = \bigg \{{\dot{P}}(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))]^2\\&\qquad + 2({\tilde{\lambda }}-z)P(t,\alpha (t-)){\dot{H}}(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))] \nonumber \\&\qquad + 2P(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))] \\&\qquad \times \Big [r_{0}(t,\alpha (t-))X(t)+A(t,\alpha (t-))q_{1}(t)+B(t,\alpha (t-))q_{2}(t)+C(t,\alpha (t-))\xi (t)\Big ]\\&\qquad + P(t,\alpha (t-)) \Big [\sigma ^{2}(t,\alpha (t-))\xi ^{2}(t)\\&\qquad +\sigma ^{2}_{1}(t,\alpha (t-))q_{1}^{2}(t)+\sigma ^{2}_{2}(t,\alpha (t-))q_{2}^{2}(t)\\&\qquad + 2\rho (t,\alpha (t-))\sigma _{1}(t,\alpha (t-))\sigma _{2}(t,\alpha (t-))q_{1}(t)q_{2}(t)\Big ]\\&\qquad + \sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t,j)[X(t)+({\tilde{\lambda }}-z)H(t,j)]^2\bigg \}dt \\&\qquad + \{\ldots \}dW(t)+\{\ldots \}dM(t). \end{aligned}$$

With tedious calculations, we have

$$\begin{aligned}&d\{P(t, \alpha (t))[X(t)+({\tilde{\lambda }}-z)H(t, \alpha (t))]^2\} \\&\quad = P(t,\alpha (t-))G(q_{1}(t),q_{2}(t))dt -P(t,\alpha (t-))E(t,\alpha (t-)){[}X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))]^{2}dt\\&\qquad + P(t,\alpha (t-))\sigma ^{2}(t,\alpha (t-))\big [\xi (t)-\xi ^{*}(t,X(t),\alpha (t-))\big ]^{2}dt\\&\qquad - \frac{C^{2}(t,\alpha (t-))}{\sigma ^{2}(t,\alpha (t-))}P(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))]^{2}dt\\&\qquad + X(t)^{2}\bigg [ {\dot{P}}(t, \alpha (t-))+2r_{0}(i)P(t,\alpha (t-))-\sum _{j=1}^l\delta _{\alpha (t-)j}P(t, j)\bigg ] dt\\&\qquad + X(t)\bigg \{2{\dot{P}}(t, \alpha (t-))({\tilde{\lambda }}-z)H(t,\alpha (t-))+2({\tilde{\lambda }}-z)P(t, \alpha (t-)){\dot{H}}(t,\alpha (t-)) \\&\qquad + 2P(t,\alpha (t-)) \Big [({\tilde{\lambda }}-z)r_{0}(t,\alpha (t-))H(t,\alpha (t-))+D(t,\alpha (t-))\Big ]\\&\qquad + 2({\tilde{\lambda }}-z)\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) H(t,j)\bigg \}dt \\&\qquad + \bigg \{{\dot{P}}(t, \alpha (t-)) ({\tilde{\lambda }}-z)^{2}H^{2}(t,\alpha (t-)) + 2({\tilde{\lambda }}-z)^{2}P(t,\alpha (t-)){\dot{H}}(t,\alpha (t-))H(t,\alpha (t-))\\&\qquad + 2P(t,\alpha (t-))({\tilde{\lambda }}-z)H(t,\alpha (t-))D(t) +({\tilde{\lambda }}-z)^{2}\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) H^{2}(t,j)\bigg \} dt \\&\qquad + \{\ldots \}dW(t)+\{\ldots \}dM(t). \end{aligned}$$

After some simple calculations and rearranging terms, we obtain

$$\begin{aligned}&d\{P(t, \alpha (t))[X(t)+({\tilde{\lambda }}-z)H(t, \alpha (t))]^2\} \\&\quad =P(t,\alpha (t-))G(q_{1}(t),q_{2}(t))dt \\&\qquad + P(t,\alpha (t-))\sigma ^{2}(t,\alpha (t-))\big [\xi (t)-\xi ^{*}(t,X(t),\alpha (t-))\big ]^{2} dt \\&\qquad + X(t)\bigg \{2{\dot{P}}(t, \alpha (t-))({\tilde{\lambda }}-z)H(t,\alpha (t-))+2({\tilde{\lambda }}-z)P(t, \alpha (t-)){\dot{H}}(t,\alpha (t-)) \\&\qquad + 2P(t, \alpha (t-))[({\tilde{\lambda }}-z)r_{0}(t,\alpha (t-))H(t,\alpha (t-))+D(t, \alpha (t-)] \\&\qquad + 2({\tilde{\lambda }}-z)\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) H(t,j)\bigg \}dt \\&\qquad + \bigg \{{\dot{P}}(t, \alpha (t-))({\tilde{\lambda }}-z)^{2}H^{2}(t,\alpha (t-))+2({\tilde{\lambda }}-z)^{2}P(t, \alpha (t-)){\dot{H}}(t,\alpha (t-))H(t,\alpha (t-)) \\&\qquad + 2P(t, \alpha (t-))({\tilde{\lambda }}-z)H(t,\alpha (t-))D(t, \alpha (t-) \\&\qquad + ({\tilde{\lambda }}-z)^{2}\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) H^{2}(t,j)\bigg \}dt \\&\qquad + \{\ldots \}dW(t)+\{\ldots \}dM(t) \end{aligned}$$

and then

$$\begin{aligned}&d\{P(t, \alpha (t))[X(t)+({\tilde{\lambda }}-z)H(t, \alpha (t))]^2\} \nonumber \\&\quad = P(t,\alpha (t-))G(q_{1}(t),q_{2}(t))dt \nonumber \\&\qquad + P(t,\alpha (t-))\sigma ^{2}(t,\alpha (t-))\big [\xi (t)-\xi ^{*}(t,X(t),\alpha (t-))\big ]^{2}dt\nonumber \\&\qquad +0\times X(t)^{2}dt+0\times X(t)dt \nonumber \\&\qquad + \bigg \{{\dot{P}}(t,\alpha (t-))({\tilde{\lambda }}-z)^{2}H^{2}(t,\alpha (t-)) + \ 2({\tilde{\lambda }}-z)^{2}\nonumber \\&\qquad P(t, \alpha (t-)){\dot{H}}(t,\alpha (t-))H(t,\alpha (t-)) \nonumber \\&\qquad + 2P(t, \alpha (t-))({\tilde{\lambda }}-z)H(t,\alpha (t-))D(t,\alpha (t-)) \nonumber \\&\qquad + ({\tilde{\lambda }}-z)^{2}\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) H^{2}(t,j)\bigg \}dt \nonumber \\&\qquad + \{\ldots \}dW(t)+\{\ldots \}dM(t) \nonumber \\&\quad = P(t,\alpha (t-))G(q_{1}(t),q_{2}(t))dt \nonumber \\&\qquad + P(t,\alpha (t-))\sigma ^{2}(t,\alpha (t-))\big [\xi (t)-\xi ^{*}(t,X(t),\alpha (t-))\big ]^{2} dt \nonumber \\&\qquad + \bigg \{({\tilde{\lambda }}-z)^{2}\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) \big [H(t,\alpha (t-))-H(t,j)\big ]^{2}\bigg \}dt \nonumber \\&\qquad + \{\ldots \}dW(t)+\{\ldots \}dM(t), \end{aligned}$$

(7.1)

where

$$\begin{aligned} G(q_{1}(t),q_{2}(t))= & {} \sigma ^{2}_{1}(t,\alpha (t-))q_{1}^{2}(t)+\sigma ^{2}_{2}(t,\alpha (t-))q_{2}^{2}(t)\\&+ 2A(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))]q_{1}(t)\\&+ 2B(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))]q_{2}(t)\\&+ 2\rho (t,\alpha (t-))\sigma _{1}(t,\alpha (t-))\sigma _{2}(t,\alpha (t-))q_{1}(t)q_{2}(t)\\&+ E(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))]^{2}, \end{aligned}$$

and M(t) is a zero expected value martingale. Note that Eqs. (4.1) and (4.2) are used in the derivation of (7.1). Integrating (7.1) from 0 to T and taking expectations, we get

$$\begin{aligned}&J(x_0, i_0, q_{1}(\cdot ),q_{2}(\cdot ),\xi , {\tilde{\lambda }})\\&\quad = {\mathbb {E}}[X(T)+{\tilde{\lambda }}-z]^2-{\tilde{\lambda }}^2 \\&\quad = {\mathbb {E}}\int _{0}^{T}P(t,\alpha (t-))G(q_{1}(t),q_{2}(t))dt\\&\qquad + \ {\mathbb {E}}\int _{0}^{T}P(t,\alpha (t-))\sigma ^{2}(t,\alpha (t-))\big [\xi (t)-\xi ^{*}(t,X(t),\alpha (t-))\big ]^{2}dt \\&\qquad + \ {\mathbb {E}}\int _{0}^{T} ({\tilde{\lambda }}-z)^{2}\left\{ \sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) \big [H(t,\alpha (t-))-H(t,j)\big ]^{2}\right\} dt \\&\qquad + \ P(0, i_0)[x_0+({\tilde{\lambda }}-z)H(0, i_0)]^{2}-{\tilde{\lambda }}^{2} \\&\quad \ge ({\tilde{\lambda }}-z)^{2}{\mathbb {E}}\int _{0}^{T} \left\{ \sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j) \big [H(t,\alpha (t-))-H(t,j)\big ]^{2}\right\} dt \\&\qquad + \ P(0, i_0)[x_0+({\tilde{\lambda }}-z)H(0, i_0)]^{2}-{\tilde{\lambda }}^{2} \\&\quad = \big [P(0, i_0)H^{2}(0, i_0)+\theta (i_{0})-1\big ]({\tilde{\lambda }}-z)^{2} \\&\qquad + \ 2\big [P(0, i_0)H(0,i_0)x_{0}-z\big ]({\tilde{\lambda }}-z) \\&\qquad + \ P(0, i_0)x_{0}^{2}-z^{2}, \end{aligned}$$

where $\theta (i_{0})$ is given in (4.9). Note that Lemma 4.1 is used in the above calculations, that is, $G(\cdot ,\cdot )$ attains its minimum value 0 at $(q_{1}^{*}(\cdot ),q_{2}^{*}(\cdot ))$. For the optimal strategy $\pi ^{*}(t,x,i)$ of (4.7), the inequality becomes an equality, and hence we obtain (4.8). $\square$

Appendix B: The proof of Proposition 4.1

Plugging the optimal feedback control (4.7) into the dynamic of X(t) in (2.7), applying the generalized Itô’s formula for Markov-modulated processes, and using the dynamics of P(t, i) and Q(t, i) in (4.1)-(4.2), we obtain

$$\begin{aligned}&d\{P(t, \alpha (t))[X(t)+({\tilde{\lambda }}-z)H(t, \alpha (t-))]\} \\&\quad =\bigg \{{\dot{P}}(t,\alpha (t-))[X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))] \\&\qquad + \ ({\tilde{\lambda }}-z)P(t,\alpha (t-)){\dot{H}}(t, \alpha (t-)) + P(t,\alpha (t-)) \\&\qquad \times \bigg [r_{0}(t,\alpha (t-))X(t) + \bigg (\frac{A(t,\alpha (t-))K_1(t,\alpha (t-)) + B(t,\alpha (t-)) K_2(t,\alpha (t-))}{\sigma _{1}^{2}(t,\alpha (t-))\sigma _{2}^{2}(t,\alpha (t-))(1-\rho ^{2}(t,\alpha (t-)))} \\&\qquad - \ \frac{C^{2}(t,\alpha (t-))}{\sigma ^{2}(t,\alpha (t-))}\bigg ) \times \bigg (X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))\bigg )+D(t,\alpha (t-))\bigg ]\biggr \}dt \\&\qquad + \sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t,j)[X(t)+({\tilde{\lambda }}-z)H(t,j)] dt\\&\qquad + \{\ldots \}dW(t)+\{\ldots \}dM(t) \\&\quad = \bigg \{ \bigg [-2r_{0}(t, \alpha (t-))P(t, \alpha (t-))-\sum _{j=1}^{l}\delta _{ij}P(t, j)\bigg ]\big [X(t)+ ({\tilde{\lambda }}-z)H(t,\alpha (t-))\big ]\\&\qquad + \ r_{0}(t, \alpha (t-))({\tilde{\lambda }}-z)P(t,\alpha (t-))H(t, \alpha (t-))-P(t,\alpha (t-))D(t, \alpha (t-))\\&\qquad - \ ({\tilde{\lambda }}-z)\sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t, j)\big [H(t, j)-H(t, \alpha (t-))\big ] \\&\qquad + \ P(t,\alpha (t-))r_{0}(t, \alpha (t-))X(t)+P(t,\alpha (t-))D(t, \alpha (t-))\biggr \}dt\\&\qquad + \ \sum _{j=1}^{l}\delta _{\alpha (t-)j}P(t,j)[X(t)+({\tilde{\lambda }}-z)H(t,j)] dt\\&\qquad + \ \{\ldots \}dW(t)+\{\ldots \}dM(t) \\&\quad =\biggr \{-r_{0}(t,\alpha (t-))P(t,\alpha (t-))\big [X(t)+({\tilde{\lambda }}-z)H(t,\alpha (t-))\big ]\biggr \}dt\\&\qquad + \ \{\ldots \}dW(t)+\{\ldots \}dM(t), \end{aligned}$$

where

$$\begin{aligned} K_1(t,\alpha (t-))= & {} B(t,\alpha (t-))\rho (t,\alpha (t-)) \sigma _{1}(t,\alpha (t-))\sigma _{2}(t,\alpha (t-))\\&-A(t,\alpha (t-))\sigma _{2}^{2}(t,\alpha (t-)), \\ K_2(t,\alpha (t-))= & {} A(t,\alpha (t-))\rho (t,\alpha (t-)) \sigma _{1}(t,\alpha (t-))\sigma _{2}(t,\alpha (t-))\\&-B(t,\alpha (t-))\sigma _{1}^{2}(t,\alpha (t-)). \end{aligned}$$

Then arguments similar to those in the proof of Theorem 4.2 in Chen and Yam (2013) can be used to reach the desired result. $\square$