Application of Generalized Geometric Itô-Lévy Process to Investment-Consumption-Insurance Optimization Problem under Inflation Risk

We consider a problem of maximizing the utility of an agent who invests in a stock, money market account and an index bond incorporating life insurance, deterministic income, and consumption. The stock is assumed to be a generalized geometric It?-Levy process. Assuming a power utility function, we determine the optimal investment-consumption-insurance strategy under inflation risk for the investor in a jump-diffusion setting using martingale approach.


Introduction
Modern Portfolio Theory has attracted many researchers since the break through publications of Markowitz [1] and Merton [2] [3]. The extensions with regards to the market imperfections have significantly improved both financial theory and practice. Zhang, et al. [4] and Chen, et al. [5] proposed mean-semivariance diversified models for uncertain portfolio selection. The proposed methods are based on hybrid intelligent algorithm. The inclusion of life insurance in the analysis of optimal portfolio selection has also been vehemently studied lately. Huang, Milevsky & Wang [6]

Model Description
We consider an investor (wage earner) investing in a market consisting of three investment opportunities, in a money market account, a stock and an index bond which has the same risk source as the price level process, in a finite time horizon [ ] 0,T . The investor is also subjected to insurance market with a risk of inflation which will represented by Consumer Price Index (CPI), which can be regarded as a price level process. We assume a filtered probability space is governed by the Brownian motion ρ is the correlation parameter (see Protter [16] and the dynamics of stock ( ) are given by the generalized geometric Itô-Lévy process is the mean rate of return, N t ζ is the differential notation of the random measure Moreover, if we assume that r t is the is the real interest rate. As in Kwak and Lim [14], we assume that the index bond is freely traded so it fulfills the demand for hedging inflation risk. We assume that the policyholder has to make decisions regarding investment, consumption, life insurance and inflation. As in Kronborg & Steffensen [11], Guambe and Kufakunesu [10] and Wang et al. [9], we consider the investor whose lifetime is a non-negative random variable τ defined on a probability space ( ) Moreover, the probability that the life time t τ > is . The instantaneous force of mortality or hazard function ( ) p t for the investor to be alive at time t is given by The conditional survival probability of the investor is given by and conditional survival probability density of the death of the policyholder is given by We also assume that the wage earner pays premiums at a rate ( ) q t at time t from the bond for a life insurance contract. The life insurance firm will pay his/her death to the beneficiaries. When policyholder dies, the total wealth payable to the beneficiaries is where t W is the wealth process at time t. We also assume that wage earner works and converts labor and time into wages and income. That is, he/she In addition, some portion of the income is consumed and the remainder is strategically allocated to purchasing the stock and saving in the money market account.
is the time-t actuarial value of the future labour income,  With respect to wealth Equation (13), the Conditions (15), (16) and (18) for some t  -adapted processes , then H is a martingale under  .

Optimal Strategy
We determine the optimal investment-consumption-insurance through solving unrestricted pension problem using martingale approach. For more information regarding this choice of approach see [10], [11] and references thereof. The policyholder seeks for the strategy satisfying the following: γ ∈ ∞ in terms of cash flow or wealth w at time t. The policyholder therefore has a constant relative risk aversion (CRRA) coefficient γ .
As γ tends to one, the utility function becomes logarithmic. A bequest para- capture the importance of bequest relative to retirement income. The time preference coefficient 0 1 ρ ≤ ≤ reflects the policyholder's preference over early income or later income. Using Equations (6) and (7), we rewrite the policyholder's optimization problem in Equation (23) as (for more details see [11]) of which can be written in stochastic differential equation (SDE) form as Hence, the candidate optimal strategy ( ) From Equation (28) and the Itô's formula (see [18], Theorem 1.14 and/or