Optimal Investment-Consumption and Life Insurance Strategy with Mispricing and Model Ambiguity

In this paper, we consider the optimal investment-consumption and life insurance strategy for a wage earner who has uncertain labor income described by an Ornstein-Uhlenbeck process. In addition to consumption and purchasing life insurance, the wage earner invests his wealth in the financial market, which consists of a risk-free asset, a market index, and a pair of risky assets with mispricing. Our aim is to maximize the expected utilities obtained from consumption, bequest, or his wealth at the end of the decision horizon. With the dynamic programming approach, we obtain explicit solutions for the optimization problem by solving the corresponding HJB equation. Finally, several numerical examples are presented to illustrate our results.


Introduction
Since Merton's pioneering work (1969Merton's pioneering work ( , 1971)), in which the optimal investment and consumption strategy is obtained through the stochastic dynamic programming approach, related problems have been extensively studied.The issue of household investment and consumption attracts attention shortly after that.In a discrete model by Hakansson (1969) and a continuous model by Richard (1975), respectively, the problem of optimal lifetime consumption, investment, and life insurance purchase is examined.They take into account the household's life insurance strategy in addition to investment and consumption.The justification is that buying life insurance is frequently regarded as a prudent precaution to safeguard the financial interests of the household's dependents in the event of his premature or accidental death.In this vein, in recent decades, the topic of uncertain lifetime or death risk has been taken into account in the study of optimal investment, consumption, and life insurance strategies.For instance, Ye (2007) describes an optimization problem for a wage earner with unpredictable lifespans in terms of consumption, investment, and life insurance.Pliska and Ye (2007) investigate the optimal investment-consumption-insurance strategy for a wage earner with an unbounded random lifetime.Moore and Young (2006) further examine the optimization of insurance schemes for individuals, taking into account bequests and personal consumption needs.
In this paper, using an Ornstein-Uhlenbeck process to depict a wage earner's uncertain labor income, we examine the optimal life insurance and investing strategies.In contrast to the previous research, we take into account a special financial market with mispricing. 1 That is, the market consists of two mispriced stocks with comparable or identical contingent claim values that are traded on different exchanges at different prices.We also take into account the wage earner's aversion to the ambiguity of this model arising from the drift terms.By taking full advantage of the mispricing, investors can earn abnormal returns by using the straightforward trading method of buying the less costly class and selling the more expensive class when one's bid price exceeds its ask price by a certain amount.This tactic is known as the "long-short" (L-S) method, and it has been considered by Liu and Longstaff (2004) and Jurek and Yang (2006).Recently, Yi et al. (2015a, b) and Gu et al. (2017) emphasize the benefit of mispricing for medium-and long-term investment strategies in their studies of the optimal reinsurance-investment problem for an insurer with mispricing and model ambiguity.Finally, note that families sometimes set aside a portion of their wealth for medium-and longterm investments in order to have some cash on hand for unplanned needs and the care of elderly relatives.Thus, in this paper, we are motivated to investigate the optimal investment strategy for the wage earner.
Model ambiguity refers to a scenario where the decision maker's beliefs are not captured by a particular probability measure and the probabilities are neither known nor calculable.The significance of model ambiguity has been recognized by the researchers.For instance, Asano and Osaki (2020), Nishimura andOzaki (2004), andDow andWerlang (1992) demonstrate its significant impact on investment strategies from the perspective of empirical and financial theory.Recently, model ambiguity has been investigated by more and more scholars.Anderson et al. (1999) introduces ambiguity-aversion into the Lucas model.The model framework is then updated by Uppal and Wang (2003) to include the degree of uncertainty.Maenhout (2004Maenhout ( , 2006) ) introduce a penalty function to prevent the alternative model from deviating too much from the reference model and derive closed-form expressions for the inter-temporal consumption problem.Recently, a large number of scholars have used a similar technique to study problems such as optimal reinsurance-investment, DC pension, optimal investmentconsumption problems, and much more.Yi et al. (2015a, b) and Gu et al. (2017) discuss the optimal reinsurance-investment strategy; they find that ambiguity aversion has a significant impact on an insurer's optimal strategy; Zeng et al. (2018) and Wang and Li (2018) consider the optimal investment strategy for the members of DC pension plans, and show that ignoring model ambiguity may lead to significant utility losses.As a result of the aforementioned investigations, we make the assumption in this paper that the wage earner is ambiguity-averse with regard to the drift terms of the model and consider the optimal investment, consumption, and life insurance problems for the wage earner.
The contribution of this paper is threefold.First, we build a continuous-time life cycle decision model that incorporates stock mispricing and model ambiguity simultaneously.Second, using the dynamic programming approach and the HJB technique, we obtain closedform expressions for the value function, the optimal investment-consumption-life insurance strategy.Third, we obtain some economic insights through several numerical examples.For example, we show that model ambiguity leads to less investment in the stock, while pricing error has the opposite effect.
The outline of this paper is as follows: Sect. 2 introduces the financial market, the insurance market, and the wealth process of the wage earner.Model ambiguity is incorporated into the model, and the optimization problem for the wage earner is addressed.In Sect.3, the optimization problem is solved using the dynamic programming approach.We write down the HJB equation and derive explicit optimal strategies and the value function from it.Section 4 presents numerical analyses to illustrate our theoretical results.Section 5 concludes the paper.

Model Setting
In this paper, a wage earner is assumed to have a stochastic labor income.In order to mitigate the risk of life uncertainty, he pays a portion of his income for a life insurance contract.He is also permitted to invest in the financial market.Here, we contribute to investigating the wage earner's optimization problem to determine the optimal consumption, life insurance, and investment strategy for him.Let ( , F , {F t } 0≤t≤T , P) be a completed probability space, where {F t } 0≤t≤T is a filtration satisfying the usual conditions, P is defined as the reference probability measure, and T is the terminal time of the wage earner's decision horizon, which may be explained, for instance, as the wage earner's time of retirement.

Financial and Insurance Market
The financial market consists of one risk-free asset, one market index and and a pair of stocks with mispricing.The price process of the risk-free asset {S 0 (t)} 0≤t≤T satisfies where the risk-free rate r > 0 is constant.The price process of the market index follows a geometric Brownian motion in which μ(t) and σ m are nonnegative parameters that represent the market index's risk premium and volatility rate; B m (t) is a standard Brownian motion.In the financial market, there exists a pair of stocks P1 and P2 , which are indexed to the same company but traded in different exchanges, so they have different prices,2 which are coupled by a pricing error . We assume that ( P1 (t), P2 (t)) satisfies the following stochastic differential equations (c.f.Liu and Timmermann 2013):  2) and (3), we can derive the dynamics of the pricing error X (t) as the following Ornstein-Uhlenbeck (or mean reverting) process: (4) Here, λ 1 + λ 2 represents the speed of mean-reversion of X (t), i.e. the rate of X (t) reverting to its mean value 0. It is noted that λ 1 + λ 2 = 0; otherwise, there exist two assets with the same expected rate of return but higher risk than the market index, which is unreasonable.Besides, we make the assumption that λ 1 + λ 2 > 0.3

Life Insurance and Wealth Process
The investor's time of death is random.It is depicted by a non-negative random variable τ with the mortality rate λ(t) given as following: Thus, the distribution function of τ is given by where f (t) is the probability density function and is defined by The probability that the wage earner survives up to time t is Similarly, we can define the conditional survival probability F(s, t) of the wage earner being alive up to time s, given that he is alive at time t and t ≤ s: The conditional probability density function is At any time t < τ, the wage earner purchases life insurance with premium p(t) to hedge the uncertain mortality risk.According to the insurance contract, once the wage earner dies at time t, his dependents will receive a lump-sum death benefit p(t) η(t) , where η(t)(≥ λ(t)) is the premium-insurance ratio.In this paper, without affecting our main results we assume that η(t) = λ(t) for simplicity.In other words, there are no frictions in the insurance market, and the insurer won't impose a risk loading for providing life insurance; for examples, see Shen and Sherris (2018) and Wang et al. (2021).
The wage earner receives uninsurable labor income Y (t), which is assumed to follow the following Ornstein-Uhlenbeck (i.e., O-U) process: where b is the long-term mean of Y (t) and l > 0 is the speed of convergence or divergence from the steady state.σ y1 and σ y2 are positive volatility rates.The O-U process is a continuoustime version of the first-order autoregressive process that has been extensively used to model the dynamics of labor income (see, for example, Deaton 1992).While we cannot guarantee that the solution to the O-U process is always positive, we are able to choose a set of parameters to ensure that the probability of receiving negative labor income is slim, and that a simulated path of the income process is always above zero for the given set of parameters (see Wang et al. (2021)).
The wage earner makes a profit by investing in the financial market.At time t < τ, let π m (t) and π i (t) be the amount of wealth invested in the market index and stock i.Let c(t) be the instantaneous consumption rate, p(t) be the life insurance premium, and denote W (t) as the wage earner's wealth at time t.Denote π t = (π m (t), π 1 (t), π 2 (t), c(t), p(t)) as the investment, consumption and life insurance strategy.With the strategy π , the dynamics of the wage earner's wealth process is described by: In the sequel, we use u as a new control in place of π m in π.
If the wage earner passes away at time t, his dependents will receive a legacy I (t), including his wealth and the lump-sum death benefit:

Optimization Problem
This subsection is dedicated to identifying the optimal investment-consumption-life insurance strategy for the wage earner to maximize his expected utility from consumption and terminal wealth if the wage earner survives retirement time T , or a bequest if he dies before T .Thus, the wage earner's optimization problem is defined as where U (•) is the wage earner's utility function, is the conditional expectation, δ is the subjective discount rate, and I is the identity function.In this paper, the wage earner is assumed to have a CARA utility: where γ > 0 is the the absolute risk aversion coefficient.Equation ( 11) is an optimization problem with a random terminal time.Based on the distribution of the random variable τ , we can convert the above problem to one with a fixed terminal time (see Lemma 1 in Pliska and Ye (2007) or Jin et al. (2020)): In this paper, we assume that the wage earner is an ambiguity aversion investor (i.e., AAI).In other words, he is skeptical of statistical estimation accuracy and possible specification errors in reference models with the reference probability measure P.Even though the real probability measure is unknown, he believes it is within a certain range of the reference measure.In line with Maenhout (2004Maenhout ( , 2006) ) and Flor and Larsen (2014), we solve the wage earner's optimization problem under an alternative probability measure Q that is equivalent to P. We denote the set of alternative probability measures by Q:

123
According to Maenhout (2004), for each Q ∈ Q there exists a progressively measurable process h where h (T ) is the Radon-Nikodym derivative that is defined by We assume that h satisfies then h (T ) is a martingale under probability measure P.
According to Girsanov's theorem, under the probability measures is a 4-dimensional standard Brownian.Therefore, under the probability measure Q, (4), ( 7) and ( 13) become in which u(t) is given in (9).
In the following, we will consider the wage earner's optimization problem under the new probability measure Q ∈ Q.However, the alternative probability measure Q should not deviate too much from the reference probability measure P, since P is the best prior probability 123 generated under the information set F t .Thus, a penalty term should be incorporated into the objective function to prevent the alternative measurer Q from deviating too far.Inline with Maenhout (2004), the relative entropy between Q and P is used to define the penalty function, which takes the following form: Here, E Q t,w,x,y [•] denotes the conditional expectation operator with respect to the alternative probability measure Q, given Thus, when the wage earner is ambiguity averse, the robust optimization problem is defined as where (•, •) models the wage earner's ambiguity aversion to the financial market models.As mentioned above, since the term 1 2 (s,W (s),X (s),Y (s)) h(s) 2 stands for a penalty to the wage earner's model ambiguity, is supposed to be non-negative.In fact, captures the wage earner's degree of ambiguity about the model.A larger indicates that a given deviation from the reference probability measure P is less penalized, i.e. the wage earner has less faith in P. In this sense, the wage earner's ambiguity aversion is increasing in .For analicical tractability, inline with Maenhout (2004) and Yi et al. (2015a, b), we assume where k is a non-negative constant describing the degree of the wage earner's ambiguity aversion.Thus, the wage earner's ambiguity aversion is increasing in k.Specially, when k = 0, the wage earner completely trusts the reference probability measure P and sets Q = P. Equation ( 17) reduces to the standard optimization problem.14) has a unique solution {W π (t)}.Moreover, the probability measure in the worst case.
The set of all admissible strategies is denoted by . 123 Using ( 5)-( 6), the robust optimization problem with CARA utility can be rewritten as: Obviously, we have the terminal condition J (T , w, x, y) = − 1 γ e −γ w .

Solution to the Optimization Problem
In this section, we solve the robust optimization problem (18) using the dynamic programming approach.With the wealth process, pricing error process, and the income process given by ( 14), ( 15) and ( 16), the corresponding HJB equation of optimization problem ( 18) is given by with the boundary condition V (T , w, x, y) = − 1 γ e −γ w .V t , V w , V x , V y , V yy , V ww , V x x , V wx , and V wy represent the value function's partial derivative with respect to the corresponding variables.
The solution to ( 19) is given by the proposition below.
The proof is given in the Appendix.According to a classic verification procedure (see e.g.Yi et al. 2015a, b), we can prove that the solution to the HJB Eq. ( 19) coincides with the value function of the optimization problem (18).
Theorem 3.2 shows that the wage earner's ambiguity aversion to the drift terms makes him more conservative, leading to less investment in the stocks and the market index (i.e.

| π *
. Therefore, with the same initial wealth at time t = 0, the ambiguity-aversion wage earner will spend less in his consumption comparing to the case that he is ambiguity-neutral.

Numerical Illustration
In this section, we present several numerical examples to further explore our theoretical results in Sect.3. Specifically, we show the impact of k, λ, γ, x, e, σ, σ m on the wage earner's investment, consumption, and life insurance strategies. 4Referring to Pliska and Ye (2007), Wei et al. (2020) and Wang et al. (2020), we set the basic values of the parameters as:

Sensitivity Analysis of the Optimal Consumption and Life Insurance Strategies
In this subsection, we study the effects of ambiguity aversion level k, mortality rate λ, pricing error x and absolute risk aversion coefficient γ on optimal consumption strategy c * (t) and optimal life insurance strategy p * (t).
Figure 1 shows the effects of k and λ on c * (t) and p * (t).Firstly, it is observed that the optimal consumption strategy c * (t) and the optimal life insurance strategy p * (t) decrease with the ambiguity aversion level k.Intuitively, as the investor's ambiguity aversion to the diffusion modeling risk grows, so does his uncertainty about the investment income.As a result, he becomes more conservative in his consumption and life insurance spending.Secondly, it is shown that, as the mortality rate λ increases, the optimal consumption strategy c * (t) decreases slightly and the optimal life insurance strategy p * (t) increases remarkably.
In fact, the investor's remaining years of life decreases as λ increases.Thus, he consumes less and purchases more life insurance to guarantee a sufficient legacy for his family.This is completely in line with reality, and it is also the reason that more and more middle-aged and elderly people tend to buy more life insurance.

(a) (b)
Fig. 2 Effects of x and γ on c * (t) and p * (t) Figure 2 shows the effects of x and γ on c * (t) and p * (t).We first analyse the effect of pricing error on optimal consumption strategy and optimal life insurance strategy.From the left panel, we can find that the optimal consumption strategy c * (t) and the optimal life insurance strategy p * (t) increase with the pricing error x.With a larger x, the wage earner obtains more investment profits by arbitraging in the two stocks.That is, buying at a low price and selling at a high price.With more profits from investment, the investor can spend more money for consumption and purchasing life insurance.In addition, it is shown that, as the investor becomes more risk averse, both the optimal consumption strategy c * (t) and the optimal life insurance strategy p * (t) decrease.These results are intuitive and realistic.

Sensitivity Analysis of the Optimal Investment Strategies
In this subsection, we study the effects of ambiguity aversion k, mortality rate λ, pricing error x, absolute risk aversion coefficient γ and some other parameters (σ m , β, σ and e ) on optimal investment strategies.
Fig. 3 shows the effects of k and λ on π * 1 (t) and π * 2 (t). Figure 3(a) and (b) show that, as the ambiguity aversion level k increases, π * 1 (t) increases and π * 2 (t) decreases respectively.Note that the investor exploits the pricing error by holding long position on the underpriced stock 2 and short position on the overpriced stock 1 simultaneously.When the investor becomes more ambiguity averse, he becomes more conservative and reduces the positions on both stocks.We also observe that, as the mortality rate λ increases, the optimal investment strategies |π * 1 (t)| and |π * 2 (t)| decrease.This is consistent with the phenomenon that older people tend to hold fewer shares.
Figure 4 shows the effects of x and γ on π * 1 (t) and π * 2 (t).From Fig. 4(a) and (b), we can see that the optimal investment strategy |π * 1 (t)| and π * 2 (t) increase with the pricing error x and decrease with the risk aversion coefficient γ , respectively.
Figure 5 shows the effects of k and γ on π * m (t) and π * 1 (t) + π * 2 (t).As shown in Fig. 5(a) and (b), the optimal investment strategies π * m (t) and π * 1 (t) + π * 2 (t) increase with the increase of the ambiguity aversion level k and the absolute risk aversion coefficient γ .That is, being more ambiguity averse or more risk averse, the investor tends to reduce his short selling in the market index, and decreases his share holdings.
Figure 6 shows the effects of some other parameters on π * m (t) and π * 1 (t) + π * 2 (t).Specifically, Fig. 6(a) describes the effects of σ m and β on π * m (t), and Fig. 6(b) describes the effects of σ and e on π * 1 (t) + π * 2 (t).From Fig. 6(a), we find that the optimal investment strategy π * m (t) increases with the increase of the market index's volatility rate σ m and the parameter β.As mentioned above, βσ m d B m (t) represents the systematic risk of the market.The systematic risk of the market rises as σ m or β increases, thus the wage earner becomes more cautious and tends to cut back on his short selling in the market index.As shown in Fig. 6(b), the optimal investment strategy π * 1 (t) + π * 2 (t) increases with the increase of the volatility rate σ and e (i.e., the idiosyncratic risk of stocks).Moreover, when the common shock to both stocks reaches a specific level, such as σ = 0.3, the investing strategy π * 1 (t) + π * 2 (t) is less sensitive to the individual risk of stocks, it is mainly determined by the systematic risk.

Conclusion
In this paper, we consider the optimal investment, consumption, and life insurance strategies for a wage earner with model ambiguity.In particular, the wage earner's income is described by an Ornstein-Uhlenbeck process.Furthermore, the wage earner considers not only consumption and the purchase of life insurance, but also investing his wealth in the financial market, which consists of one risk-free asset, a market index, and a pair of risky assets with mispricing.By using dynamic programming methods, we establish and solve the corresponding HJB equation and derive the explicit optimal strategies and the value function under the CARA utility.Finally, we present some numerical examples to illustrate our results.
With some numerical analysis, we have the following findings.Firstly, the ambiguity aversion level and the absolute risk aversion coefficient have a negative effect on his holdings in the financial market.More ambiguity-averse and more risk-averse, the wage earner becomes more conservative when he makes decisions.Secondly, the mortality rate has a negative impact on optimal consumption and investment strategies while having a positive impact on optimal life insurance strategies.Third, the pricing error has a positive effect on optimal strategies, indicating that as the pricing error increases, the wage earner becomes more aggressive in stock investment.

Fig. 1
Fig. 1 Effects of k and λ on c * (t) and p * (t)