# On optimal investment-consumption and life insurance with capital constraints

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## Outline of the thesis

This thesis treats various optimal portfolios and life insurance problems under jump-diffusion setup. In the first part (Chapter 3), we consider a jump-diffusion problem with stochastic volatility. This problem has been solved in Mnif [68] via dynamic programming approach. The application of this approach in a jump-diffusion setting, results in a nonlinear parabolic partial differential equation (PPDE) which in general the solution is not obtained. In his paper, Mnif proves the existence of a smooth solution by reducing a nonlinear PPDE to a semi-linear one under certain conditions. To overcome these limitations, we propose a martingale approach developed by Karatzas et al. [50] and Karatzas and Shreve [52] in a diffusion process to solve the unrestricted problem. Then we solve a constrained optimization problem, where the constraint is of American put type. Considering a jump-diffusion model, a market is incomplete and consequently we have many martingale measures. We obtain the optimal investment, consumption and life insurance strategy by the convex optimization method.

The Unrestricted problem

In this section, we solve our main optimization problem using the martingale duality method. Consider the concave, non-decreasing, upper semicontinuous and differentiable w.r.t. the second variable utility functions Uk : [0, T] × R+ → R+ , k = 1, 2, 3 . Let ρ(t) be a deterministic function representing the policyholder’s time preferences. The policyholder chooses his strategy (c(t), π(t), p(t)) in order to optimize the expected utility from consumption, legacy upon death and terminal pension. His strategy, therefore, fulfils the following: The feasible strategy (3.22) means that it is allowed to draw an infinite utility from the strategy (π, c, p) ∈ A0 , but only if the expectation over the negative parts of the utility function is finite. It is clear that for a positive utility function, the sets A and A0 are equal ( see eg., Kronborg and 3.3. The Unrestricted problem 36 Steffensen [55]). In order to solve the unrestricted control problem (3.21) one can use the Hamilton-Jacobi-Bellman (HJB) equation (e.g., Mnif [68]) or the combination of HJB equation with backward stochastic differential equation (BSDE) with jumps (Guambe and Kufakunesu [41]). In this Chapter, we use the duality martingale approach applied in (Karatzas et al. [51], CastanedaLeyva and Hern´andez-Hern´andez [13], Kronborg and Steffensen [55]). This is due to the incompleteness of the market and the restricted problem in the next section, where its terms are derived from the martingale method in the unrestricted problem. From (2.16) and (2.17), we can rewrite the policyholder’s optimization problem (3.21) as (See, for example, Kronborg and Steffensen [55]):

### On the optimal investment-consumption and life insurance selection problem with stochastic volatility

The problem of a wage earner who wants to invest and protect his dependent for a possible premature death has gained much concern in recent times. Since the research paper on portfolio optimization and life insurance purchase by Richard [85] appeared, a number of works in this direction have been reported in the literature. For instance, Pliska and Ye [83] studied an optimal consumption and life insurance contract for a problem described by a risk-free asset. Duarte et al. [29] considered a problem of a wage earner who invests and buys a life insurance in a financial market with n diffusion risky shares. Similar works include (Guambe and Kufakunesu [41], Huang et al. [47], Liang and Guo [60], Shen and Wei [88], among others). In all the above-mentioned papers, a single life insurance contract was considered. Recently, Mousa et al. [72], extended Duarte et al. [29] to consider a wage earner who buys life insurance contracts from M > 1 life insurance companies. Each insurance company offers pairwise distinct contract. This allows the wage earner to compare the premiums insurance ratio of the companies and buy the amount of life insurance from the one offering the smallest premium-payout ratio at each time.
Using a dynamic programming approach, they solved the optimal investment, consumption and life insurance contracts in a financial market comprised by one risk-free asset and n risky shares driven by diffusion processes. In this chapter, we extend their work to a jump-diffusion setup with stochastic volatility. This extension is motivated by the following reasons: First, the existence of high frequency data on the empirical studies carried out by Cont [17], Tankov [94] and references therein, have shown that the analysis of price evolution reveals some sudden changes that cannot be explained by models driven by diffusion processes. Another reason is related to the presence of volatility clustering in the distribution of the risky share process, i.e., large changes in prices are often followed by large changes and small changes tend to be followed by small changes. To enable a full capture of these and other aspects, we consider a jump diffusion model with stochastic volatility similar to that in Mnif [68]. Using Dynamic programming approach, Mnif [68] proved the existence of a smooth solution of a semi-linear integro-Hamilton-Jacobi-Bellman (HJB) for the exponential utility function.
Zeghal and Mnif [100] considered the same problem for power utility case. Under some particular assumptions, they also derived the backward stochastic differential equation (BSDE) associated with the semi-linear HJB. The drawback of the dynamic programming approach is that it requires the system to be Markovian. To overcome this limitation, we propose a maximum principle approach to solve this stochastic volatility jump-diffusion problem. This approach allows us to solve this problem in a more general setting. We prove a sufficient and necessary maximum principle in a general stochastic volatility problem. Then we apply these results to solve the wage earner investment, consumption and life insurance problem described earlier.

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Maximum principle for stochastic optimal control problem with stochastic volatility

As in the previous Chapter, let T < ∞ be a finite time horizon investment period, which can be viewed as a retirement time of an investor. Consider two independent Brownian motions {W1(t); W2(t), 0 ≤ t ≤ T} associated to the complete filtered probability space (ΩW , F W , {FW t }, P W ). Furthermore, we consider a Poisson process N independent of W1 and W2, associated with the complete filtered probability space (ΩN , F N , {F N t }, P N ) with the intensity measure dt × dν(z), where ν is the σ-finite Borel measure on R \ {0}. A P N -martingale compensated Poisson random measure is given by: N˜(dt, dz) := N(dt, dz) − ν(dz)dt . We define the product space: (Ω, F, {Ft}0≤t≤T , P) := (ΩW × Ω N , F W ⊗ F N , {FW ⊗ F N }, P W ⊗ P N ) where {Ft}t∈[0,T] is a filtration satisfying the usual conditions.

#### Optimal investment-consumption and life insurance selection problem under inflation

The problem of asset allocation with life insurance consideration is of great interest to the investor because it protects their dependents if a premature death occurs. Since the optimal portfolio, consumption and life insurance problem by Richard [85] in 1975, many works in this direction have been reported in the literature. (See, e.g., Pliska and Ye [83], Guambe and Kufakunesu [41], Han and Hu [42], among others). In this chapter, we discuss an optimal investment, consumption and life insurance problem using the backward stochastic differential equations (BSDE) with jumps approach. Unlike the dynamic programming approach applied, for instance, in Han and Hu [42], this approach allows us to solve the problem in a more general non-Markovian case. For more details on the theory of BSDE with jumps, see e.g., Delong [26], Cohen and Elliott [16], 77 5.1. Introduction 78 Morlais [71], and references therein. Our results extend, for instance, the paper by Cheridito and Hu [14] to a jump diffusion setup and we allow the presence of life insurance and inflation risks. Inflation is described as a percentage change of a particular reference index. The inflation-linked products may be used to protect the future cash flow of the wage earner against inflation, due to its rapid escalation in some developing economies. Therefore, it make sense to model the inflation-linked products using jumpdiffusion processes.
For more details on the inflation-linked derivatives, see e.g., Tiong [96], Mataramvura [61] and references therein. We consider a model described by a risk-free asset, a real zero coupon bond, an inflationlinked real money account and a risky asset under jump-diffusion processes. These type of processes are more appropriate for modeling the response to some important extreme events that may occur since they allow capturing some sudden changes in the price evolution, as well as, the consumer price index that cannot be explained by models driven by Brownian information. Such events happen due to many reasons, for instance, natural disasters, political situations, etc. The corresponding quadratic-exponential BSDE with jumps relies on the results by Morlais [71], Morlais [70], where the existence and uniqueness properties of the quadratic-exponential BSDE with jumps have been proved. Thus, we are also extending the utility maximization problem in Morlais [70] by including consumption and life insurance. Similar works include Hu et. al. [46], Xing [98], Siu [91], Øksendal and Sulem [78], among others. This chapter is organized as follows: in Section 5.2, we introduce the inflation risks and the related assets: the real zero coupon bond, the inflationlinked real money account, and the risky asset. We also introduce the insurance market and we state the main problem under study. Section 5.3 is the main section of this paper, we present the general techniques of the BSDE approach and we prove the main results in the exponential and power utility function. Finally, in Section 5.4, we give some concluding remarks.

Contents :

• 1 Introduction
• 1.1 Background
• 1.2 Outline of the thesis
• 1.3 Published papers and preprints
• 2 Stochastic calculus and portfolio dynamics
• 2.1 Brownian motion and L´evy processes
• 2.2 Jump-diffusion processes
• 2.3 Martingales for jump-diffusion processes and the Girsanov Theorem
• 2.4 Backward stochastic differential equations
• 2.5 Portfolio dynamics under jump-diffusion processes
• 2.6 Life insurance contract
• 2.7 Utility functions
• 3 On optimal investment-consumption and life insurance with capital constraints
• 3.1 Introduction
• 3.2 The Financial Model
• 3.3 The Unrestricted problem
• 3.3.1 Results on the power utility case
• 3.4 The restricted control problem
• 4 On the optimal investment-consumption and life insurance selection problem with stochastic volatility
• 4.1 Introduction
• 4.2 Maximum principle for stochastic optimal control problem
• with stochastic volatility
• 4.3 Application to optimal investment- consumption and life insurance selection problem
• 5 Optimal investment-consumption and life insurance selection problem under inflation
• 5.1 Introduction
• 5.2 Model formulation
• 5.3 The BSDE approach to optimal investment, consumption and insurance
• 5.3.1 The exponential utility
• 5.3.2 The power utility case
• 5.4 Conclusion
• 6 Risk-based optimal portfolio of an insurer with regime switching and noisy memory
• 6.1 Introduction
• 6.2 Model formulation
• 6.3 Reduction by the filtering theory
• 6.4 Risk-based optimal investment problem
• 6.5 The BSDE approach to a game problem
• 6.6 A quadratic penalty function case
• 7 Conclusion and future research
• Bibliography

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Stochastic optimal portfolios and life insurance problems in a L´evy market