Research on household investment consumption and life insurance purchase under partial information

Abstract

This paper studies how a couple in a family can reasonably allocate the family wealth to investment, consumption and life insurance in terms of of partial information, so as to maximize the family wealth. After constructing an appropriate mathematical model, the optimal strategy is mainly discussed before and after the death of the first family member, the corresponding value function is obtained, as well as the explicit solution of optimal strategy under the logarithmic utility function is given.

Keywords

Investment-consumption-life insurance optimal strategy partial information HJB equation logarithmic utility function

1. Introduction

With the rapid economic development of economy, the wealth level of each family has increased dramatically. In order to better manage assets and increase income, a family regularly invests its assets to achieve the purpose of asset appreciation and preservation. With the booming financial and insurance industries, the investment of household assets should consider not only the financial market, but also the insurance market. In recent years, life insurance investment has been accepted by more and more families because of their characteristics of small investment and high risk aversion. In recent years, especially life insurance investment. Then, how to rationally distribute family assets, invest, consume and buy life insurance has gradually become a problem faced by every family. This problem has gradually attracted the attention of numerous scholars. Some of them began to use mathematical tools to consider constructing mathematical models to address such problems. Yaari [1] was the first to systematically study this problem, whose research model mainly considered three optimization problems, such as personal consumption, asset investment and personal life insurance, and concluded that consumers were more inclined to preserve their liquidity wealth in the absence of bequest motivation. Subsequently, Hakansson [2] and Richard [3] et al. extended Yaari’s research, and carried out a systematic study on discrete time and continuous time cases, respectively. After that, Ye [4] defined the condition as uncertainty of life insurance in the research of such kind of problems, and proposed the corresponding optimal strategy. In regard with the problems of wealth distribution optimization, Mousa et al. [5] extended the original model of only one type of life insurance for policyholders to choose from to a variety of life insurance for them to choose. Han and Hung [6] considered stochastic interest rates and inflation, and made an optimal investment strategy for families with labor income and dependents. In addition to the aforementioned, there were also Kraft [7], Kwak [8], Gu [9], Hata [10], Kasumo [11], Chen [12], Wang [13], Yang [14, 15] etc., who have made contributions in this field. In recent years, when studying this kind of optimization problems, a great deal of scholars began to consider establishing more realistic models. One of the main improvement methods is to consider the situation of incomplete information. Since only a part of the information flow can be observed in real life, the investment portfolio problem under the partial information has become the focus of many scholars. Bai and Guo [16] studied the stochastic control problems under partially observable conditions with the goal of maximizing the expected utility of terminal wealth, and finally gave the optimal strategies in both exponential and logarithmic cases. Liu [17] applied the stochastic maximum principle to solve the optimal consumption and investment strategy under complete information, and used the backward separation technique to solve the optimal consumption and investment strategy under partial information. Zhang et al. [18] studied the problems of investment optimization, who not only considered the unobservable situation of stock returns, but also noted that personal returns wete unobservable as well. They also considered the problem, that was, the condition of incomplete information. Moreover, they comprehensively applied Kalman filtering and nonlinear filtering to obtain the explicit solution of Zakai equation. By calculation, they got the stochastic optimal control problem with partial information, and obtained the explicit solution of the problem. There were Wang [19], Huang [20], Hata [21], Zhu [22], Wan [23], Yu [24] etc., who have studied such problems.

Through combing the existing literature, it is found that there is no literature that considers the family optimization problem considering the lifespan correlation among different members under the condition of partial information. Therefore, based on the previous research results, this paper proposes the consumption, investment and life insurance purchase problems of couples in the family under partial information. Herein, it is assumed that the lifespans of the couples are correlated, and a copula model is used to simulate this correlation. This paper mainly discusses the optimal strategy of asset allocation before and after the death of any family member.

2. Problem description

This paper will make a detailed and professional mathematical description of the problem to be studied, and construct a mathematical model of the problem to be studied. First of all, the mathematical model in the case of complete information is introduced. Secondly, this paper takes into account the actual situation, considers the incomplete information situation, improves the model mentioned above, and obtains a new mathematical model.

2.1 Model construction

The family considered in this paper consists of a couple, both of whom have labor income and are assumed to have the right to manage all their assets. Since the survival time of both spouses is uncertain, they can hedge the risk of death by purchasing life insurance.

Let $T<\infty$ be the time when both husband and wife workers retire at the same time. $B(\cdot)$ donates the standard Brownian motion, and $t_{i}(i=1,2)$ represents the death time of two laborers of the husband and wife. It is assumed that $B(\cdot)$ and $(t_{1},t_{2})$ definitions are in the $(\Omega,\mathbb{F},F,P)$ probability space and they are mutually independent, in which $F=\{\mathbb{F}_{t}\}_{t\in[0,T]}$ represents the partial flow of information generated by $B(\cdot)$ , $t_{1}$ and $t_{2}$ jointly. Let $t_{i}$ have a marginal probability distribution function and $F_{i}(t)=P\{t_{i}\leqslant t\}=1-e^{-\int_{0}^{t}{\lambda_{i}(s)ds}},(i=1,2)$ , in which $\lambda_{i}(\cdot)$ represents the mortality rate of the ith policyholders.

It is assumed that $(t_{1},t_{2})$ joint probability distribution function is $F(t_{1},t_{2})=C(F_{1}(t),F_{2}(t))$ , where $C(\cdot,\cdot)$ is the copula function. Suppose $(t_{1},t_{2})$ is a joint probability density function, that is, $f(\cdot,\cdot)$ . Suppose two policyholders earn a certain amount of labor incomes $I_{1}(\cdot)$ and $I_{2}(\cdot)$ , respectively, before retirement. Assume that both husband and wife invest part of their property in the financial market, where the financial market includes two assets. One is a risk-free asset $S_{0}(t)$ , and the other is a risky asset $S_{1}(t)$ . Their price processes respectively satisfy the Eq. (1)

$\displaystyle\left\{\begin{array}[]{l}\frac{dS_{0}(t)}{S_{0}(t)}=r(t)dt\\ \\ \frac{dS_{1}(t)}{S_{1}(t)}=\mu(t)dt+\sigma(t)dB_{1}(t)\\ \\ d\mu(t)=a[\bar{\mu}-\mu(t)]dt+\textit{bdB}_{2}(t)\\ \end{array}\right.$ (1)

In the above-mentioned model, $r(\cdot)$ is the interest rate of risk-free assets (generally referred to bonds). $\mu(\cdot)$ and $\sigma(\cdot)$ term the rate of return and volatility of risky assets (generally referred to stocks).

Let $c_{i}(\cdot),k_{i}(\cdot)$ and $\pi_{i}(\cdot)$ denote the consumption of policyholder $i(i=1,2)$ , the life insurance premium and the assets invested in the risk market, respectively, and the asset $X_{i}(\cdot)$ of the ith insured person can be obtained to satisfy the SDE:

$\displaystyle dX_{i}(t)=[X_{i}(t)-\pi_{i}(t)]\frac{dS_{0}(t)}{S_{0}(t)}+\pi_{i% }(t)\frac{dS_{1}(t)}{S_{1}(t)}-[c_{i}+k_{i}-I_{i}]dt.$

Substitute $\frac{dS_{0}(t)}{S_{0}(t)}$ and $\frac{dS_{1}(t)}{S_{1}(t)}$ into the $X_{i}(\cdot)$ formula to get the following formula:

$\displaystyle dX_{i}(t)=\{r(t)X_{i}(t)+\pi_{i}(t)[\mu(t)-r(t)]+I_{i}(t)-c_{i}(% t)-k_{i}(t)\}dt+\pi_{i}(t)\sigma(t)dB_{1}(t),$

where the range of $t$ is $[0,\min\{t_{i},T\}]$ .

Then the family wealth is recorded as $X(\cdot)$ ,

$\displaystyle dX(t)=\left\{\begin{array}[]{ll}{dX_{1}(t)+dX_{2}(t)}&\text{if }% 0\leqslant t\leqslant\min\{t_{1},t_{2},T\}\\ {dX_{3-i}(t)}&\text{if }t_{i}\leqslant t_{3-i}\text{ and }{t_{i}\wedge T% \leqslant t\leqslant t_{3-i}\wedge T}\\ \end{array}\right.$

where $i$ takes 1or 2, and $X(\cdot)$ satisfies the equation as follows:

$\displaystyle dX(t)=\{r(t)X(t)+\pi(t)[\mu(t)-r(t)]-{\rm I}_{\{t<t_{1}\}}[c_{1}% (t)+k_{1}(t)-I_{1}(t)]{}\!-{\rm I}_{\{t<t_{2}\}}[c_{2}(t)+k_{2}(t)-I_{2}(t)]\}% dt+\sigma(t)\pi(t)dB(t)$ $\displaystyle\pi(t)=\pi_{1}{\rm I}_{\{t<t_{1}\}}+\pi_{2}{\rm I}_{\{t<t_{2}\}}$

The range of $t$ is $[0,(t_{1}\vee t_{2})\wedge T]$ , where the ${\rm I}_{\{t<t_{1}\}}$ and ${\rm I}_{\{t<t_{2}\}}$ are characteristic functions.

For the death of the ith family member at time $t_{i}$ , the insurance agent will pay an insurance premium to the beneficiary with an amount of $\frac{k_{i}(t_{i})}{\xi_{i}(t_{i})}$ , where $\xi_{i}(\cdot)$ is the premium-insurance ratio. Therefore, if $t_{i}<t_{3-i}$ , here, then $i$ takes 1or 2. That is, if the ith family member dies at time $t_{i}$ , then the wealth process of the family is $X(t_{i})=X(t_{i}^{-})+\frac{k_{i}(t_{i})}{\xi_{i}(t_{i})}$ .

2.2 Part of the information and wealth process

For the real financial markets, when investors invest their household assets in risky assets such as stocks, they often rely on the information flow generated by past stock prices, rather than the entire information flow. Thence, this part defines part of the information flow, and gives the wealth process under this information flow.

It is assumed that the entire information flow is $\mathbb{F}_{t}=\sigma\{B_{1}(\varepsilon),B_{2}(\varepsilon):0\leqslant% \varepsilon\leqslant t\}$ , and the partial information flow is $\mathbb{F}_{t}^{s}=\sigma\{S_{1}(\varepsilon):0\leqslant\varepsilon\leqslant t\}$ . The conditional expectation of $\mu(\cdot)$ is denoted as $m(\cdot)$ , which its conditional variance to be denoted as $\gamma(\cdot)$ . The conditional expectation and conditional variance of $\mu(\cdot)$ of the partial information flow are:

$\displaystyle m(\cdot)=E[\mu(t)\left|{\mathbb{F}_{t}^{s}}\right.],\gamma(\cdot% )=E[(\mu(t)-m(t))^{2}\left|{\mathbb{F}_{t}^{s}}\right.].$ $\displaystyle\left\{{{\begin{array}[]{l}{dm(t)=a[\bar{\mu}-\mu(t)]dt+\gamma(t)% \sigma^{-1}d\hat{B}(t)}\\ m(0)=m_{0}\\ \end{array}}}\right.$ (2) $\displaystyle\left\{{{\begin{array}[]{l}{d\hat{B}(t)=\frac{1}{\sigma}\{[\mu(t)% -m(t)]dt+\sigma dB_{1}(t)\}}\\ \hat{B}(0)=0\\ \end{array}}}\right.$ (3) $\displaystyle\left\{{{\begin{array}[]{l}{\frac{d\gamma(t)}{dt}=b^{2}-2a\gamma(% t)-\frac{\gamma^{2}(t)}{\sigma^{2}(t)}}\\ \gamma(0)=\gamma_{0}\\ \end{array}}}\right.$ (4)

Therefore, the wealth process under partial information is:

$\displaystyle dX(t)=\{rX(t)+\pi[m(t)-r]-{\rm I}_{\{t<t_{1}\}}[c_{1}+k_{1}-I_{1% }]-{\rm I}_{\{t<t_{2}\}}[c_{2}+k_{2}-I_{1}]\}dt{}\!+\pi\sigma(t)dB_{1}(t)$ (5)

where $m(\cdot)$ and $\hat{B}(\cdot)$ are given by two mathematical Eqs (2) and (3), respectively. And $\gamma(t)$ is given by the Riccati equation aforementioned.

Let $u(\cdot)=(c_{1}(\cdot),c_{2}(\cdot),k_{1}(\cdot),k_{2}(\cdot),\pi(\cdot))$ , it can be defined that the set of all admissible policies is

$\displaystyle{\rm A}=\left\{u(\cdot)\in(R^{+})^{2}\times R^{3}\left|E\int_{0}^% {T}\pi^{2}(t)dt<\infty,E\int_{0}^{T}c_{i}(t)dt<\infty,\right.\right.\left.E% \int_{0}^{T}k_{i}(t)dt<\infty,i=1,2\right\}.$

For $u(\cdot)\in{\rm A}(0,x)$ , the boundary value condition is $X(t)=x$ , and the family wealth process satisfying this condition satisfies as follows:

$\displaystyle J(x,m,t,u(\cdot))=E_{t}\left\{\int_{t}^{t_{1}\wedge T}{\omega_{1% }e^{-\delta s}U(c_{1}(s))}ds+\int_{t}^{t_{2}\wedge T}{\omega_{2}e^{-\delta s}U% (c_{2}(s))}ds\right.{}\!+\omega_{3}{\rm I}_{\{t_{1}\vee t_{2}\leqslant T\}}e^{% -\delta(t_{1}\vee t_{2})}U\left[X(t_{1}\vee t_{2})+\sum\limits_{i=1}^{2}{\frac% {k_{i}(t_{i})}{\xi_{i}(t_{i})}}{\rm I}_{\{t_{i}=t_{1}\vee t_{2}\}}\right]{}\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left.\phantom{\int_{t}^{t_{1}\wedge T}}+% \omega_{4}{\rm I}_{\{t_{1}\vee t_{2}>T\}}e^{-\delta T}U(X(T))|{\mbox{F}_{t}^{s% }}\right\}$

where $\delta>0$ represents the discount factor, $U(\cdot)$ is the utility function, and $\omega_{i}$ reflects the importance of one utility to another, where $\omega_{i}$ satisfies the $\omega_{i}\geqslant 0,\sum\limits_{i=1}^{4}{\omega_{i}=1}$ condition.

3. Solving the optimal strategy

Next, this paper will focus on how to distribute the household wealth rationally, that is, to explore the optimal strategy that can maximize the total wealth utility of the entire family. There are two more common situations in practice. The first is that the death of two family members occurs after both of them retire, and the second is that one of the two family members dies before retirement. In comparation with the actual situation, this section later mainly studies the optimal investment strategy of a couple before and after the death of the first family member. By calculating the optimal strategy, the influence of the death time of the first family member on the optimal strategy is compared.

3.1 The optimal policy problem after one of the members dies

Suppose $T_{1}$ to be the lesser of the time to death of the two members of the family, that is, $T_{1}=t_{1}\wedge t_{2}$ . Consider the condition $T_{1}=t_{3-i}<t_{i},(i=1,2)$ . After time $T_{1}$ , the optimization problem for the ith policyholder can be expressed as:

$\displaystyle\tilde{J}_{i}(t,x,m;c_{i}(\cdot),k_{i}(\cdot),\pi_{i}(\cdot))=E_{% t}\left\{\int_{t}^{t_{i}\wedge T}{\omega_{i}e^{-\delta s}U(c_{i}(s))}ds+\omega% _{3}{\rm I}_{\{t_{i}\leqslant T\}}e^{-\delta t_{i}}U\right.{}\!\!\left.\left(X% (t_{i})+\frac{k_{i}(t_{i})}{\xi_{i}(t_{i})}\right)+\omega_{4}{\rm I}_{\{t_{i}>% T\}}e^{-\delta T}U(X(T))|{\mbox{F}_{t}^{s}}\right\}$

The wealth of each family member here satisfies the Eq. (5).

Theorem 1 For $U(x)=\ln x$ , $V(\cdot)=\alpha(t)\ln[x+\beta(t)]+g(t,m)$ , if $T_{1}=t_{3-i}$ , then after $T_{1}$ time, the value function of the $i$ policyholder is $V_{i}(\cdot)=\alpha_{i}(t)\ln[x+\beta_{i}(t)]+g_{i}(t,m)$ , and the optimal strategy is:

$\displaystyle c_{i}^{*}=\frac{\omega_{i}(x+\beta_{i}(t))}{\alpha_{i}(t)e^{% \delta t}},k_{i}^{*}=\frac{\lambda_{i}(t)\omega_{3}[x+\beta_{i}(t)]}{\alpha_{i% }(t)e^{\delta t}}-x\xi_{i},\pi_{i}^{*}=\frac{[m(t)-r]}{\sigma^{2}}[x+\beta_{i}% (t)].$

In the above three formulas, $\alpha(\cdot)$ and $\beta(\cdot)$ can be shown as the following formulas:

$\displaystyle\alpha_{i}(t)=\exp\int_{t}^{T}{\lambda_{i}}(s)ds\left\{e^{-\delta t% }+\int_{t}^{T}{\frac{1+\lambda_{i}(s)}{\exp[\omega_{3}\delta t+\int_{T}^{s}{% \omega_{3}\lambda_{i}(v)dv}]}}ds\right\},$ $\displaystyle\beta_{i}(t)=\int_{t}^{T}{I_{i}}(s)e^{-\int_{t}^{s}{[r(v)+\xi_{i}% (v)]dv}}ds.$

Let $g_{i}(t,m)=\phi m^{2}+\varphi m+\psi$ , then $\phi,\varphi,\psi$ are the following forms:

$\displaystyle\phi=\int_{t}^{T}{\frac{\alpha_{i}(t)}{2\sigma^{2}}e^{\int_{t}^{s% }{[2a-\lambda_{i}(\nu)]d\nu}}ds}.$ $\displaystyle\varphi=-\int_{t}^{T}{\frac{\alpha_{i}(t)r(t)}{\sigma^{2}(t)}}e^{% [a(s-t)+\int_{t}^{s}{\lambda_{i}(v)dv]}}ds.$ $\displaystyle\psi=\int_{t}^{T}{\zeta(t)}e^{\int_{t}^{s}{\lambda_{i}(v)dv}}ds.$ $\displaystyle\zeta(t)=\alpha_{i}(t)[r+\xi_{i}]+\frac{\alpha_{i}(t)\gamma^{2}(t% )}{2\sigma^{2}(t)}-\lambda_{i}(t)\omega_{3}e^{-\delta t}\ln[\lambda_{i}(t)e^{-% \delta t}]-[\omega_{i}+\lambda_{i}\omega_{3}]e^{-\delta t}{}\!-\omega_{3}% \lambda_{i}e^{-\delta t}\ln[\alpha_{i}(t)e^{\delta t}\xi_{i}]-\omega_{3}e^{-% \delta t}\ln[\alpha_{i}(t)e^{\delta t}].$

Proof:

For 1 or 2, define the value function $V_{i}(t,x,m)=\mathop{\sup}\limits_{c_{i},k_{i},\pi}\tilde{J}_{i}(t,x,m,c_{i}(% \cdot),k_{i}(\cdot),\pi(\cdot))$ , and then apply Theorems 3.3.1 and 3.3.2 in Ye [4]. It can be shown that $V_{i}$ is the classical solution (if any) of the following HJB equation:

$\displaystyle V_{i,t}-\lambda_{i}(t)V_{i}+\mathop{\sup}\limits_{c_{i},k_{i},% \pi_{i}}\left\{[rx+\pi_{i}(t)(m(t)-r)+I_{i}(t)-c_{i}(t)-k_{i}(t)]\phantom{% \left[x+\frac{k_{i}(t)}{\xi_{i}(t)}\right]}\right.$ $\displaystyle V_{i,x}+am(t)V_{i,m}+\frac{1}{2}\sigma^{2}(t)\pi_{i}^{2}(t)V_{i,% xx}+\pi_{i}(t)\gamma(t)V_{i,mx}$

(6) $\displaystyle\quad{}+\left.\frac{1}{2}\frac{\gamma^{2}(t)}{\sigma^{2}(t)}V_{i,% mm}+\lambda_{i}(t)\omega_{3}e^{-\delta t}\ln\left[x+\frac{k_{i}(t)}{\xi_{i}(t)% }\right]+\omega_{i}e^{-\delta t}\ln c_{i}\right\}=0$ $\displaystyle V_{i}(T,x,m)=e^{-\delta T}\ln x,$

where $V_{i,t}(\cdot),V_{i,m}(\cdot)$ and $V_{i,x}(\cdot)$ represents the first-order partial derivative of $V_{i}(\cdot)$ with respect to $t, x$ and $m$ respectively, and $V_{i,t}(\cdot),V_{i,m}(\cdot)$ and $V_{i,x}(\cdot)$ respectively represents the second-order partial derivative of $V_{i}(\cdot)$ pertaining to $m$ and $x$ .

Try to find the partial derivative of $c_{i}$ with respect to Eq. (3.1) and make it equal to 0 to get:

$\displaystyle-V_{i,x}+\omega_{i}e^{-\delta t}\frac{1}{c_{i}(t)}=0,$ (7) $\displaystyle c_{i}^{}(t)=\frac{\omega_{i}}{V_{i,x}(t,x,m)e^{\delta t}}$

Try to find the partial derivative of $k_{i}$ with respect to Eq. (3.1) and make it equal to 0 to get:

$\displaystyle-V_{i,x}+\lambda_{i}(t)\omega_{3}e^{-\delta t}\left[x+\frac{k_{i}% (t)}{\xi_{i}(t)}\right]^{-1}\frac{1}{\xi_{i}}=0,x+\frac{k_{i}^{}(t)}{\xi_{i}(% t)}=\frac{\lambda_{i}(t)\omega_{3}}{V_{i,x}\xi_{i}e^{\delta t}},$ (8) $\displaystyle k_{i}^{}=\frac{\lambda_{i}(t)\omega_{3}}{V_{i,x}(t,x,m)e^{% \delta t}}-x\xi_{i}$

Try to find the partial derivative of $k_{i}$ with respect to Eq. (3.1) and make it equal to 0 to get:

$\displaystyle[m-r(t)]V_{i,x}+\sigma^{2}\pi_{i}V_{i,xx}+\gamma(t)V_{i,mx}=0,$ (9) $\displaystyle\pi_{i}^{}=-\frac{[m-r(t)]V_{i,x}(t,x,m)+\gamma(t)V_{i,mx}(t,x,m% )}{\sigma^{2}V_{i,xx}(t,x,m)}$

Substitute Eqs (3.1)–(3.1) into Eq. (3.1) and simplify them to get:

$\displaystyle V_{i,t}-\lambda_{i}(t)V_{i}+r(t)xV_{i,x}+I_{i}(t)V_{i,x}+x\xi_{i% }(t)V_{i,x}$ $\displaystyle\quad{}-\frac{1}{2\sigma^{2}(t)V_{i,xx}}\{[m-r(t)]V_{i,x}\}^{2}-% \omega_{i}e^{-\delta t}-\lambda_{i}(t)\omega_{3}e^{-\delta t}+amV_{i,m}$ (10) $\displaystyle\quad{}+\frac{1}{2}\frac{\gamma^{2}(t)}{\sigma^{2}(t)}V_{i,mm}+% \lambda_{i}(t)\omega_{3}e^{-\delta t}\ln\frac{\lambda_{i}(t)\omega_{3}e^{-% \delta t}}{V_{i,x}\xi_{i}(t)}+\frac{\omega_{3}}{e^{\delta t}}\ln\frac{\omega_{% i}e^{-\delta t}}{V_{i,x}}=0$

The Eq. (3.1) is the second order partial differential equation of $V_{i}$ , whose solution is set as: $V_{i}=\alpha_{i}(t)\ln[x+\beta_{i}(t)]+g_{i}(t,m)$ .

Obviously $V_{i}(\cdot)$ is a second-order differentiable function pertaining to variables $x$ and $m$ . Find $V_{i,t}(\cdot)$ , $V_{i,m}(\cdot)$ , $V_{i,x}(\cdot)$ , $V_{i,xx}(\cdot)$ , $V_{i,mx}(\cdot)$ and $V_{i,mm}(\cdot)$ for $V_{i}(\cdot)$ , and then substitute them into Eq. (3.1) to get a new first-order ordinary differential equation about $\alpha_{i}(\cdot)$ and $\beta_{i}(\cdot)$ .

Use the undetermined coefficient method to get:

$\displaystyle\left\{{{\begin{array}[]{l}{\alpha_{i,t}(t)-\alpha_{i}(t)\lambda_% {i}(t)+\omega_{3}\lambda_{i}(t)e^{-\delta t}+\omega_{3}e^{-\delta t}=0}\\ \alpha_{i}(T)=e^{-\delta t}\\ \end{array}}}\right.$

Solved: $\alpha_{i}(t)=\exp\int_{t}^{T}{\lambda_{i}}(s)ds\{e^{-\delta t}+\int_{t}^{T}{% \frac{1+\lambda_{i}(s)}{\exp[\omega_{3}\delta t+\int_{T}^{s}{\omega_{3}\lambda% _{i}(v)dv}]}}ds\}$ .

$\displaystyle\left\{{{\begin{array}[]{l}{\beta_{i,t}(t)-\beta_{i}(t)[r(t)+\xi_% {i}(t)]+I_{1}\mbox{(}t\mbox{) = }0}\hfill\\ \beta_{i}(T)=0\\ \end{array}}}\right.$

Solved: $\beta_{i}(t)=\int_{t}^{T}{I_{i}}(s)e^{-\int_{t}^{s}{[r(v)+\xi_{i}(v)]dv}}ds$ .

$\displaystyle g_{i,t}(t,m)-\lambda_{i}(t)g_{i}(t,m)+amg_{i,m}(t,m)+\frac{1}{2}% \frac{\gamma^{2}(t)}{\sigma^{2}(t)}g_{i,mm}(t,m)+\alpha_{i}(t)[r+\xi_{i}(t)]$ $\displaystyle\quad{}+\frac{\alpha_{i}(t)}{2\sigma^{2}(t)}[m-r]^{2}-[\omega_{i}% +\lambda_{i}(t)\omega_{3}]e^{-\delta t}+\lambda_{i}(t)\omega_{3}e^{-\delta t}% \ln[\lambda_{i}(t)e^{-\delta t}]$ (11) $\displaystyle\quad{}-\omega_{3}\lambda_{i}(t)e^{-\delta t}\ln[\alpha_{i}(t)e^{% \delta t}\xi_{i}]-\omega_{3}e^{-\delta t}\ln[\alpha_{i}(t)e^{\delta t}]=0$

Suppose that the form of the solution of the above-mentioned equation is $g_{i}(t,m)=\phi m^{2}+\varphi m+\psi$ , where $\phi$ , $\varphi$ and $\psi$ are all functions of variable $t$ . It follows from the $g_{i}(\cdot)$ expression that $g_{i}(\cdot)$ is second-order differentiable with respect to $m$ and first-order differentiable with respect to $t$ . Find the partial derivative of it and substitute it into Eq. (3.1), simplify them to get:

$\displaystyle\phi_{t}m^{2}+m\varphi_{t}+\psi_{t}-\lambda_{i}\phi m^{2}-\lambda% _{i}\varphi m-\lambda_{i}(t)\psi+2a\phi m^{2}$ $\displaystyle\quad{}+a\varphi m+\frac{\gamma^{2}(t)}{\sigma^{2}(t)}\phi+\alpha% _{i}(t)[r(t)+\xi_{i}(t)]+\frac{\alpha_{i}(t)}{2\sigma^{2}(t)}[m-r(t)]^{2}-[% \omega_{i}+\lambda_{i}(t)\omega_{3}]e^{-\delta t}$ (12) $\displaystyle\quad{}+\lambda_{i}(t)\omega_{3}e^{-\delta t}\ln[\lambda_{i}(t)e^% {-\delta t}]-\lambda_{i}(t)\omega_{3}e^{-\delta t}\ln[\alpha_{i}(t)e^{\delta t% }\xi_{i}(t)]-\omega_{3}e^{-\delta t}\ln[\alpha_{i}(t)e^{\delta t}]=0$

where $\phi_{t},\varphi_{t}$ and $\psi_{t}$ are the derivatives of $\phi$ , $\varphi$ and $\psi$ with respect to $t$ , respectively.

Here is the available equation from Eq. (3.1) $\phi:\left\{{{\begin{array}[]{l}{\phi_{t}-\lambda_{i}(t)\phi+2a\phi+\frac{% \alpha_{i}(t)}{2\sigma^{2}(t)}=0}\\ \phi(t)=0\\ \end{array}}}\right.$ ,

Solved $\phi=\int_{t}^{T}{\frac{\alpha_{i}(t)}{2\sigma^{2}}e^{\int_{t}^{s}{[2a-\lambda% _{i}(\nu)]d\nu}}ds}$ .

Here is the available equation from Eq. (3.1) $\varphi:\left\{{{\begin{array}[]{l}{\varphi_{t}-\lambda_{i}(t)\varphi+a\varphi% -\frac{\alpha_{i}(t)r(t)}{\sigma^{2}(t)}=0}\\ \varphi(t)=0\\ \end{array}}}\right.$ ,

Solved $\varphi=-\int_{t}^{T}{\frac{\alpha_{i}(t)r(t)}{\sigma^{2}(t)}}e^{[a(s-t)+\int_% {t}^{s}{\lambda_{i}(v)dv]}}ds$ .

Here is the available from Eq. (3.1) $\psi:\left\{{{\begin{array}[]{l}\psi_{t}-\lambda_{i}(t)\psi+\zeta(t)=0\\ {\psi(t)=0}\\ \end{array}}}\right.$ ,

Solved $\psi=\int_{t}^{T}{\zeta(t)}e^{\int_{t}^{s}{\lambda_{i}(v)dv}}ds$ , where $\zeta(t)$ is expressed as follows:

$\displaystyle\zeta(t)=\alpha_{i}(t)[r+\xi_{i}]+\frac{\alpha_{i}(t)\gamma^{2}(t% )}{2\sigma^{2}(t)}-\lambda_{i}(t)\omega_{3}e^{-\delta t}\ln[\lambda_{i}(t)e^{-% \delta t}]-[\omega_{i}+\lambda_{i}\omega_{3}]e^{-\delta t}{}\!-\omega_{3}% \lambda_{i}e^{-\delta t}\ln[\alpha_{i}(t)e^{\delta t}\xi_{i}]-\omega_{3}e^{-% \delta t}\ln[\alpha_{i}(t)e^{\delta t}]$

Herein, the optimal investment, consumption and life insurance purchase strategies calculated by Theorem 1 are numerically simulated by numerical integration. Figure 1 illustrates the variation of $\beta_{i}(t)$ with respect to time $t$ . Figures 2–4 show the changes of $k_{i}^{}$ , $c_{i}^{}$ and $\pi_{i}^{}$ with respect to $t$ and $m(t)$ , respectively. It is assumed that both husband and wife start working at the age of 25. The initial salary is 60,000 yuan/year, which increases at a rate of 3.2%. Please refer to Table 1 for more detial.

Table 1
Parameter value

Pram T $r(t)$ $\sigma$ $\delta$ $\lambda_{i}(t)$ $\eta_{i}(t)$ $\omega_{i}$

Value 60 0.03 exp (0.01) $t$ 0.18 0.03 0.001 $+\exp$ ( $-9.5+0.1t$ ) $1.05\lambda(t)$ 0.25

Figure 1.
Changes in human capital value.

Figure 2.
Changes in optimal life insurance purchase.

Figure 3.
Changes in optimal consumption.

For $\beta_{i}(t)$ , it can be regarded as the discounted value of this person’s income at some future time. Therefore, it can intuitively interpret $\beta_{i}(t)$ as a person’s human capital. It can be seen from Fig. 1 that $\beta_{i}(t)$ reaches a maximum value at $t=$ 37. At $t\in(25,37)$ , the personal human capital of family members increases strictly monotonously with time, that is, the value of human capital increases with the increase of working hours. It also continues to grow, reaching its maximum at around the age of 37. At $t\in(37,60)$ , with the passage of time, human capital shows a rapid downward trend.

It can be clearly seen from Fig. 2 that $k_{i}^{}$ decreases slowly with time at $t\in(25,55)$ , and $k_{i}^{}$ increases rapidly with time at $t\in(55,60)$ .

As can be seen from Fig. 3, $c_{i}^{}$ grows slowly with time at $t\in(25,55)$ , and increases rapidly with time at $t\in(55,60)$ .

Herein, the value range of $m(t)$ is $(0.08,0.20)$ , and $r=0.03$ .

As can be seen from Fig. 4, $\pi_{i}^{*}$ increases linearly with time $t$ .

Figure 4.
Change of optimal investment strategy.

3.2 Optimization problem before no member dies

Pram	T	$r(t)$	$\sigma$	$\delta$	$\lambda_{i}(t)$	$\eta_{i}(t)$	$\omega_{i}$
Value	60	0.03 exp (0.01) $t$	0.18	0.03	0.001 $+\exp$ ( $-9.5+0.1t$ )	$1.05\lambda(t)$	0.25

It is assumed that the joint distribution function and joint probability density function of death time $t_{1}$ and $t_{2}$ are $F(\cdot,\cdot)$ and $f(\cdot,\cdot)$ respectively. If $(t_{1},t_{2})$ and the information flow generated by Brownian motion are independent, then:

$\displaystyle J(x,m,t;u(\cdot))=\frac{1}{1-F_{T_{1}}(t)}E_{t}\left\{\int_{t}^{% T}{[1-F_{T_{1}}(s)]}\right\}e^{-\delta s}[\omega_{1}U(\bar{c}_{1}(s))+\omega_{% 2}U(\bar{c}_{2}(s))]ds{}\!+\int_{t}^{T}{\left[\int_{z}^{\infty}{f(s,z)}ds% \right]V_{1}\left(z,x(z)+\frac{\bar{k}_{2}(z)}{\xi_{2}(z)}\right)dz}{}\!+\int_% {t}^{T}{\left[\int_{s}^{\infty}{f(s,z)dz}\right]}V_{2}\left(s,x(s)+\frac{\bar{% k}_{1}(s)}{\bar{\xi}_{1}(s)}\right)ds{}\!+[1-F_{T_{1}}(T)]\omega_{4}e^{-\delta T% }U(X(T)|{\mathbb{F}_{t}^{s}})$

where $F_{T_{1}}(\cdot)=F_{1}(\cdot)+F_{2}(\cdot)-F(\cdot,\cdot)$ represents the distribution function of the first member at death time $T_{1}$ . Each marginal distribution is assumed to be a Gompertz distribution, that is,

$\displaystyle F_{i}(t)=1-\exp\{\exp(-m_{i}/n_{i})[1-\exp(t/n_{i})]\},$

where $m_{i}$ and $n_{i}$ are parameters.

$\displaystyle dX(t)=\left\{rX(t)+[m(t)-r]\bar{\pi}(t)+\sum\limits_{i=1}^{2}{[-% \bar{c}_{i}(t)-\bar{k}_{i}(t)+I_{i}(t)]}\right\}dt+\sigma(t)\bar{\pi}(t)B_{1}(% t).$

Herein, $m(\cdot)$ and $\hat{B}(\cdot)$ are given by two mathematical Eqs (2) and (3), respectively, and $\gamma(\cdot)$ is given by the Riccati Equation of Eq. (4).

Let $\tilde{J}(x,m,t;u(\cdot))=[1-F_{T_{1}}(t)]J(x,m,t;u(\cdot))$ , then $\tilde{V}(\cdot)=\mathop{\sup}\limits_{u}\tilde{J}(x,m,t;u(\cdot))=[1-F_{T_{1}% }(t)]\linebreak V(\cdot)$ , abbreviated as $\tilde{V}$ . $\tilde{V}$ is continuous and differentiable pertaining to variable $t$ , which is continuous and quadratic differentiable about $m$ and $x$ , satisfying the following HJB equation:

$\displaystyle\left\{{{\begin{array}[]{l}\tilde{V}_{t}+\mathop{\sup}\limits_{% \bar{c}_{1},\bar{c}_{2},\bar{k}_{1},\bar{k}_{2},\bar{\pi}}\{\frac{1}{2}\sigma^% {2}(t)\bar{\pi}^{2}\tilde{V}_{xx}+e^{-\delta t}[1-F_{T_{1}}][\omega_{1}U(\bar{% c}_{1})]+\omega_{2}U(\bar{c}_{2})\}\\ \quad+[rx+(m(t)-r)\bar{\pi}-\bar{c}_{1}-\bar{k}_{1}+I_{1}(t)-\bar{c}_{2}-\bar{% k}_{2}+I_{2}(t)]\tilde{V}_{x}+am(t)\tilde{V}_{m}\\ \quad+\tilde{\pi}\gamma(t)\tilde{V}_{mx}+\frac{1}{2}\frac{\gamma^{2}(t)}{% \sigma^{2}(t)}\tilde{V}_{mm}+e^{-\delta t}\ln(x+\frac{\bar{k}_{2}}{\xi_{2}(t)}% +\beta_{1}(t))\int_{t}^{\infty}{f(s,t)ds}\\ \quad+e^{-\delta t}\ln(x+\frac{\bar{k}_{1}}{\xi_{1}(t)}+\beta_{2}(t))\int_{t}^% {\infty}{f(t,z)dz}\}=0\\ \tilde{V}(T,x,m)=[1-F_{T_{1}}(T)]\omega_{4}e^{-\delta T}U(X(T))\\ \end{array}}}\right.$

Theorem 2 For $U(x)=\ln x$ , $\tilde{V}(\cdot)=\alpha(t)\ln[x+\beta(t)]+g(t,m)$ , in finite time $[0,T]$ , the optimal strategy is:

$\displaystyle\bar{c}_{i}^{*}=\frac{\omega_{i}[1-F_{T_{1}}(t)]}{\alpha(t)e^{% \delta t}}[x+\beta(t)],$

where $i$ is 1or 2.

$\displaystyle\bar{k}_{1}^{*}=\frac{e^{-\delta t}\int_{t}^{\infty}{f(t,z)dz}}{% \alpha(t)}[x+\beta(t)]-\xi_{1}(t)x-\beta_{2}(t)\xi_{1}(t),$ $\displaystyle\bar{k}_{2}^{*}=\frac{e^{-\delta t}\int_{t}^{\infty}{f(s,t)ds}}{% \alpha(t)}[x+\beta(t)]-\xi_{2}(t)x-\beta_{1}(t)\xi_{2}(t),$ $\displaystyle\bar{\pi}^{*}=\frac{[m(t)-r]\alpha(t)+\gamma(t)[x+\beta(t)]}{% \sigma^{2}(t)\alpha(t)}(x+\beta(t)).$

Among them $\alpha(t)=-\int_{t}^{T}{[e^{-\delta t}\int_{t}^{\infty}{f(t,z)}dz+e^{-\delta t% }\int_{t}^{\infty}{f(s,t)ds}]}dt$ ,

$\displaystyle\beta(t)=\int_{t}^{T}{[I_{1}(s)+I_{2}(s)+\xi_{1}(s)\beta_{2}(s)+% \xi_{2}(s)\beta_{1}(s)]e^{-\int_{t}^{s}{[r(v)+\xi_{1}(v)+\xi_{2}(v)]dv}}}ds,$

$g_{t}(t,m)=Fm^{2}(t)+Nm(t)+P$ , where $F$ , $N$ and $P$ are all functions of variable $t$ .

$\displaystyle F=\int_{t}^{T}{\frac{\alpha(t)}{2\sigma^{2}(t)}}e^{2a(s-t)}ds,N=% -\int_{t}^{T}{\frac{\alpha(t)\gamma(t)}{\sigma^{2}(t)}}e^{a(s-t)}ds,P=-\int_{t% }^{T}{Y(s)ds},$ $\displaystyle Y(t)=\alpha(t)[r+\xi_{1}]+\alpha(t)(r+\xi_{2})+\frac{\alpha(t)}{% 2\sigma^{2}(t)}[m-r]^{2}-e^{-\delta t}[1-F_{T_{1}}(t)](\omega_{1}+\omega_{2}){% }\!-e^{-\delta t}\left[\int_{t}^{\infty}f(t,z)dz+\int_{t}^{\infty}f(s,t)ds% \right]{}\!+\left[e^{-\delta t}\ln\int_{t}^{\infty}f(t,z)dz-\delta te^{-\delta t% }-e^{-\delta t}\ln\alpha(t)\xi_{1}\right]\int_{t}^{\infty}{f(t,z)dz}{}\!+\left% [e^{-\delta t}\ln\int_{t}^{\infty}{f(s,t)ds-\delta te^{-\delta t}-e^{-\delta t% }\ln\alpha(t)\xi_{2}}\right]\int_{t}^{\infty}{f(s,t)}ds.$

The proof process of this theorem will not be given in detail, and the specific proof method can be proved by itself according to the method selected in Theorem 1.

It can be seen from the conclusion of Theorem 2 that, $\beta(t)$ can be understood firstly as the sum of the human capital of family. Secondly, with the growth of the total human capital of family members, the total investment and consumption of families show a trend of gradual growth. Thirdly, if the total household human capital is assumed to remain unchanged, the purchase strategy of life insurance of one family member will decrease with the increase of human capital of another family member. It is understandable that because one party’s human capital increases, the whole family will reduce the purchase of life insurance for the party whose human capital decreases.

4. Conclusion

On the basis of previous research, this paper extends the optimal strategy of personal wealth distribution to the optimal strategy of couples and families, and synchronously improves all the information in the model to partial information. By constructing a mathematical model, the corresponding HJB equation is derived using a dynamic programming method. According to the two situations before and after the death of family members, considering the logarithmic utility function, the optimal strategy under the new pattern is obtained, and the homologous value function is acquired. The establishment of the model in this paper is more in line with the actual market situation, which makes the theoretical derivation more practical significantly. By comparing the results, it can be found that the time of death of family members does have an impact on the optimal strategy.

Footnotes

Acknowledgments

The authors acknowledge the Key Scientific Research Project of Henan Province (Grant: 22B470007), the School-level science and technology topics (Grant: 2022-KJYB-011).

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