Portfolio Selection and Risk Control for an Insurer With Uncertain Time Horizon and Partial Information in an Anticipating Environment

Abstract

This paper is devoted to the study of an optimal investment and risk control problem for an insurer. The risky asset process and the insurance liability process are governed by stochastic differential equations with jumps and anticipative parameters. The insurer can only get access to partial information about the financial market and the insurance business to make decisions. Taking into account endogenous and exogenous factors, we assume the time horizon is uncertain. With the aim of expected logarithmic utility maximization, we adopt the forward stochastic calculus and the Malliavin calculus to derive a characterization of the optimal strategy. In some particular cases, we obtain the optimal strategies in closed-form and get some new insights.

Proof of Theorem 3.2

Let $(\pi (s), \kappa (s))$ be an admissible strategy. By conditions (1) and (2) in Definition 2.1, we have

$$\begin{aligned} D_t^1\pi (s)=D_t^1\kappa (s)=D_t^2\kappa (s)=0, \end{aligned}$$

$$\begin{aligned} D_{t,z}^1\pi (s)=D_{t,z}^2\kappa (s)=0, \end{aligned}$$

for all $t>s$. By the product rule and the chain rule for the Malliavin derivatives, we have

$$\begin{aligned} D_{s+}^1\left( \sigma (s)\pi (s)-q_1(s)\kappa (s)\right) =\pi (s)D_{s+}^1\sigma (s)-\kappa (s)D_{s+}^1q_1(s), \end{aligned}$$

$$\begin{aligned} D_{s+}^2\left( q_2(s)\kappa (s)\right) = \kappa (s)D_{s+}^2q_2(s), \end{aligned}$$

$$\begin{aligned}&D_{s+,z}^1\log (1+\pi (s)\gamma _1(s,z)) \nonumber \\&=\log \left( 1+\pi (s)\gamma _1(s,z)+D_{s+,z}^1(\pi (s)\gamma _1(s,z))\right) -\log (1+\pi (s)\gamma _1(s,z)) \nonumber \\&=\log \left( 1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z)\right) -\log (1+\pi (s)\gamma _1(s,z)), \end{aligned}$$

and

$$\begin{aligned}&D_{s+,z}^2\log (1-\kappa (s)\gamma _2(s,z)) \nonumber \\&=\log \left( 1-\kappa (s)\gamma _2(s,z)-D_{s+,z}^2(\kappa (s)\gamma _2(s,z))\right) -\log \left( 1-\kappa (s)\gamma _2(s,z)\right) \nonumber \\&=\log \left( 1-\kappa (s)\gamma _2(s,z)-\kappa (s)D_{s+,z}^2\gamma _2(s,z)\right) -\log \left( 1-\kappa (s)\gamma _2(s,z)\right) . \end{aligned}$$

Then by conditions (5) and (6) in Definition 2.1, Lemma 3.1, the Fubini theorem and the equalities above, we can deduce that

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)\int _0^t(\sigma (s)\pi (s)-q_1(s)\kappa (s))d^-W_s^1dt\bigg ]\\&=\int _0^TE\bigg [\int _0^tD_s^1\delta (t)(\sigma (s)\pi (s)-q_1(s)\kappa (s))ds +\delta (t)\int _0^t\big (\pi (s)D_{s+}^1\sigma (s)-\kappa (s)D_{s+}^1q_1(s)\big )ds\bigg ]dt\\&=E\bigg [\int _0^T\bigg (\int _s^TD_s^1\delta (t)dt\cdot (\sigma (s)\pi (s)-q_1(s)\kappa (s))+\int _s^T\delta (t)dt\cdot \left( \pi (s)D_{s+}^1\sigma (s)-\kappa (s)D_{s+}^1q_1(s)\right) \bigg )ds\bigg ]\\&=E\bigg [\int _0^T\bigg [\Big (\sigma (s)\int _s^TD_s^1\delta (t)dt+D_{s+}^1\sigma (s)\cdot \int _s^T\delta (t)dt\Big )\pi (s)\\&\quad -\Big (q_1(s)\int _s^TD_s^1\delta (t)dt+D_{s+}^1q_1(s)\cdot \int _s^T\delta (t)dt\Big )\kappa (s)\bigg ]ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\int _0^T \delta (t)\int _0^tq_2(s)\kappa (s)d^-W_s^2dt\bigg ]\\&=\int _0^TE\bigg [\int _0^tD_s^2\delta (t)q_2(s)\kappa (s)ds+\delta (t)\int _0^t\kappa (s)D_{s+}^2q_2(s)ds\bigg ]dt\\&=E\bigg [\int _0^T\bigg (q_2(s)\int _s^TD_s^2\delta (t)dt+D_{s+}^2q_2(s)\int _s^T\delta (t)dt\bigg )\kappa (s)ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\log (1+\pi (s)\gamma _1(s,z))\tilde{N}^1(d^-s,dz)dt\bigg ]\\&=\int _0^TE\bigg [\int _0^t\int _{\mathbb R_0}D_{s,z}^1\delta (t)\Big (\log (1+\pi (s)\gamma _1(s,z))+D_{s+,z}^1\log (1+\pi (s)\gamma _1(s,z))\Big )v_1(dz)ds\\&\quad +\delta (t)\int _0^t\int _{\mathbb R_0}D_{s+,z}^1\log (1+\pi (s)\gamma _1(s,z))v_1(dz)ds\bigg ]dt\\&=\int _0^TE\bigg [\int _0^t\int _{\mathbb R_0}D_{s,z}^1\delta (t)\cdot \log (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z))v_1(dz)ds\\&\quad +\delta (t)\int _0^t\int _{\mathbb R_0}\Big (\log (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z))-\log (1+\pi (s)\gamma _1(s,z))\Big )v_1(dz)ds\bigg ]dt\\&=E\bigg [\int _0^T\int _{\mathbb R_0}\bigg (\int _s^TD_{s,z}^1\delta (t)dt+\int _s^T\delta (t)dt\bigg )\log (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z)) v_1(dz)ds\bigg ]\\&\quad -E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\log (1+\pi (s)\gamma _1(s,z))v_1(dz)dsdt\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\log (1-\kappa (s)\gamma _2(s,z))\tilde{N}^2(d^-s,dz)dt\bigg ]\\&=E\bigg [\int _0^T\int _{\mathbb R_0}\bigg (\int _s^TD_{s,z}^2\delta (t)dt+\int _s^T\delta (t)dt\bigg ) \log \big (1-\kappa (s)\gamma _2(s,z)-\kappa (s)D_{s+,z}^2\gamma _2(s,z)\big )v_2(dz)ds\bigg ]\\&\quad -E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\log (1-\kappa (s)\gamma _2(s,z))v_2(dz)dsdt\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\bar{F}(T)\int _0^T(\sigma (s)\pi (s)-q_1(s)\kappa (s))d^-W_s^1\bigg ]\\&=E\bigg [\int _0^TD_s^1\bar{F}(T)\cdot (\sigma (s)\pi (s)-q_1(s)\kappa (s))ds\bigg ] +E\bigg [\bar{F}(T)\int _0^T\Big (\pi (s)D_{s+}^1\sigma (s)-\kappa (s)D_{s+}^1q_1(s)\Big )ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\bar{F}(T)\int _0^Tq_2(s)\kappa (s)d^-W_s^2\bigg ]\\&=E\bigg [\int _0^TD_s^2\bar{F}(T)q_2(s)\kappa (s)ds\bigg ]+E\bigg [\bar{F}(T)\int _0^T\kappa (s)D_{s+}^2q_2(s)ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\bar{F}(T)\int _0^T\int _{\mathbb R_0}\log (1+\pi (s)\gamma _1(s,z))\tilde{N}^1(d^-s,dz)\bigg ]\\&=E\bigg [\int _0^T\int _{\mathbb R_0}D_{s,z}^1\bar{F}(T)\Big (\log (1+\pi (s)\gamma _1(s,z))+D_{s+,z}^1\log (1+\pi (s)\gamma _1(s,z))\Big )v_1(dz)ds\\&\quad +\bar{F}(T)\int _0^T\int _{\mathbb R_0}D_{s+,z}^1\log (1+\pi (s)\gamma _1(s,z))v_1(dz)ds\bigg ]\\&=E\bigg [\int _0^T\int _{\mathbb R_0}D_{s,z}^1\bar{F}(T)\cdot \log (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z))v_1(dz)ds\\&\quad +\bar{F}(T)\int _0^T\int _{\mathbb R_0}\Big (\log (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z))-\log (1+\pi (s)\gamma _1(s,z))\Big )v_1(dz)ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\bar{F}(T)\int _0^T\int _{\mathbb R_0}\log (1-\kappa (s)\gamma _2(s,z))\tilde{N}^2(d^-s,dz)\bigg ]\\&=E\bigg [\int _0^T\int _{\mathbb R_0}D_{s,z}^2\bar{F}(T)\cdot \log (1-\kappa (s)\gamma _2(s,z)-\kappa (s)D_{s+,z}^2\gamma _2(s,z))v_2(dz)ds\\&\quad +\bar{F}(T)\int _0^T\int _{\mathbb R_0}\Big (\log (1-\kappa (s)\gamma _2(s,z)-\kappa (s)D_{s+,z}^2\gamma _2(s,z)) -\log (1-\kappa (s)\gamma _2(s,z))\Big )v_2(dz)ds\bigg ]. \end{aligned}$$

Moreover, by conditions (5) and (6) in Definition 2.1 and the Fubini theorem, we have

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)\int _0^t\Big [(\mu (s)-r(s))\pi (s)+(\lambda (s)-p(s))\kappa (s)\\&-\frac{1}{2}(\sigma (s)\pi (s)-q_1(s)\kappa (s))^2 -\frac{1}{2}q_2^2(s)\kappa ^2(s)\Big ]dsdt\bigg ]\\&=E\bigg [\int _0^T\int _s^T\delta (t)dt\cdot \Big [(\mu (s)-r(s))\pi (s)+(\lambda (s)-p(s))\kappa (s)\\&\quad -\frac{1}{2}(\sigma (s)\pi (s)-q_1(s)\kappa (s))^2 -\frac{1}{2}q_2^2(s)\kappa ^2(s)\Big ]ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\big [\log (1+\pi (s)\gamma _1(s,z))-\pi (s)\gamma _1(s,z)\big ]v_1(dz)dsdt\bigg ]\\&=E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\log (1+\pi (s)\gamma _1(s,z))v_1(dz)dsdt\bigg ]\\&-E\bigg [\int _0^T\int _{\mathbb R_0}\int _s^T\delta (t)dt\cdot \pi (s)\gamma _1(s,z)v_1(dz)ds\bigg ], \end{aligned}$$

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\big [\log (1-\kappa (s)\gamma _2(s,z))+\kappa (s)\gamma _2(s,z)\big ]v_2(dz)dsdt\bigg ]\\&=E\bigg [\int _0^T\delta (t)\int _0^t\int _{\mathbb R_0}\log (1-\kappa (s)\gamma _2(s,z))v_2(dz)dsdt\bigg ]\\&-E\bigg [\int _0^T\int _{\mathbb R_0}\int _s^T\delta (t)dt\cdot \kappa (s)\gamma _2(s,z)v_2(dz)ds\bigg ]. \end{aligned}$$

From Eq. (1) and the equalities above, we obtain that

$$\begin{aligned}&E\bigg [\int _0^T\delta (t)J^u(t)dt+\bar{F}(T)J^u(T)\bigg ]\nonumber \\&=E\bigg [\int _0^T\bigg [\sigma (s)\Big (\int _s^TD_s^1\sigma (t)dt+D_s^1\bar{F}(T)\Big )\nonumber \\&\quad +\Big (\int _s^T\delta (t)dt+\bar{F}(T)\Big )\Big (\mu (s)-r(s)-\int _{\mathbb R_0}\gamma _1(s,z)v_1(dz)+D_{s+}^1\sigma (s)\Big )\bigg ]\pi (s)ds\bigg ]\nonumber \\&\quad +E\bigg [\int _0^T\bigg [-q_2(s)\Big (\int _s^TD_s^2\delta (t)dt+D_s^2\bar{F}(T)\Big )-q_1(s)\Big (\int _s^TD_s^1\delta (t)dt+D_s^1\bar{F}(T)\Big )\nonumber \\&\quad +\Big (\lambda (s)-p(s)+\int _{\mathbb R_0}\gamma _2(s,z)v_2(dz) -D_{s+}^2q_2(s)-D_{s+}^1q_1(s)\Big )\Big (\int _s^T\delta (t)dt+\bar{F}(T)\Big )\bigg ]\kappa (s)ds\bigg ]\nonumber \\&\quad +E\bigg [\int _0^T-\frac{1}{2}\Big (\int _s^T\delta (t)dt+\bar{F}(T)\Big )\Big [(\sigma (s)\pi (s)-q_1(s)\kappa (s))^2+q_2^2(s)\kappa ^2(s)\Big ]ds\bigg ]\nonumber \\&\quad +E\bigg [\int _0^T\int _{\mathbb R_0}\Big (\int _s^TD_{s,z}^1\delta (t)dt+\int _s^T\delta (t)dt +D_{s,z}^1\bar{F}(T)+\bar{F}(T)\Big )\nonumber \\&\quad \times \log \big (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z)\big )v_1(dz)ds\bigg ]\nonumber \\&\quad +E\bigg [\int _0^T\int _{\mathbb R_0}\Big (\int _s^TD_{s,z}^2\delta (t)dt+\int _s^T\delta (t)dt +D_{s,z}^2\bar{F}(T)+\bar{F}(T)\Big )\nonumber \\&\quad \times \log \big (1-\kappa (s)\gamma _2(s,z)-\kappa (s)D_{s+,z}^2\gamma _2(s,z)\big )v_2(dz)ds\bigg ]\nonumber \\&=E\bigg [\int _0^TC_1(s)\pi (s)ds\bigg ]+E\bigg [\int _0^TC_2(s)\kappa (s)ds\bigg ]\nonumber \\&\quad -\frac{1}{2}E\bigg [\int _0^T\bar{F}(s)\Big [(\sigma (s)\pi (s)-q_1(s)\kappa (s))^2+q_2^2(s)\kappa ^2(s)\Big ]ds\bigg ]\nonumber \\&\quad +E\bigg [\int _0^T\int _{\mathbb R_0}B_1(s,z)\log \big (1+\pi (s)\gamma _1(s,z)+\pi (s)D_{s+,z}^1\gamma _1(s,z)\big )v_1(dz)ds\bigg ]\nonumber \\&\quad +E\bigg [\int _0^T\int _{\mathbb R_0}B_2(s,z)\log \big (1-\kappa (s)\gamma _2(s,z)-\kappa (s)D_{s+,z}^2\gamma _2(s,z)\big )v_2(dz)ds\bigg ]. \end{aligned}$$